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\section{Introduction}\label{sec:intro}
In this paper we deal with the problem of hedging general contingent claims under illiquidity. Stochastic liquidity cost is incurred by hedging with multiple assets with possibly different levels of liquidity. Our main motivation comes from energy markets. Consider for example an agent hedging an Asian-style call option written on the average spot price $S = (S_{u})_{0 \leq u \leq T_2}$ of energy. Such an option has the payoff
\begin{equation}\label{asian:payoff}
\left( \frac{1}{T_{2} - T_{1}} \int_{T_{1}}^{T_2} S_{u} du - K \right)^{+}
\end{equation}
for a strike $K$, with a so-called \emph{delivery period} $[T_{1}, T_2]$. The instruments available for hedging such options are futures delivering over the same or a different time period. Hedging is a challenge though since these futures are either not trading in their delivery period at all \citep[such a setting was considered in][]{BenthDetering} or are very illiquid such that hedging incurs large transaction costs. In addition, futures are usually very illiquid for $t\ll T_1$, so their liquidity has a delicate time-structure. In the market, multiple futures with different delivery periods (week, month, quarter, year) and different levels of liquidity are available as hedging instruments. The results of our paper can be applied to hedging options in energy markets with multiple futures by accounting for their different levels of liquidity. The Asian-style option is a particular example but other payoffs as for example Quanto options \cite[see][]{benthQuanto} can be handle
d equally.
\par
The effect of illiquidity on hedging and optimal trading is a very active research topic in mathematical finance. Still, there is neither an agreed notion of liquidity risk, which is roughly speaking the additional risk due to timing and size of a trade, nor a standard approach for hedging under liquidity costs. A good overview on existing liquidity models in continuous and discrete time can be found in \cite{goekay.roch.soner:2010}.
\par
There are basically two different approaches how to model liquidity risk. The first one is a class of models incorporating feedback effects, that is when the trade volume has a lasting impact on the asset price \citep[see e.g.][]{bank.baum:2004}, also known as permanent price impact or lasting impact. The second approach considers smaller agents whose transactions have no lasting impact on the price of the underlying \cite[see e.g.][and the references therein]{cetin.jarrow.protter:2004}.
\par
In this paper we stay within the small agents approach and understand liquidity costs as the transaction costs incurred from the hedging strategy by trading through a fast recovering limit order book. In particular we follow the arbitrage-free model suggested by \cite{cetin.jarrow.protter:2004}, who introduced the so called \textit{supply curve} model. There, the asset price is a function of the trade size and the authors developed an extended arbitrage pricing theory.
\par
In addition to the vast majority of papers on illiquid markets dealing with optimal execution, there are also many papers investigating hedging under illiquidity, most of them consider super-replication \cite[see for example][]{bank.baum:2004, cetin.soner.touzi:2010, goekay.soner:2012}. As super-replication is often too expensive we use a quadratic risk criterion. In the classical frictionless theory without transaction costs, there are two main approaches for quadratic hedging \cite[see][for a survey]{schweizer:2001}. First, the \emph{mean-variance} approach, which was introduced in \cite{foellmer.sondermann:1986}, relies on self-financing strategies which produce as final outcome the portfolio $V_{T} := c + \int_{0}^{T} X_{u} dS_{u}$ for some initial capital $c \in \mathbb{R}$ and a trading strategy $X$ in the risky asset $S = (S_{u})_{0 \leq u \leq T}$. The goal of this method is to look for the best approximation of a contingent claim $H$ by the terminal portfolio value $
V_{T}$, that is minimizing the quadratic hedge error
\begin{equation}
{\mathbb E} \left[ \left( H - \left( c + \int_{0}^{T} X_{u} dS_{u} \right) \right)^{2} \right]
\end{equation}
under the real world probability measure and over an appropriately constrained set of strategies. This is also called global risk-minimization. In discrete time, this problem was solved in \cite{schweizer:1995} in a general setting and relaxing the assumptions imposed earlier in \cite{schael:1994}. Later on, this was extended to the multidimensional case with proportional transaction cost in \cite{motoczynski:2000} and \cite{beutner:2007} where the authors show existence of an optimal strategy. The papers \cite{rogers.singh:2010}, \cite{agliardi.gencay:2014} and \cite{bank.soner.voss:2017} can be seen as an extension under illiquidity of the mean-variance hedging criterion, which is based on minimizing the global risk against random fluctuations of the stock price incurring low liquidity costs.
\par
A second quadratic method for hedging in an incomplete market is \emph{local risk-minimization} first introduced in \cite{schweizer:1988} and later extended in \cite{lamberton.pham.schweizer:1998} by accounting for proportional transaction costs in the discrete time case. For discrete time $k=0, \dots, T$ this method does not insist on the self-financing condition but instead the goal is to find a strategy $(X,Y)=(X_{k+1}, Y_{k})_{k=0, \dots, T}$ with book value $V_{k} = X_{k+1}S_{k} + Y_{k}$ (the risk-free asset is assumed constantly equal to $1$) such that $H = V_{T}$, the cost process $C_{k} = V_{k} - \sum_{m=1}^{k} X_{m} (S_{m} - S_{m-1})$ is a martingale and the variance of the incremental cost is minimized. Here, the strategy $X_{k+1}$ represents the number of shares held in the risky asset and $Y_{k}$ the units held in the bank account in the time interval $(k, k+1]$. In our paper, we extend the work of \cite{schweizer:1988} by considering a multidimensional asset pri
ce process in discrete time. Secondly we extend the local risk-minimizing quadratic criterion to an illiquid market in the spirit of \cite{rogers.singh:2010} and \cite{agliardi.gencay:2014}.
\par
In contrast to the existing literature our approach and setting is designed to address the above mentioned problem in energy markets. For this we need a multi-dimensional setup to allow for hedging with multiple futures. Second, the assets price dynamics has to be general enough to capture the characteristics of energy markets and we need a time dependent liquidity structure. Our risk criterion is chosen such that it allows for more explicit formulas for the optimal strategy than existing approaches. Furthermore, as shown in a case study, they are also computationally tractable. Our main result is the existence of a locally risk-minimizing strategy under illiquidity requiring only mild conditions on the asset price. These conditions are quite technical but they can be reduced to conditions on the covariance matrix of the price process, which can be checked easily for most processes relevant in practice. Furthermore, the strategies can be calculated backwards in time by using
a \emph{least-squares Monte Carlo} algorithm.
\par
\par
Our setup allows us to explore the tradeoff between liquidity and hedge quality of available hedge instruments. For example, consider the Asian-style option (\ref{asian:payoff}) in a market where different futures with different delivery periods and different liquidity levels are available for hedging. In such a situation, there are futures with delivery period well matching the delivery period of the option payout resulting in a strong correlation between the future and the option to hedge. However, in certain time periods these {\em hedge-optimal} futures are very illiquid and futures which are less correlated but more liquid might be better for hedging. Our framework allows us to explore this tradeoff and provide market-makers with a more profound tool for risk management.
\par
The paper is structured as follows. Section \ref{sec:model} explains the model framework and describes the basic problem. In Section \ref{sec:linear:supply} we focus on a linear supply curve and impose necessary assumptions on the price process to prove our main existence Theorem~\ref{theo:existence}. Sufficient conditions to check the assumptions are also provided. Section \ref{sec:applications} considers an application to energy markets. Optimal strategies under illiquidity are simulated by means of a least-squares Monte Carlo method. This allows us to explore the tradeoff between liquidity and hedging performance of futures available for hedging.
\section{The Model}
\label{sec:model}
Given a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ consider a financial market consisting of $d+1$ assets.
We denote by $\mathbb{P}$ the {\em objective} probability measure and by $\mathbb{F}$ the filtration $\mathbb{F} = (\mathcal{F}_k)_{k = 0,1, \dots, T} $, which describes the flow of information. We shall use the indices $k = 0,1, \dots, T$ to refer to a discrete time grid with time points $t_{0} < t_{1} < \dots < t_{T}$ and sometimes use both interchangeably. An $\mathbb{F}$-adapted, nonnegative $d$-dimensional stochastic process $S = (S_{k})_{k=0,1, \dots, T}$ describes the discounted price of $d$ risky assets (typically futures or stocks). We use $S_{k}^{j}$ to refer to the price of asset $j$ at time $t_{k}$. Furthermore, a riskless asset (typically a bond) exists whose discounted price is constantly $1$.
\par
Similar as in \cite{cetin.jarrow.protter:2004} we assume that a hedger observes an exogenously given nonnegative $d$-dimensional \textit{supply curve} $S_{k}(x)$ where $S_{k}(x)^{j} := S_{k}^{j}(x^{j})$ represents the $j$-th stock price per share at time $k$ for the purchase (if $x^{j} > 0$) or sale (if $x^{j} < 0$) of $|x^{j}|$ shares. We call $S(0) = S$ the \textit{marginal price}.
The supply curve determines the actual price that market participants pay or receive respectively for a transaction of size $x$ at time $k$. This curve is also assumed to be independent of the participants past actions which implies no lasting impact of the trading strategy on the supply curve. The only assumption that we need for the moment is measurability of the supply curve w.r.t. the filtration $\mathbb{F}$ and that it is non-decreasing in the number of shares $x$, i.e. for each $k$ and $j$, $S_{k}(x)^{j} \leq S_{k}(y)^{j}, \, \mathbb{P}-a.s.$ for $x^{j} \leq y^{j}$. This will ensure that the liquidity costs are non-negative.
\par
In \cite{cetin.jarrow.protter:2004} the authors develop a continuous time version of such a supply curve model and an extended arbitrage pricing theory. They show that the existence of an equivalent local martingale measure $\mathbb{Q}$ for the marginal price process $S$ rules out arbitrage. A similar results can easily be seen to hold in our setting as liquidity cost is always positive.
\par
However, even a unique martingale measure (and state space restrictions in a discrete setting) do not necessarily ensure completeness if one incorporates illiquidity. Since we cannot hedge perfectly, we want to minimize locally the risk of hedging under illiquidity according to an optimality criterion introduced in Definition \ref{defi:local risk minimizing strategy under illiquidity}.
\par
In the following we shall consider an investor who aims at hedging an $\mathcal{F}_T$-measurable claim $H$. For $x\in \mathbb{R}^d$, let $\abs{x}$ denote the Euclidian norm and $x^{*}$ the transpose of $x$. Further, $\langle x , y \rangle$ denotes the inner product of $x,y\in \mathbb{R}^d$. Adapting \cite{schweizer:1988} we define the investor's possible trading strategies. For this we denote by $\mathbb{L}_{T}^{p} (\mathbb{R}^d)$ (in short $\mathbb{L}_{T}^{p, d}$),
the space of all $\mathcal{F}_T$-measurable
random variables $Z : \Omega \to \mathbb{R}^d$ satisfying
$\norm{Z}^{p} = \mathbb{E}( \abs{Z}^{p} ) < \infty$. We abbreviate $\Delta S_{k} := S_{k} - S_{k-1}$. Furthermore, we denote by $\Theta_{d}(S)$ the space of all $\mathbb{R}^{d}$-valued predictable strategies $X = (X_{k})_{k=1,2, \dots, T+1}$ so that $ X_{k}^{*} \Delta S_{k} \in \mathbb{L}_{T}^{2, 1}$ and $\Delta X_{k+1}^{*} [ S_{k}(\Delta X_{k+1}) - S_{k}(0) ] \in \mathbb{L}_{T}^{1, 1}$ for $k=1, 2, \dots, T$
\begin{defi}
\label{defi:trading strategy}
A pair $\varphi = (X,Y) $ is called a \textit{trading strategy} if:
\begin{enumerate}[label=(\roman*)]
\item $Y = (Y_{k})_{k=0,1, \dots, T}$ is a real-valued $\mathbb{F}$-adapted process.
\item \label{item:trading strategy L2int}
$X\in \Theta_{d}(S)$.
\item $V_{k}(\varphi) := X_{k+1}^{*} S_{k} + Y_{k} \in \mathbb{L}_{T}^{2, 1}$ for $k=0, 1, \dots, T$.
\end{enumerate}
\end{defi}
\par
We call $V_{k}(\varphi)$ the \textit{marked-to-market value} or the \textit{book value} of the portfolio $(X_{k+1},Y_{k})$ at time $k$. We interpret $X_{k+1}^{j}$ as the number of shares held in the risky asset $S_{k}^{j}$ and $Y_{k}$ the units held in the riskless asset (bank account) in the time interval $( k, k+1 ]$. Note that with a non-flat supply curve, there is no unique value for a portfolio, as the value that can be realized depends on the liquidation strategy.
\subsection{Cost And Risk process}
\label{sec:cost and risk process}
Consider an $\mathbb{L}_{T}^{2, 1}$-\textit{contingent claim} of the form $H = \bar X_{T+1}^{*} S_{T} + \bar Y_{T}$, where $ \bar X_{T+1}^{*}S_{T} \in \mathbb{L}_{T}^{2, 1}$, $ \bar X_{T+1} \in \mathbb{L}_{T}^{2, d}$ and the components of the pair $(\bar X_{T+1}, \bar Y_{T})$ are $\mathcal{F}_{T}$-measurable random variables describing the quantity in risky assets and bonds respectively that the option seller is committed to provide to the buyer at the expiration date $T$ of the financial contract $H$.\footnote{For example, in the $1$-dimensional case one could set $\bar X_{T+1} = 0$ and $Y_{T}=(S_{T} - K)^{+}$ for a call option with strike $K$ without physical delivery.}
\par
Assuming that at time $k \in \{ 1, 2, \dots, T \}$ an order of $\Delta Y_{k}:= Y_{k} - Y_{k-1}$ bonds and $\Delta X_{k+1}:=X_{k+1} - X_{k}$ shares is made, then the \textit{total outlay} (under liquidity costs) is
\begin{equation}
\label{eq:total outlay}
\Delta Y_{k} + \Delta X_{k+1}^{*} S_{k}(\Delta X_{k+1})
= \Delta Y_{k} + \Delta X_{k+1}^{*} S_{k} + \Delta X_{k+1}^{*} [ S_{k}(\Delta X_{k+1}) - S_{k}(0) ].
\end{equation}
Note that $S_{k}(0)= S_{k}$ is the marginal price, such that the last term can be seen as the transaction cost resulting from market illiquidity. Furthermore, using the definition of the book value the previous equation can be written as
\begin{equation}
\label{eq:total outlay 2}
\Delta Y_{k} + \Delta X_{k+1}^{*} S_{k}(\Delta X_{k+1})
= \Delta V_{k} (\varphi) - X_{k}^{*} \Delta S_{k} + \Delta X_{k+1}^{*} [ S_{k}(\Delta X_{k+1}) - S_{k}(0) ]
\end{equation}
For a self-financing trading strategy the total outlay at time $k$ would be zero.
\\
Now, by defining $\hat C_{0}(\varphi) := V_{0}(\varphi)$, the \textit{initial cost}\footnote{For simplicity we do not account for any liquidity costs paid to set up the initial portfolio.}, we can define the \textit{cost process under illiquidity} $\hat C(\varphi) = (\hat C_{k}(\varphi))_{k=0,1, \dots, T}$ as
\begin{equation}
\hat C_{k}(\varphi)
:= \sum_{m=1}^{k} \Delta Y_{m} + \sum_{m=1}^{k} \Delta X_{m+1}^{*} S_{m}(\Delta X_{m+1}) + V_{0}(\varphi).
\end{equation}
It quantifies the cumulative costs of the strategy $\varphi = (X, Y)$. A simple calculation using the definition of $V_{k} (\varphi)$ shows that
\begin{equation}
\label{eq:cost process}
\hat C_{k}(\varphi) =V_{k} (\varphi) - \sum_{m=1}^{k} X_{m}^{*} \Delta S_{m} + \sum_{m=1}^{k} \Delta X_{m+1}^{*} [ S_{m}(\Delta X_{m+1}) - S_{m}(0) ],
\end{equation}
which will be needed later. If we can ensure that the cost process is square integrable, then we can define the \textit{quadratic risk process under illiquidity} $\hat R (\varphi)= (\hat R_{k}(\varphi))_{k=0,1, \dots, T}$ by
\begin{equation}
\label{eq:quadratic risk process}
\hat R_{k}(\varphi)
:= \condE{ (\hat C_{T}(\varphi) - \hat C_{k}(\varphi) )^{2} } \,.
\end{equation}
At this point we would like to mention that the classical local risk-minimization approach aims at finding a locally risk-minimizing strategy $\varphi = (X, Y)$ such that $V_{T}(\varphi) = H$ with $X_{T+1} = \bar X_{T+1}$ and $Y_{T} = \bar Y_{T}$ (see Section \ref{sec:description of the basic problem} for more details).
\par
We denote by $C(\varphi) = (C_{k}(\varphi))_{k=0,1, \dots, T}$ the classical cost process without liquidity costs (i.e., $S(x)=S(0)$), that is
\begin{equation}
C_{k}(\varphi)
:= V_{k} (\varphi) - \sum_{m=1}^{k} X_{m}^{*} \Delta S_{m},
\end{equation}
and obtain the relation
\begin{equation}
\hat C_{T}(\varphi) - \hat C_{k}(\varphi)
= C_{T}(\varphi) - C_{k}(\varphi) + \sum_{m=k+1}^{T} \Delta X_{m+1}^{*} [ S_{m}(\Delta X_{m+1}) - S_{m}(0) ].
\end{equation}
Furthermore, we denote by $ R (\varphi)= (R_{k}(\varphi))_{k=0,1, \dots, T}$ the classical risk process, defined as in (\ref{eq:quadratic risk process}) but with $\hat C$ replaced by $C$.
One could also define the \textit{linear risk process under illiquidity}
\begin{equation}
\label{eq:linear risk process}
\bar R_{k}(\varphi)
:= \condE{ | \hat C_{T}(\varphi) - \hat C_{k}(\varphi) | }
\end{equation}
which is motivated in \cite{coleman.li.patron:2003}.
\begin{remark}
A linear local risk-minimization criterion might be more natural than a quadratic one from a financial perspective. The $L^2$-norm overemphasizes large values even if these values occur with small probability. Nevertheless, by minimizing over the $L^2$-norm, it is possible to get explicit results.
\end{remark}
A combination of the two, that means measuring the quadratic difference of the classical cost process and linearly the liquidity costs, yields the \textit{quadratic-linear risk process under illiquidity}.
\begin{equation}
\label{eq:quadratic linear risk process}
T_{k}(\varphi)
:= \condE{ ( C_{T}(\varphi) - C_{k}(\varphi) )^{2} }
+ \condE{ \sum_{m=k+1}^{T} \Delta X_{m+1}^{*} [ S_{m}(\Delta X_{m+1}) - S_{m}(0) ] }.
\end{equation}
As we will see later on, by minimizing the expression in (\ref{eq:quadratic linear risk process}) we will be able to construct an explicit representation of the LRM-strategy under illiquidity where large values of liquidity costs are not overemphasized by the $L^2$-norm.
\subsection{Description of the basic problem}
\label{sec:description of the basic problem}
The aim of the classical local risk-minimization is to minimize locally the conditional mean square incremental cost of a strategy. Our criterion is targeting on minimizing locally the risk against random fluctuations of the stock price but at the same time reducing liquidity costs. It balances low liquidity costs against poor replication. Such an approach is similar to \cite{agliardi.gencay:2014} or \cite{rogers.singh:2010} and yields a tractable problem.
In minimizing locally the risk process at time $k$, we only minimize in $Y_{k}$ and $X_{k+1}$ in order to make the current optimal choice of the strategy by fixing the holdings at past or future times. Definition \ref{defi:local perturbation} and Definition \ref{defi:local risk minimizing strategy under illiquidity} give us the optimality criterion that the minimization problem is based on.
\begin{defi}
\label{defi:local perturbation}
A \textit{local perturbation} $ \varphi ' = ( X', Y')$ of a strategy $\varphi = (X, Y)$ at time $k \in \{ 0,1, \dots, T-1 \}$ is a trading strategy such that $X_{m+1} = X'_{m+1}$ and $Y_{m} = Y'_{m}$ for all $m \neq k$.
\end{defi}
By a slight abuse of notation let
\begin{equation}
\label{eq:quadratic linear risk process new}
T_{k}^{\alpha}(\varphi)
:= \condE{ ( C_{T}(\varphi) - C_{k}(\varphi) )^{2} }
+ \alpha \condE{ \Delta X_{k+2}^{*} [ S_{k+1}(\Delta X_{k+2}) - S_{k+1}(0) ] }.
\end{equation}
We specify in Definition \ref{defi:local risk minimizing strategy under illiquidity} what we call local risk minimizing (LRM) strategy under illiquidity for some $\alpha \in \mathbb{R^{+}}$.
\begin{defi}
\label{defi:local risk minimizing strategy under illiquidity}
A trading strategy $\varphi = (X, Y)$ is called \textit{locally risk-minimizing under illiquidity} if
\begin{equation}
T_{k}^{\alpha} (\varphi)
\leq T_{k}^{\alpha} (\varphi ' )
\quad \quad \ensuremath{\mathbb{P}\text{ -- a.s.}}
\end{equation}
for any time $k \in \{ 0,1, \dots, T-1 \}$ and any local perturbation $\varphi '$ of $\varphi$ at time $k$.
\end{defi}
Definition \ref{defi:local risk minimizing strategy under illiquidity} assumes that for any strategy the classical cost process $C$ is square-integrable and the liquidity costs are integrable. By Definition \ref{defi:trading strategy} this is ensured. Note also that in Definition \ref{defi:local risk minimizing strategy under illiquidity} we have only taken into account the liquidity costs at the current time. This is equivalent to minimizing over $T_{k}$ in equation (\ref{eq:quadratic linear risk process}) since we minimize only locally.
\begin{remark}
The choice $\alpha = 1$ represents an equal concern about the risk to be hedged as incurred by market price fluctuations and the cost of hedging incurred by liquidity costs. Otherwise, $\alpha < 1$ means a major risk aversion to the risk of miss-hedging and $\alpha > 1$ a major risk aversion to the cost of illiquidity. One could also generalize by having a deterministic $\mathbb{R}$-valued process $\alpha = (\alpha_{k})_{k=0, 1, \dots, T}$ and trivially our results will still hold.
\end{remark}
So in the following we assume $\alpha$ is given and we aim at finding a locally risk-minimizing strategy $\varphi = (X, Y)$ under illiquidity such that $V_{T}(\varphi) = H$ with $X_{T+1} = \bar X_{T+1}$ and $Y_{T} = \bar Y_{T}$.
Some useful Lemmas follow, which even in the multi-dimensional case, can be shown by means very similar to those used in \cite{lamberton.pham.schweizer:1998}. For completeness we provide their proofs in Appendix~\ref{appendix}.
The first Lemma shows that a main property of a local risk-minimizing strategy, namely that its cost process is a martingale, generalizes to our setting. The reason is that a strategy $\varphi$ can be perturbed to $\varphi '$ such that $C(\varphi ')$ is a martingale by changing only the $\mathcal{F}_{k}$-measurable risk free investment. This in turn reduces the first term in (\ref{eq:quadratic linear risk process new}) but leaves the second term unchanged.
\begin{lem}
\label{lem:cost process martingale}
For a LRM-strategy $\varphi$ under illiquidity, the cost process $C(\varphi)$ is a martingale. Furthermore, from the martingale property of the cost process we get the representation,
\begin{equation}
\label{eq:cost process martingale}
R_{k}(\varphi)
= \condE{ R_{k+1}(\varphi) }
+ \condVar{ \Delta C_{k+1}(\varphi) }
\quad \quad \ensuremath{\mathbb{P}\text{ -- a.s.}}
\end{equation}
for $k = 0,1, \dots, T-1$.
\end{lem}
So, for $\varphi$ a LRM-strategy under illiquidity, the quadratic-linear risk process (QLRP) under illiquidity has the representation
\begin{equation}
\label{eq:quadratic linear risk process modified}
T_{k}^{\alpha}(\varphi)
= \condE{ R_{k+1}(\varphi) }
+ \condVar{ \Delta C_{k+1}(\varphi) }
+ \alpha \condE{ \Delta X_{k+2}^{*} [ S_{k+1}(\Delta X_{k+2}) - S_{k+1}(0) ] }.
\end{equation}
\par
The next lemma provides a representation for the QLRP process of a perturbed strategy.
\begin{lem}
\label{lem:quadratic linear risk process modified}
If $C(\varphi)$ is a martingale and $\varphi '$ a local perturbation of $\varphi$ at time $k$ then
\begin{align}
\label{eq:quadratic linear risk process modified}
T_{k}^{\alpha}(\varphi ')
= & \condE{ R_{k+1}(\varphi) }
+ \condE{ (\Delta C_{k+1}(\varphi '))^{2} } \nonumber
\\ & + \alpha \condE{ (X_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X_{k+2} - X'_{k+1}) - S_{k+1}(0) ] } \,.
\end{align}
\end{lem}
\begin{remark}
Since $R_{k+1}(\varphi)=R_{k+1}(\varphi ')$ for any local perturbation $\varphi '$ of $\varphi$ at time $k$, it follows from equation (\ref{eq:quadratic linear risk process modified}) that one needs to minimize over
\begin{equation}
\condVar{ \Delta C_{k+1}(\varphi) }
+ \alpha \condE{ \Delta X_{k+2}^{*} [ S_{k+1}(\Delta X_{k+2}) - S_{k+1}(0) ] }
\end{equation}
at time $k$ (see also Proposition \ref{prop:minimizing variance}).
\end{remark}
\begin{prop}
\label{prop:minimizing variance}
A trading strategy $\varphi = (X, Y)$ is LRM under illiquidity if and only if the two following properties are satisfied:
\begin{enumerate}[label=(\roman*)]
\item \label{prop:1}$C(\varphi)$ is a martingale.
\item \label{prop:2}For each $k \in \{ 0,1, \dots, T-1 \}, X_{k+1}$ minimizes
\begin{align}
& \condVar{ V_{k+1}(\varphi) - (X'_{k+1})^{*} \Delta S_{k+1} } \nonumber
\\& \quad \quad + \alpha \condE{ (X_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X_{k+2} - X'_{k+1}) - S_{k+1}(0) ] }
\end{align}
over all $\mathcal{F}_{k}$-measurable random variables $X'_{k+1}$ so that $(X'_{k+1})^{*} \Delta S_{k+1} \in \mathbb{L}_{T}^{2,1}$ and $(X_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X_{k+2} - X'_{k+1}) - S_{k+1}(0) ] \in \mathbb{L}_{T}^{1,1}$.
\end{enumerate}
\end{prop}
Proposition \ref{prop:minimizing variance} is quite general since it holds for any supply curve. For the existence and recursive construction of a LRM-strategy under illiquidity we will consider in the next section a special case of the supply curve that is motivated from a multiplicative limit order book. For this model we can construct explicitly the optimal strategy and we are able to state conditions that ensure that the optimal strategy belongs to the space $\Theta_{d}(S)$.
\section{Linear supply curve}\label{sec:linear:supply}
When trading through a limit order book (LOB) in an illiquid environment, liquidity costs are related to the depth of the order book. We do not take into account any feedback effects from hedging strategies, so we assume that the speed of resilience, i.e., the ability of the order book to recover itself after a trade, is infinite.
We choose the form of the supply curve $S_{k}(x) = (S_{k}^{1}(x^{1}), \dots, S_{k}^{d}(x^{d}))^{*}$ to be
\begin{equation}
S_{k}^{j}(x^{j}) = S_{k}^{j} + x^{j} \varepsilon_{k}^{j} S_{k}^{j}
\end{equation}
and assume that the price process $S$ is a non-negative semimartingale and $\varepsilon = ( \varepsilon_{k} )_{k = 0,1, \dots, T} $ is a positive deterministic $\mathbb{R}^{d}$-valued process. Note that it is possible for $S_{k}(x)$ to take negative values for some $x$, but in practice this is unlikely to happen for small values of $\varepsilon_k$ and reasonable values of $x$.
\par
Now let us describe a (multiplicative) limit order book for the specific form of the supply curve. A symmetric, $1$-dimensional, time independent (for simplicity) LOB is represented by a density function $q$, where $q(x)dx$ is the bid or ask offers at price level $xS_{k}$. Denote by $F(\rho) = \int_{1}^{\rho} q(x) dx$ the quantity to buy up through the LOB at price $\rho S_{k}$. If an investor makes an order of $x = F(\rho)$ shares through the LOB at time $k$ then some limit orders are eaten up and the quoted price is shifted up to $S_{k}(x)^{+} := g(x)S_{k}$ where $g(x)$ solves the equation $x = \int_{1}^{g(x)} q(y) dy$, that is $g(x) = F^{-1}(x)$.\footnote{Note the multiplicative way of shifting up the price. In an additive LOB this would be of the form $S_{k}(x)^{+} := S_{k} + g(x)$ as for example in \cite{roch:2011}. For a description of multiplicative and additive limit order books see for example \cite{lokka:2012}.} Since we do not account for any price impact, then
after the trade, the price returns to $S_{k}$.\footnote{In the literature, this is the so-called resilience effect, measuring the proportion of new bid or ask orders filling up the LOB after a trade. In our case we have infinite resilience.} The cost of an order of $x$ shares will be $S_{k} \int_{1}^{g(x)} \rho dF(\rho)$ which for an appropriate choice of the function $q$, should be equal to $xS_{k}(x) = xS_{k} + \varepsilon x^{2} S_{k}$. Choosing an, independent from price, density
\begin{equation}
q(x) = \frac{1}{2 \varepsilon}
\end{equation}
does the job. The process $\varepsilon$ can be thought as a measure of illiquidity. For $\varepsilon$ tending to zero the market becomes more liquid and the liquidity cost vanishes.
\begin{remark}
Recall that the supply curve $S_{k}(x)$ is increasing in the transaction size $x$ which ensures non-negative liquidity cost, that is $x[S_{k}(x) - S_{k}(0)] \geq 0$. The specific choice of a linear supply curve implies $\varepsilon S_{k} | x |^{2}$ liquidity costs for a transaction of size $x$ at time $k$. Note that it is essential to assume that the marginal price process $S$ is non-negative in order to avoid negative liquidity costs.
Note that when the price process $S_{k}$ increases, then naturally also the liquidity cost increases but not the availability of assets in the LOB since the depth of the order book $q_{k}(y) = \frac{1}{2 \varepsilon_{k}}$ depends only on $\varepsilon_{k}$.
\end{remark}
Proposition \ref{prop:minimizing variance} tells us how to construct an optimal strategy according to the LRM-criterion under illiquidity. Going backward in time we need to minimize at time $k$
\begin{align}
\label{eq:minimizing expression}
& \condVar{ V_{k+1}(\varphi) - (X'_{k+1})^{*} \Delta S_{k+1} } \nonumber
\\& \quad \quad \quad \quad
+ \alpha \condE{ \sum_{j=1}^{d} \varepsilon_{k+1}^{j} S_{k+1}^{j} (X_{k+2}^{j} - (X'_{k+1})^{j})^{2} }
\end{align}
over all appropriate $X'_{k+1}$ (see Definition \ref{defi:trading strategy}) and choose $Y_{k}$ so that the cost process $C$ becomes a martingale.
Before continuing let us first introduce some notation:
\begin{align}
A_{k; j}^{0} &:= \condVar{ \Delta S_{k+1}^{j} }
\quad & \quad
A_{k; j}^{\varepsilon} &:= \condE{ \varepsilon_{k+1}^{j} S_{k+1}^{j} }
\quad & \quad
A_{k; j} &:= A_{k; j}^{0} + A_{k; j}^{\varepsilon} \nonumber
\\
b_{k; j}^{0} &:= \condCov{ V_{k+1}, \Delta S_{k+1}^{j} }
\quad & \quad
b_{k; j}^{\varepsilon} &:= \condE{ \varepsilon_{k+1}^{j} S_{k+1}^{j} X_{k+2}^{j} }
\quad & \quad
b_{k; j} &:= b_{k; j}^{0} + b_{k; j}^{\varepsilon} \nonumber
\\
D_{k; j, i} &:= \condCov{ \Delta S_{k+1}^{j}, \Delta S_{k+1}^{i} }
& \text{ for } i \neq j &
\end{align}
for all $i,j = 1, \dots, d$ and $k=0,\dots,T-1$.
\par
Furthermore, we can rewrite the expression (\ref{eq:minimizing expression}) by defining the function $f_{k} : \mathbb{R}^{d} \times \Omega \rightarrow \mathbb{R}^{+}$ as
\begin{align}
\label{eq:minimizing function}
f_{k}(c, \omega)
&= \sum_{j=1}^{d} | c_{j} |^{2} A_{k; j}(\omega)
- 2 \sum_{j=1}^{d} c_{j} b_{k; j}(\omega)
+ \sum_{j \neq i} c_{j} c_{i} D_{k; j, i}(\omega) \nonumber
\\
&+ \condVar{ V_{k+1} }(\omega)
+ \sum_{j=1}^{d} \condE{
\varepsilon_{k+1}^{j} S_{k+1}^{j} | X_{k+2}^{j} |^{2} }(\omega).
\end{align}
Fixing $\omega$ one can easily calculate the gradient of the multidimensional function $f_{k}$. We need to solve $grad(f_{k}) = 0$ to calculate the candidates of extreme points which translates into solving a linear equation system of the form
\begin{equation}\label{c:equation}
F_{k} \, c = b_{k}
\end{equation}
where $F_{k} \in \mathbb{R}^{d \times d}$ with $F_{k; i, j} = D_{k; i, j}$ for $i \neq j$, $F_{k; i, j} = A_{k; j}$ for $i = j$ and $b_{k} = ( b_{k;1}, \dots, b_{k; d} ) \in \mathbb{R}^{d}$. Let $F_{k}^{\varepsilon} = diag(A_{k; 1}^{\varepsilon}, \dots, A_{k; d}^{\varepsilon})$ and denote by $F_{k}^{0}$ the matrix $F_{k}$ with $\varepsilon_{k+1}^{j} = 0$ for all $j$, that is the covariance matrix of the marginal price process $S$. Then the symmetric matrix $F_{k}$ is the sum of two real symmetric, positive semidefinite matrices $F_{k} = F_{k}^{0} + F_{k}^{\varepsilon}$. This implies that the matrix $F_{k}$ is also positive semidefinite\footnote{In fact, $F_{k}$ is positive definite if $\varepsilon_{k+1}^{j}$ is positive for all $j=1, \dots, d$.} and therefore also the Hessian matrix which calculates as $H_{f_{k}}(c) = 2 F_{k}$. So, assuming that the covariance matrix $F_{k}^{0}$ is positive definite, this implies that $F_{k}$ is invertible and equation (\ref{c:equation}) h
as a unique solution. Furthermore, since also the Hesse matrix is positive definite the function $c \rightarrow f_{k}(c, \omega)$ is strictly convex, which implies that $c^{*} := F_{k}^{-1} b_{k}$ is a global minimizer. Furthermore, since the matrix $F_{k}^{-1}$ and $b_{k}$ are both $\mathcal{F}_{k}$-measurable it is clear that the minimizer $c^{*}$ is also $\mathcal{F}_{k}$-measurable.
\subsection{Properties of the marginal price process $S$}
\label{sec:assumptions}
In order to show that the optimal strategy $c^{*}$ calculated above belongs to the space $\Theta_{d}(S)$, we need slightly stronger assumptions on the matrix $F_{k}$, which can be reduced to assumptions on the covariance matrix of $S$. We will impose these assumptions now. It will turn out that they hold for independent increments as well as for independent returns.
\begin{defi}
\label{defi:bounded mean-variance tradeoff}
We say that $S$ has \textit{bounded mean-variance tradeoff} process if for some constant $C > 0$
\begin{equation}
\frac{( \condE{ \Delta S_{k+1}^{j} } )^{2} }{ \condVar{ \Delta S_{k+1}^{j} } }
\leq C
\quad \ensuremath{\mathbb{P}\text{ -- a.s.}} \quad \text{for all } j = 1, \dots, d
\end{equation}
uniformly in $k$ and $\omega$.
\end{defi}
\begin{defi}
\label{defi:modified bounded mean-variance tradeoff}
We say that $S$ has \textit{modified above bounded mean-variance tradeoff} process if for some constant $C > 0$
\begin{equation}
\frac{( \condE{ S_{k+1}^{j} } )^{2} }{ \condVar{ S_{k+1}^{j} } }
\leq C
\quad \ensuremath{\mathbb{P}\text{ -- a.s.}} \quad \text{for all } j = 1, \dots, d
\end{equation}
uniformly in $k$ and $\omega$. Furthermore $S$ has \textit{modified below bounded mean-variance tradeoff} process if for some constant $\tilde C > 0$
\begin{equation}
\frac{( \condE{ S_{k+1}^{j} } )^{2} }{ \condVar{ S_{k+1}^{j} } }
\geq \tilde C
\quad \ensuremath{\mathbb{P}\text{ -- a.s.}} \quad \text{for all } j = 1, \dots, d
\end{equation}
uniformly in $k$ and $\omega$. If both bounds hold then we say that $S$ has \textit{modified bounded mean-variance tradeoff}.
\end{defi}
\begin{remark}
Note that for the case of $S$ being a submartingale then the property of modified above bounded mean-variance tradeoff implies that of bounded mean-variance tradeoff, since by using $(a + b)^{2} \leq 2a^{2} + 2b^{2}$ we can estimate
\begin{align}
(\condE{ \Delta S_{k+1}^{j} } )^{2}
\leq 2 (\condE{ S_{k+1}^{j} } )^{2} + 2 | S_{k} |^{2}
\leq 4 (\condE{ S_{k+1}^{j} } )^{2}
\end{align}
where we have also used the fact that $S_{k}^{j}$ is positive.
\end{remark}
\begin{defi}
\label{defi:diagonal condition F}
We say that $S$ satisfies the \textit{F-diagonal condition} if for some constant $C > 0$
\begin{equation}
\label{eq:diagonal condition F ineq 1}
\sqrt{\condVar{ \Delta S_{k+1}^{j} } }
+ \frac{ \condE{ S_{k+1}^{j} } }{ \sqrt{\condVar{ S_{k+1}^{j} } } }
\geq C
\quad \ensuremath{\mathbb{P}\text{ -- a.s.}} \quad \text{for all } j = 1, \dots, d
\end{equation}
uniformly in $k$ and $\omega$ and if for some constant $\tilde C > 0$
\begin{equation}
\label{eq:diagonal condition F ineq 2}
\frac{ \sqrt{\condVar{ S_{k+1}^{j} } } }{ \condE{ S_{k+1}^{j} } }
+ \frac{ 1 }{ \sqrt{\condVar{ \Delta S_{k+1}^{j} } } }
\geq \tilde C
\quad \ensuremath{\mathbb{P}\text{ -- a.s.}} \quad \text{for all } j = 1, \dots, d
\end{equation}
uniformly in $k$ and $\omega$.
\end{defi}
\begin{remark}
The name \textit{F-diagonal condition} in Definition \ref{defi:diagonal condition F} comes from the diagonal terms of the matrix $F$, since
\begin{align}
\frac{ F_{k; j, j}^{0} }{ | F_{k; j, j} |^{2} }
&= \left( \sqrt{\condVar{ \Delta S_{k+1}^{j} } }
+ \frac{ \condE{ \varepsilon_{k+1}^{j} S_{k+1}^{j} } }{ \sqrt{\condVar{ S_{k+1}^{j} } } } \right)^{-2}
\nonumber \\
| F_{k; j, j}^{\varepsilon} |^{2} \frac{ F_{k; j, j}^{0} }{ | F_{k; j, j} |^{2} }
&= \left( \frac{ \sqrt{\condVar{ S_{k+1}^{j} } } }{ \condE{ \varepsilon_{k+1}^{j} S_{k+1}^{j} } }
+ \frac{ 1 }{ \sqrt{\condVar{ \Delta S_{k+1}^{j} } } } \right)^{-2} \,.
\end{align}
\end{remark}
Writing $S_{k+1}^{j} = S_{k}^{j}( 1 + \rho_{k+1}^{j} )$ for $j = 1, \dots, d$ , we denote by $\rho = (\rho_{k})_{k=0,1, \dots T}$ the $d$-dimensional return process of $S$.
\par
The next two Propositions \ref{prop:independent increments} and \ref{prop:independent returns} give sufficient conditions for the previous properties on the marginal price process $S$ to hold.
\begin{prop}
\label{prop:independent increments}
For $S$ satisfying $\tilde C \leq \condVar{ \Delta S_{k+1}^{j} } \leq C$ for some positive constants $C, \tilde C$ and for all $j = 1, \dots, d$, then the $F$-diagonal condition holds. In particular, if $S$ has independent increments then $S$ has bounded mean-variance tradeoff and satisfies the $F$-diagonal condition.
\end{prop}
\begin{proof}
The claim follows directly from the fact that $\tilde C \leq \condVar{ \Delta S_{k+1}^{j} } \leq C$.
\end{proof}
\begin{prop}
\label{prop:independent returns}
For $S$ having modified bounded mean-variance tradeoff then the $F$-diagonal condition holds. In particular, if $S$ has independent returns then $S$ has bounded mean-variance tradeoff and satisfies the $F$-diagonal condition.
\end{prop}
\begin{proof}
The claim follows directly from the fact that $S$ has modified bounded mean-variance tradeoff.
\end{proof}
\begin{remark}
Consider the $1$-dimensional Black-Scholes model of a geometric Brownian motion $W$, that is
\begin{equation}
S_{kh} = S_{0} \exp{(bkh + \sigma W_{kh})}
\end{equation}
with discretization time step $\Delta t = h$. Then the return process $\rho_{k}$ can be defined by,
\begin{equation}
1 + \rho_{k} = \frac{ S_{kh} }{ S_{(k-1)h} }
\end{equation}
and is lognormally distributed. This is also a process of i.i.d. random variables. By Proposition \ref{prop:independent returns}, $S$ has bounded mean-variance tradeoff and satisfies the $F$-diagonal condition.
\end{remark}
\par
\subsection{Some preliminaries}
\label{sec:existence and recursive construction of an optimal strategy under illiquidity}
Now let us state some useful Lemmas needed in the proof of Theorem \ref{theo:existence} in order to show that the integrability conditions are fulfilled. In what follows we will use the notation
\begin{align}
\alpha_{k;i,j} &:= F_{k; j, j}^{0} F_{k; i, i}^{0} | F_{k; j, i}^{-1} |^{2}
\quad \quad
&& \alpha_{k;i,j}^{\varepsilon} := F_{k; j, j}^{0} | F_{k; i, i}^{\varepsilon} |^{2} | F_{k; j, i}^{-1} |^{2}
\nonumber \\
\beta_{k;i,j} &:= F_{k; i, i}^{0} | F_{k; j, i}^{-1} |^{2}
\quad \quad
&& \beta_{k;i,j}^{\varepsilon} := | F_{k; i, i}^{\varepsilon} |^{2} | F_{k; j, i}^{-1} |^{2}
\end{align}
for $i,j=1, \dots, d$ and $k=0,\dots , T$ when the inverse matrix $F_{k}^{-1}$ of $F_{k}$ exists.
\par
In the following we will denote by $M_{k; i, j}$ the matrix $F_{k}$ without the $i$-th row and $j$-th column. Recall also from linear algebra that if the inverse of a symmetric matrix $F_{k}$ exists then $F_{k; j, i}^{-1} = \frac{ (-1)^{i+j} \det(M_{k; i, j}) }{ \det(F_{k}) }$ which we use in Lemma \ref{lem:the alpha, beta, gamma terms}.
\begin{lem}
\label{lem:F-matrix}
For all $d \in \mathbb{N}_{\geq 2}$:
\begin{align}
\det(M_{k; i, j})^{2}
& \leq C F_{k; j, j}^{0} F_{k; i, i}^{0} \prod_{\substack{ l=1\\ l \neq i,j}}^{d} | F_{k; l, l} |^{2}
\quad && \text{ for all } i, j = 1, \dots, d \text{ with } i \neq j
\label{eq:lemma F-matrix ineq 1}
\\
| F_{k; j, j} |^{2} \det(M_{k; j, j})^{2}
& \leq \tilde C \det(F_{k}^{A})^{2}
\quad && \text{ for all } j = 1, \dots, d
\label{eq:lemma F-matrix ineq 2}
\\
F_{k; j, j} F_{k; i, i} \det(M_{k; i, j})^{2}
& \leq \bar C \det(F_{k}^{A})^{2}
\quad && \text{ for all } i, j = 1, \dots, d
\label{eq:lemma F-matrix ineq 3}
\end{align}
for some positive constants $C, \tilde C$ and $\bar C$ where $F_{k}^{A} := \diag(A_{k; 1}, \dots, A_{k; d})$.
\end{lem}
\begin{proof}
First note that the last inequality (\ref{eq:lemma F-matrix ineq 3}) follows from the first two. Indeed for the case $i \neq j$ and since $F_{k; j, j}^{0} \leq F_{k; j, j}$ (since $\varepsilon_{k+1}^{j}$ and $S_{k+1}^{j}$ are non-negative) for all $j$, then from inequality (\ref{eq:lemma F-matrix ineq 1}) we have
\begin{equation}
\det(M_{k; i, j})^{2}
\leq C F_{k; j, j} F_{k; i, i} \prod_{\substack{ l=1\\ l \neq i,j}}^{d} | F_{k; l, l} |^{2} \,.
\end{equation}
Since the matrix $F_{k}^{A}$ is a diagonal matrix then it is clear that now inequality (\ref{eq:lemma F-matrix ineq 3}) follows for $i \neq j$. The case $i=j$ follows directly from inequality (\ref{eq:lemma F-matrix ineq 2}).
\par
For showing the inequalities (\ref{eq:lemma F-matrix ineq 1}) and (\ref{eq:lemma F-matrix ineq 2}) for $d=2$ is trivial. We will show for the case $d=3$ the inequality (\ref{eq:lemma F-matrix ineq 1}). Inequality (\ref{eq:lemma F-matrix ineq 2}) follows then analogously. Let w.l.o.g. $i=1$. For $j=2$ we have
\begin{align}
\det(M_{k; 1, 2})^{2}
= (D_{k; 1, 2} A_{k; 3} - D_{k; 2, 3} D_{k; 1, 3})^{2}
\leq 2 | D_{k; 1, 2} |^{2} | A_{k; 3} |^{2} + 2 | D_{k; 2, 3} |^{2} | D_{k; 1, 3} |^{2}
\end{align}
where we have used the inequality $(a + b)^{2} \leq 2 a^{2} + 2 b^{2}$. Now, applying the conditional Cauchy-Schwarz inequality we get,
\begin{align}
\det(M_{k; 1, 2})^{2}
\leq 2 A_{k; 1}^{0} A_{k; 2}^{0} | A_{k; 3} |^{2} + 2 A_{k; 2}^{0} A_{k; 3}^{0} A_{k; 1}^{0} A_{k; 3}^{0}
\leq 4 A_{k; 1}^{0} A_{k; 2}^{0} | A_{k; 3} |^{2} \,.
\end{align}
The case $j=3$ follows analogously and so inequality (\ref{eq:lemma F-matrix ineq 1}) holds.
\par
A generalization of the proof for an arbitrary $d$ can be done using the Laplace's formula and the symmetry of the matrices $F_{k}$ and $F_{k}^{0}$.
\end{proof}
The next Definition of the \textit{$F$-property} is crucial, not only for extending the LRM-criterion of \cite{schweizer:1988} to the illiquid case (i.e. $\varepsilon \neq 0$) but also (especially) for the extension to the multidimensional case. In the $1$-dimensional case the $F$-property translates to $\condVar{ \Delta S_{k+1} } + \condE{ \varepsilon_{k+1}S_{k+1} } \geq 0$ for a one-dimensional price process $S$ which is always fulfilled.\footnote{Recall the assumption that the price process $S$ and the process $\varepsilon$ are both non-negative.} Also if we are dealing with independent components, i.e., $S^{i}$ and $S^{j}$ are independent for $i \neq j$, then it reduces to $\det(F_{k}^{A}) \geq 0$ which also always holds since the matrix $F_{k}^{A}$ is positive semi-definite. So the next property is essentially linked to the covariance matrix of the multidimensional price process $S$. We will see later on in Section \ref{sec:the reduction of the F-property to the covariance matrix} that this property can be reduced to a property on the covariance matrix of $S$. In what follows, $C$ denotes a generic positive constant that might change from line to line.
\begin{defi}
\label{defi:the F-property}
We say that the process $S$ has the $F$-property if there exists some $\delta \in (0, 1)$ such that
\begin{equation}
\det(F_{k}) - (1 - \delta) \det(F_{k}^{A})
\geq 0
\end{equation}
for all $k = 0, 1, \dots, T$ where $F_{k}^{A} := \diag(A_{k; 1}, \dots, A_{k; d})$.
\end{defi}
%
\begin{lem}
\label{lem:alpha beta terms boundedness}
Assume that $S$ has the $F$-property and satisfies the $F$-diagonal condition. Then the terms $\alpha_{k;i,j}$, $\beta_{k;i,j}$, $\alpha_{k;i,j}^{\varepsilon}$ and $\beta_{k;i,j}^{\varepsilon}$ are uniformly bounded in $k$ and $\omega$ for all $i,j = 1, \dots, d$.
\end{lem}
\begin{proof}
For the first term $\alpha_{k;i,j}$ we have
\begin{align}
\alpha_{k;i,j}
= F_{k; j, j}^{0} F_{k; i, i}^{0} \frac{ \det(M_{k; i, j})^{2} }{ \det(F_{k})^{2} }
\leq C \frac{ \det(F_{k}^{A})^{2} }{ \det(F_{k})^{2} }
\leq C \frac{ 1 }{ (1 - \delta)^{2} }
\end{align}
by using first the inequality (\ref{eq:lemma F-matrix ineq 3}) from Lemma \ref{lem:F-matrix} and then the $F$-property. For the second term $\beta_{k;i,j}$ we can estimate for the case $i = j$
\begin{align}
\beta_{k;i,j}
= F_{k; i, i}^{0} \frac{ \det(M_{k; i, i})^{2} }{ \det(F_{k})^{2} }
\leq C \frac{ F_{k; i, i}^{0} }{ | F_{k; i, i} |^{2} } \frac{ \det(F_{k}^{A})^{2} }{ \det(F_{k})^{2} }
\leq C \frac{ 1 }{ (1 - \delta)^{2} }
\end{align}
using inequality (\ref{eq:lemma F-matrix ineq 2}) from Lemma \ref{lem:F-matrix} and then the $F$-property and inequality (\ref{eq:diagonal condition F ineq 1}). For the case $i \neq j$ and using inequality (\ref{eq:lemma F-matrix ineq 1}) from Lemma \ref{lem:F-matrix} we have
\begin{align}
\det(F_{k})^{2} \beta_{k;i,j}
= F_{k; i, i}^{0} \det(M_{k; i, j})^{2}
\leq C F_{k; j, j}^{0} | F_{k; i, i}^{0} |^{2} \prod_{\substack{ l=1\\ l \neq i,j}}^{d} | F_{k; l, l} |^{2}
\leq C \frac{ F_{k; j, j}^{0} }{ | F_{k; j, j} |^{2} } \det(F_{k}^{A})^{2}
\end{align}
and from the $F$-property and inequality (\ref{eq:diagonal condition F ineq 1}), $\beta_{k;i,j}$ is uniformly bounded. Furthermore and by the same arguments as for the term $\beta_{k;i,j}$ we have for the case $i=j$
\begin{align}
\alpha_{k;i,j}^{\varepsilon}
= | F_{k; i, i}^{\varepsilon} |^{2} F_{k; i, i}^{0} \frac{ \det(M_{k; i, i})^{2} }{ \det(F_{k})^{2} }
\leq C | F_{k; i, i}^{\varepsilon} |^{2} \frac{ F_{k; i, i}^{0} }{ | F_{k; i, i} |^{2} } \frac{ \det(F_{k}^{A})^{2} }{ \det(F_{k})^{2} }
\leq C \frac{ 1 }{ (1 - \delta)^{2} }
\end{align}
using the $F$-property and inequality (\ref{eq:diagonal condition F ineq 2}). For $i \neq j$ we can estimate
\begin{align}
\det(F_{k})^{2} \alpha_{k;i,j}^{\varepsilon}
= | F_{k; i, i}^{\varepsilon} |^{2} F_{k; j, j}^{0} \det(M_{k; i, j})^{2}
&\leq C | F_{k; i, i}^{\varepsilon} |^{2} F_{k; i, i}^{0} | F_{k; j, j}^{0} |^{2} \prod_{\substack{ l=1\\ l \neq i,j}}^{d} | F_{k; l, l} |^{2}
\nonumber \\
&\leq C | F_{k; i, i}^{\varepsilon} |^{2} \frac{ F_{k; i, i}^{0} }{ | F_{k; i, i} |^{2} } \det(F_{k}^{A})^{2}
\end{align}
and from the $F$-property and inequality (\ref{eq:diagonal condition F ineq 2}), $\alpha_{k;i,j}^{\varepsilon}$ is also uniformly bounded. For the last term $\beta_{k;i,j}^{\varepsilon}$ we have for $i = j$
\begin{align}
\beta_{k;i,j}^{\varepsilon}
= | F_{k; i, i}^{\varepsilon} |^{2} \frac{ \det(M_{k; i, i})^{2} }{ \det(F_{k})^{2} }
\leq C \frac{ | F_{k; i, i}^{\varepsilon} |^{2} }{ | F_{k; i, i} |^{2} } \frac{ \det(F_{k}^{A})^{2} }{ \det(F_{k})^{2} }
\leq C \frac{ 1 }{ (1 - \delta)^{2} }
\end{align}
by the $F$-property. Moreover for $i \neq j$
\begin{align}
\det(F_{k})^{2} \beta_{k;i,j}^{\varepsilon}
= | F_{k; i, i}^{\varepsilon} |^{2} \det(M_{k; i, j})^{2}
&\leq C | F_{k; i, i}^{\varepsilon} |^{2} F_{k; i, i}^{0} F_{k; j, j}^{0} \prod_{\substack{ l=1\\ l \neq i,j}}^{d} | F_{k; l, l} |^{2}
\nonumber \\
&= C | F_{k; i, i}^{\varepsilon} |^{2} \frac{ F_{k; i, i}^{0} }{ | F_{k; i, i} |^{2} } \frac{ F_{k; j, j}^{0} }{ | F_{k; j, j} |^{2} } \det(F_{k}^{A})^{2}
\end{align}
where from the $F$-property and the $F$-diagonal condition the last term $\beta_{k;i,j}^{\varepsilon}$ is uniformly bounded. We also made use of the fact that the process $\varepsilon$ is deterministic and that we have a finite number of hedging times.
\end{proof}
\begin{lem}
\label{lem:the alpha, beta, gamma terms}
Assume that $F_{k}^{-1}$ exists for $k \in \{ 0, 1, \dots, T \}$ and $S$ has bounded mean-variance tradeoff. Let $(X,Y)$ be any trading strategy. Then there exists some constant $C > 0$ such that
\begin{eqnarray}
&& {\mathbb E}[ ( (F_{k}^{-1} b_{k})_{j} \Delta S_{k+1}^{j} )^{2} ] \label{strat:bound:1} \nonumber\\
&\leq &C {\mathbb E}[ \condVar{ V_{k+1} } \sum_{i=1}^{d} \alpha_{k;i,j}
+ \sum_{i=1}^{d}( c(\varepsilon_{k+1}) \alpha_{k;i,j} + \alpha_{k;i,j}^{\varepsilon}) \condE{ | X_{k+2}^{i} |^{2} } ]
\\
&& {\mathbb E}[ ( (F_{k}^{-1} b_{k})_{j} )^{2} ] \nonumber \\
&\leq & C {\mathbb E}[ \condVar{ V_{k+1} } \sum_{i=1}^{d} \beta_{k;i,j}
+ \sum_{i=1}^{d}( c(\varepsilon_{k+1}) \beta_{k;i,j} + \beta_{k;i,j}^{\varepsilon}) \condE{ | X_{k+2}^{i} |^{2} } ] \label{strat:bound:2}
\end{eqnarray}
for all $j = 1, \dots, d$ where $(F_{k}^{-1} b_{k})_{j}$ is the $j$-th component of the vector $(F_{k}^{-1} b_{k})$. The term $c(\varepsilon_{k+1})$ denotes a positive constant depending on the process $\varepsilon$ at time $k+1$ such that for $\varepsilon_{k+1} \to 0$, $c(\varepsilon_{k+1})$ converges to zero.
\end{lem}
\begin{proof}
First note that from the definition of the variance and using bounded mean-variance tradeoff, it follows directly that
\begin{equation}
\label{eq:1 lem:the alpha, beta, gamma terms}
\condE{ | \Delta S_{k+1}^{j} |^{2} }
= \condVar{ \Delta S_{k+1}^{j} } + ( \condE{ \Delta S_{k+1}^{j} } )^{2}
\leq C A_{k; j}^{0} \,.
\end{equation}
Furthermore, denoting $F = F_{k}$ and $b = b_{k}$ we have from the tower property and using inequality (\ref{eq:1 lem:the alpha, beta, gamma terms})
\begin{align}
{\mathbb E}[ ( (F^{-1} b)_{j} \Delta S_{k+1}^{j} )^{2} ]
&= {\mathbb E}[ ( (F^{-1} (b^{0} + b^{\varepsilon}))_{j} )^{2} \condE{ | \Delta S_{k+1}^{j} |^{2} } ]
\nonumber \\
&\leq 2C {\mathbb E}[ \sum_{i=1}^{d} | F_{j, i}^{-1} |^{2} (| b_{i}^{0} |^{2} + | b_{i}^{\varepsilon} |^{2}) F_{j, j}^{0} ]
\end{align}
Moreover, using the conditional Cauchy-Schwarz-Inequality for the term $b_{i}^{0}$ and the conditional inequality $({\mathbb E}[ XY | \mathcal{G} ])^{2} \leq {\mathbb E} [X^{2} | \mathcal{G} ] {\mathbb E} [Y^{2} | \mathcal{G} ]$ on the term $b_{i}^{\varepsilon}$ together with the definition of the variance yields
\begin{align}
&{\mathbb E}[ ( (F^{-1} b)_{j} \Delta S_{k+1}^{j} )^{2} ]
\nonumber \\
&\leq C {\mathbb E}[ \sum_{i=1}^{d} | F_{j, i}^{-1} |^{2} ( \condVar{ V_{k+1} } F_{i, i}^{0} + \condE{ | \varepsilon_{k+1}^{i} S_{k+1}^{i} |^{2} } \condE{ | X_{k+2}^{i} |^{2} } ) F_{j, j}^{0} ]
\nonumber \\
&= C {\mathbb E}[ \sum_{i=1}^{d} | F_{j, i}^{-1} |^{2} ( \condVar{ V_{k+1} } F_{i, i}^{0} + | \varepsilon_{k+1}^{i} |^{2} F_{i, i}^{0} \condE{ | X_{k+2}^{i} |^{2} } + | F_{i, i}^{\varepsilon} |^{2} \condE{ | X_{k+2}^{i} |^{2} } ) F_{j, j}^{0} ] \,.
\end{align}
The other inequality follows analogously.
\end{proof}
\begin{remark}
For the Existence of a LRM-strategy under illiquidity we will use Lemma \ref{lem:alpha beta terms boundedness} together with Lemma \ref{lem:the alpha, beta, gamma terms}. For the optimal strategy $\hat X$ (under the LRM-criterion under illiquidity) we will need to show that $\hat X_{k+1}^{j} \Delta S_{k+1}^{j} \in \mathbb{L}_{T}^{2, 1}$ and $\hat X_{k+1}^{j} \in \mathbb{L}_{T}^{2, 1}$. The first integrability property shows that the strategy $\hat X$ belongs to $ \hat\Theta_{d}(S)$, the space of all $\mathbb{R}^{d}$-valued predictable strategies $X = (X_{k})_{k=1,2, \dots, T+1}$ so that $ X_{k}^{*} \Delta S_{k} \in \mathbb{L}_{T}^{2, 1}$ for $k=1, 2, \dots, T$. The second one is needed to show the first one. Nevertheless, both integrability properties are needed in order to show that the liquidity costs of the optimal strategy are integrable.
\par
In the infinite liquidity case, that is $\varepsilon = 0$, since the terms $c(\varepsilon_{k+1}) \alpha_{k;i,j}$ and $\alpha_{k;i,j}^{\varepsilon}$ vanish, we do not need the second inequality of Lemma \ref{lem:the alpha, beta, gamma terms}. This implies that in the multidimensional case without liquidity costs, one needs to show only that $\hat X \in \hat \Theta_{d}(S)$ by using bounded mean-variance tradeoff and the $F$-property.
\par
Also, in the $1$-dimensional case ($d=1$) we have
\begin{equation}
\alpha_{k;1,1} = \frac{ | A_{k; 1}^{0} |^{2} }{ | A_{k; 1} |^{2} }
\quad , \quad
\beta_{k;1,1} = \frac{A_{k;1}^{0}}{| A_{k;1} |^{2}}
\quad , \quad
\alpha_{k;1,1}^{\varepsilon} = A_{k; 1}^{\varepsilon} \frac{ A_{k; 1}^{0} }{ | A_{k; 1} |^{2} }
\quad , \quad
\beta_{k;1,1}^{\varepsilon} = \frac{ | A_{k; 1}^{\varepsilon} |^{2} }{ | A_{k; 1} |^{2} }
\end{equation}
where the terms $\alpha_{k;1,1}$, $\beta_{k;1,1}^{\varepsilon}$ are bounded by $1$ and the terms $\beta_{k;1,1}$, $\alpha_{k;1,1}^{\varepsilon}$ are uniformly bounded by the $F$-diagonal property. Moreover for $\varepsilon = 0$ one would only need to show the first inequality of Lemma \ref{lem:the alpha, beta, gamma terms} which reduces to
\begin{equation}
{\mathbb E}[ ( (F_{k}^{-1} b_{k})_{1} \Delta S_{k+1}^{1} )^{2} ]
\leq C {\mathbb E}[ | V_{k+1} |^{2} ]
\end{equation}
as in the classical $1$-dimensional case in \cite{schweizer:1988}. Recall that in this case only the assumption of bounded mean-variance tradeoff is essential.
\end{remark}
We continue with the main Theorem where we show the existence of a local risk-minimizing strategy under illiquidity and under some mild conditions on the marginal price process $S$.
\subsection{Existence and recursive construction of an optimal strategy}
\label{sec:the main theorem}
Using the assumptions imposed in the previous Section \ref{sec:assumptions} we are able to prove the existence of a local risk-minimizing strategy under illiquidity and additionally to give an explicit representation by means of a backward induction argument.
\begin{theo}[\textbf{Existence result}]
\label{theo:existence}
Assume that $S$ has the $F$-property, bounded mean-variance tradeoff and satisfies the $F$-diagonal condition. Let further the covariance matrix $F_{k}^{0}$ be positive definite at all times $k = 0, 1, \dots, T-1$. Then for any contingent claim $H = \bar X_{T+1}^{*} S_{T} + \bar Y_{T} \in \mathbb{L}_{T}^{2, 1}$ with $\bar X_{T+1}^{*} S_{T} \in \mathbb{L}_{T}^{2, 1}$ and $\bar X_{T+1} \in \mathbb{L}_{T}^{2, d}$, there exists a local risk-minimizing strategy $\hat \varphi = ( \hat X, \hat Y )$ under illiquidity with $\hat X_{T+1} = \bar X_{T+1}$ and $\hat Y_{T} = \bar Y_{T}$. Furthermore, the strategy has the representation
\begin{align}
\hat X_{k+1}
&= F_{k}^{-1} b_{k}
\quad \ensuremath{\mathbb{P}\text{ -- a.s.}} \text{ for } k = 0, \dots, T-1
\label{eq:LRM-strategy X}
\\
\hat Y_{k}
&= \condE{ \hat W_{k} } - \hat X_{k+1}^{*} S_{k}
\quad \ensuremath{\mathbb{P}\text{ -- a.s.}} \text{ for } k = 0, 1, \dots, T-1
\label{eq:LRM-strategy Y}
\end{align}
where $\hat W_{k} = H - \sum_{m=k+1}^{T} \hat X_{m}^{*} \Delta S_{m}$.
\end{theo}
\begin{proof}
The proof is a backward induction argument on $k = 0, 1, \dots, T$. First set $\hat X_{T+1} = \bar X_{T+1}$ and $\hat Y_{T} = \bar Y_{T}$. So, fix some $k \in \{ 0, 1, \dots, T-2 \}$ and assume that at times $l = k, \dots, T-2$
\begin{enumerate}[label=(\roman*)]
\item \label{item:theorem proof step 1} $ \hat X_{l+2}^{j} \Delta S_{l+2}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}} \quad \text{and} \quad \hat X_{l+2}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}} $
\item \label{item:theorem proof step 2} $| \hat X_{l+2}^{j} |^{2} S_{l+1}^{j} \in \ensuremath{\mathbb{L}_{T}^{1, 1}}$
\item \label{item:theorem proof step 3} $ \hat X_{l+2}^{*} S_{l+1} + \hat Y_{l+1} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}
\quad , \quad \hat Y_{l+1} \in \mathcal{F}_{l+1} $
\end{enumerate}
for all $j = 1, \dots, d$ holds. At time $k$ we want to minimize the expression (\ref{eq:minimizing expression}) over all $X_{k+1}^{'}$ and show that the following properties are fulfilled for all $j = 1, \dots, d$:
\begin{enumerate}[label=(\roman*)]
\item \label{item:theorem proof step 1} $ X_{k+1}^{', j} \Delta S_{k+1}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}} \quad \text{and} \quad X_{k+1}^{', j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}} $
\item \label{item:theorem proof step 2} $ | X_{k+1}^{', j} |^{2} S_{k}^{j} \in \ensuremath{\mathbb{L}_{T}^{1, 1}} $
\item \label{item:theorem proof step 3} $ (X_{k+1}^{'})^{*} S_{k} + Y_{k}^{'} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}
\quad , \quad Y_{k}^{'} \in \mathcal{F}_{k} $
\end{enumerate}
Properties \ref{item:theorem proof step 1} - \ref{item:theorem proof step 3} will then ensure that $(\hat X, \hat Y) \in \Theta_d (S)$. First we define the function $f_{k}$ as in equation (\ref{eq:minimizing function}) and note that all the terms in $f_{k}$ are integrable by induction hypothesis. Since $F_{k}$ is positive definite then there exists a unique solution to the minimization problem and an $\mathcal{F}_{k}$-measurable minimizer $\hat X_{k+1}$ can be constructed, which equals $F_{k}^{-1} b_{k}$. Furthermore define $\hat Y_{k}$ as in equation (\ref{eq:LRM-strategy Y}). Then it is clear that $\hat Y_{k}$ is $\mathcal{F}_{k}$-measurable. The fact that $\hat X_{k+1}^{*} S_{k} + \hat Y_{k} = \condE{ \hat W_{k} } \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$ follows from $H \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$, the induction hypothesis $\sum_{m=k+2}^{T} \hat X _{m}^{*} \Delta S_{m} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$ and $\hat X _{k+1}^{*} \Delta S_{k+1} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$, which we will show below.
\par
Now let us show first that $ \hat X_{k+1}^{j} \Delta S_{k+1}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$. By inequality (\ref{strat:bound:1}) of Lemma \ref{lem:the alpha, beta, gamma terms} we know that for a constant $C>0$,
\begin{eqnarray}
& & {\mathbb E}[ ( \hat X_{k+1}^{j} \Delta S_{k+1}^{j} )^{2} ] \nonumber \\
&\leq & C {\mathbb E}[ \condVar{ \hat X_{k+2}^{*} S_{k+1} + \hat Y_{k+1} } \sum_{i=1}^{d} \alpha_{k;i,j}
+ \sum_{i=1}^{d}( c(\varepsilon_{k+1}) \alpha_{k;i,j} + \alpha_{k;i,j}^{\varepsilon}) \condE{ | X_{k+2}^{i} |^{2} } ]
\end{eqnarray}
holds. Since by the induction hypothesis $\hat X_{k+2}^{*} S_{k+1} + \hat Y_{k+1}$ and $\hat X_{k+2}^{i}$ both in $\ensuremath{\mathbb{L}_{T}^{2, 1}}$ for all $i=1, \dots, d$, then it remains to show that the terms $\alpha_{k;i,j}$, $\alpha_{k;i,j}^{\varepsilon}$ are uniformly bounded in $k$ and $\omega$. This follows from Lemma \ref{lem:alpha beta terms boundedness}. Similarly one can show that $ \hat X_{k+1}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$ using inequality (\ref{strat:bound:2}) of Lemma \ref{lem:the alpha, beta, gamma terms}.
\par
Next we show that the liquidity costs $\condE{ \sum_{j=1}^{d} \varepsilon_{k+1}^{j} S_{k+1}^{j} | \hat X_{k+2}^{j} - \hat X_{k+1}^{j} |^{2} } $ are integrable. From the minimization problem of expression (\ref{eq:minimizing expression}) and since $\hat X_{k+1}$ is a minimizer, we know that (w.l.o.g. $\alpha = 1$):
\begin{align}
& \condVar{ \hat X_{k+2}^{*} S_{k+1} + \hat Y_{k+1} - (\hat X_{k+1})^{*} \Delta S_{k+1} }
+ \condE{ \sum_{j=1}^{d} \varepsilon_{k+1}^{j} S_{k+1}^{j} | \hat X_{k+2}^{j} - \hat X_{k+1}^{j} |^{2} }
\nonumber \\
& \leq \condVar{ \hat X_{k+2}^{*} S_{k+1} + \hat Y_{k+1}}
+ \condE{ \sum_{j=1}^{d} \varepsilon_{k+1}^{j} S_{k+1}^{j} | \hat X_{k+2}^{j} |^{2} }
\end{align}
holds, where the right hand side corresponds to choosing $X_{k+1}=0$.
Taking expectation on both sides and since by definition the conditional variance is non-negative, we get
\begin{align}
{\mathbb E}[ \sum_{j=1}^{d} \varepsilon_{k+1}^{j} S_{k+1}^{j} | \hat X_{k+2}^{j} - \hat X_{k+1}^{j} |^{2} ]
\leq {\mathbb E}[ | \hat X_{k+2}^{*} S_{k+1} + \hat Y_{k+1} |^{2} ]
+ {\mathbb E}[ \sum_{j=1}^{d} \varepsilon_{k+1}^{j} S_{k+1}^{j} | \hat X_{k+2}^{j} |^{2} ]
\end{align}
where we have used the fact that ${\mathbb{V}ar}(X) \leq {\mathbb E}|X|^{2}$. Now, since by the inductive hypothesis, $\hat X_{k+2}^{*} S_{k+1} + \hat Y_{k+1} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$ and $S_{k+1}^{j} | \hat X_{k+2}^{j} |^{2} \in \ensuremath{\mathbb{L}_{T}^{1, 1}}$ for all $j=1, \dots, d$ then it is clear that the liquidity cost $\sum_{j=1}^{d} \varepsilon_{k+1}^{j} S_{k+1}^{j} | \hat X_{k+2}^{j} - \hat X_{k+1}^{j} |^{2}$ is in $\ensuremath{\mathbb{L}_{T}^{1, 1}}$. In particular $ \varepsilon_{k+1}^{j} S_{k+1}^{j} | \hat X_{k+2}^{j} - \hat X_{k+1}^{j} |^{2} \in \ensuremath{\mathbb{L}_{T}^{1, 1}}$ for all $j=1, \dots, d$. This holds from the fact that the deterministic process $\varepsilon$ and the marginal price process $S$ are both non-negative by assumption.
\par
In order to complete the proof, it remains to show that $| \hat X_{k+1}^{j} |^{2} S_{k}^{j} \in \ensuremath{\mathbb{L}_{T}^{1, 1}}$. This is needed in order to complete the induction argument and be able to show that the liquidity costs in the next step are again integrable. So, from the equality
\begin{equation}
| \hat X_{k+1}^{j} |^{2} S_{k}^{j}
= - | \hat X_{k+1}^{j} |^{2} \Delta S_{k+1}^{j} + | \hat X_{k+1}^{j} |^{2} S_{k+1}^{j}
\end{equation}
we need to show that $| \hat X_{k+1}^{j} |^{2} \Delta S_{k+1}^{j}$ and $| \hat X_{k+1}^{j} |^{2} S_{k+1}^{j}$ are both in $\ensuremath{\mathbb{L}_{T}^{1, 1}}$.
Since, as already shown, the liquidity costs are integrable for all $j=1, \dots, d$ and since by induction hypothesis $ | \hat X_{k+2}^{j} |^{2} S_{k+1}^{j} \in \ensuremath{\mathbb{L}_{T}^{1, 1}}$ then the inequality
\begin{align}
0
\leq | \hat X_{k+1}^{j} |^{2} S_{k+1}^{j}
\leq 2 | \hat X_{k+2}^{j} - \hat X_{k+1}^{j} |^{2} S_{k+1}^{j} + 2 | \hat X_{k+2}^{j} |^{2} S_{k+1}^{j}
\end{align}
follows. Since $\varepsilon_{k}^j >0$ this implies that $| \hat X_{k+1}^{j} |^{2} S_{k+1}^{j}$ is integrable for all $j=1, \dots, d$.
The term $| \hat X_{k+1}^{j} |^{2} \Delta S_{k+1}^{j}$ is also integrable by the fact that $\hat X_{k+1}^{j} \Delta S_{k+1}^{j}$ and $\hat X_{k+1}^{j}$ are both in $\ensuremath{\mathbb{L}_{T}^{2, 1}}$. Indeed we have
\begin{align}
{\mathbb E}[ | \hat X_{k+1}^{j} |^{2} \Delta S_{k+1}^{j} ]
&\leq {\mathbb E}[ | \hat X_{k+1}^{j} |^{2} \mathbf{1}_{ \{ | \Delta S_{k+1}^{j} | \leq 1 \} } ]
+ {\mathbb E}[ | \hat X_{k+1}^{j} \Delta S_{k+1}^{j} |^{2} \mathbf{1}_{ \{ | \Delta S_{k+1}^{j} | \geq 1 \} } ]
\nonumber \\
& \leq {\mathbb E}[ | \hat X_{k+1}^{j} |^{2} ]
+ {\mathbb E}[ | \hat X_{k+1}^{j} \Delta S_{k+1}^{j} |^{2} ]
\end{align}
and this proves and completes the induction step at time $k$.
\par
The base case at time $k=T$ where $\hat X_{T+1}^{*} S_{T} + \hat Y_{T} = H$ is clear by the same arguments and by the assumptions on $H$ and $\bar X_{T+1}$, $\bar Y_{T}$. Indeed, since $\hat X_{T+1}^{*} S_{T} + \hat Y_{T}$ and $\hat X_{T+1}$ are both square integrable, then from Lemma \ref{lem:the alpha, beta, gamma terms} and Lemma \ref{lem:alpha beta terms boundedness} it follows that $\hat X_{T}^{j} \Delta S_{T}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$ and $\hat X_{T}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$ for all $j$. Moreover, note that with the assumptions $\hat X_{T+1}^{j} S_{T}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$, $\hat X_{T+1}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$ one can show that $| \hat X_{T+1}^{j} |^{2} S_{T}^{j} \in \ensuremath{\mathbb{L}_{T}^{1, 1}}$. By the same arguments as above, this will imply the integrability of the liquidity costs. The fact that $| \hat X_{T}^{j} |^{2} S_{T-1}^{j} \in \ensuremath{\mathbb{L}_{T}^{1, 1}}$ can be shown by using exactly the same arguments as in the proof for the inductive step.
\par
Finally, by defining
\begin{equation}
\hat Y_{T-1}
= \condE{ H - \hat X_{T}^{*} \Delta S_{T} }- \hat X_{T}^{*} S_{T-1}
\end{equation}
then it is clear that $\hat Y_{T-1}$ is $\mathcal{F}_{T-1}$-measurable and $\hat X_{T}^{*} S_{T-1} + \hat Y_{T-1} = \condE{ H - \hat X_{T}^{*} \Delta S_{T} } $ belongs to $\ensuremath{\mathbb{L}_{T}^{2, 1}}$.
\par
The martingale property of $C(\hat \varphi)$ follows from the construction of $\hat Y$ since at each time $k$ we have
\begin{equation}
\condE{ C_{T}(\hat \varphi) - C_{k}(\hat \varphi) }
= 0
\end{equation}
and so by Proposition \ref{prop:minimizing variance}, since both properties are satisfied, then the trading strategy $\hat \varphi = ( \hat X, \hat Y )$ is local risk-minimizing under illiquidity and the proof is complete.
\end{proof}
\begin{remark}
\label{rem:1-dim strategy and the book values}
In the $1$-dimensional case, the LRM-strategy $\hat \varphi = ( \hat X, \hat Y )$ under illiquidity has the representation
\begin{align}
\hat X_{k+1}
&= \frac{ \condCov{ V_{k+1}(\hat \varphi), \Delta S_{k+1} }
+ \condE{ \varepsilon_{k+1} S_{k+1} \hat X_{k+2} } }
{ \condVar{ \Delta S_{k+1} } + \condE{ \varepsilon_{k+1} S_{k+1} } }
\\
V_{k}(\hat \varphi)
&= {\mathbb E} \left [ H - \sum_{m=k+1}^{T} \hat X_{m} \Delta S_{m} \middle|\ \mathcal{F}_k \right]
\end{align}
For $\varepsilon_{k+1}$ tending to zero we get the classical local risk minimization strategy without accounting for illiquidity. Let us denote this by $\bar \varphi = ( \bar X, \bar Y )$. Also, one can easily note that in the case where $S$ is a martingale, then
$V_{k}(\hat \varphi) = \condE{ H } = V_{k}(\bar \varphi)$. That means the two book values are equal.
\par
One can easily check that when $\varepsilon_{k+1}$ goes to infinity, i.e. infinite liquidity costs, then
\begin{equation}
\hat X_{k+1}
\rightarrow
{\mathbb E} \left[ \frac{ S_{k+1} \cdots S_{T} \hat X_{T+1} }{ \mathbb{E}[ S_{k+1} | \mathcal{F}_{k} ] \cdots \mathbb{E}[ S_{T} | \mathcal{F}_{T-1} ] } \middle|\ \mathcal{F}_k \right].
\end{equation}
Consider \textit{cash settlement}, i.e. $\hat X_{T+1} = 0$ and $\hat Y_{T} = H$, where the value of the option has to be paid out in cash as it is usually the market standard. Then we clearly have $\hat X_{k+1} \rightarrow 0$ for all $k = 0, 1, \dots, T$ when $\varepsilon_{k+1} \rightarrow \infty$. From a financial point of view this makes sense since for the investor the best choice is to invest nothing to avoid infinite liquidity cost. A similar observation can be made in the $d$-dimensional case.
\end{remark}
\subsection{A sufficient condition for the $F$-property in terms of the covariance matrix $F^{0}$}
\label{sec:the reduction of the F-property to the covariance matrix}
Recall that the $F$-property from Definition \ref{defi:the F-property} was used in order to show the integrability properties of Proposition~\ref{prop:minimizing variance} for the local risk-minimizing strategy under illiquidity calculated backwards in time in the proof of Theorem~\ref{theo:existence}. In this section we show how this condition is related to the covariance matrix $F^{0}$. Before we continue let us recall the definition of a principal submatrix \cite[see][]{horn.johnson:2012}.
\par
\textbf{Definition of a principal submatrix: } In general let $P \in \mathbb{R}^{m, n}$ be a real matrix with $m$ rows and $n$ columns, and let $\alpha \subset \{1, \dots, m \}$, $\beta \subset \{1, \dots, n \}$ be index sets. Denote by $P[ \alpha, \beta ]$ the (sub)matrix of entries that lie in the rows of $P$ indexed by $\alpha$ and the columns indexed by $\beta$.
For $\alpha = \beta$ denote by $P[ \alpha ] = P[ \alpha, \alpha ]$ the (sub)matrix of entries that lie in the rows and columns of $P$ indexed by $\alpha$.
Then $P[ \alpha ]$ is called a \textit{principal submatrix} of $P$.\footnote{ A matrix $P \in \mathbb{R}^{n, n}$ has $\binom{n}{l}$ distinct principal submatrices of size $l \times l$. }
\par
The following Lemma \ref{lem:the reduced $F$-property} yields a sufficient criterion in terms of the covariance matrix $F^{0}$.
\begin{lem}
\label{lem:the reduced $F$-property}
$S$ has the $F$-property if there exists some $\delta \in ( 0, 1 )$ such that
\begin{equation}
\label{eq:0 lem:the reduced $F$-property}
\det(P_{k}^{0}) - (1 - \delta) \det(P_{k}^{A^{0}})
\geq 0
\end{equation}
for all principal submatrices $P_{k}^{0}$ of $F_{k}^{0}$ and principal submatrices $P_{k}^{A^{0}}$ of $F_{k}^{A^{0}}$ where $F_{k}^{A^{0}} := \diag(A_{k; 1}^{0}, \dots, A_{k; d}^{0})$ of size $l \times l $ where $l \in \{ 2, \dots, d \}$ and for all $k = 0, 1, \dots, T$.
\end{lem}
\begin{proof}
Let $d \in \mathbb{N}_{\geq 2}$, fix $k \in \{ 0, 1, \dots, T \}$ and omitting the time $k$ denote $F = F_{k}$.
\par
Furthermore we denote by $F^{A_{m}^{0}; A_{l}} := F^{A_{m}^{0}; A_{l}}(A_{m}^{0}, A_{m+1}^{0}, \dots, A_{l-1}^{0}, A_{l+1}, A_{l+2}, \dots, A_{d})$ for $m, l \in \{ 1, \dots, d \}$, $m < l$, the $(d-m) \times (d-m)$ symmetric matrix where for $i=j$, $j \in \{ 1, \dots, l - m \}$ we set $F_{i, j}^{A_{m}^{0}; A_{l}} = A_{m + j - 1}^{0}$ and for $j \in \{ l-m, \dots, d - m - 1 \}$ we set $F_{i, j}^{A_{m}^{0}; A_{l}} = A_{m+ j +1}$ for the diagonal elements of the matrix. Otherwise for $i \neq j$ we set $F_{i, j}^{A_{m}^{0}; A_{l}} = D_{m + i - 1, m + j - 1}$ for $i, j \in \{ 1, \dots, l - m \}$ and $F_{i, j}^{A_{m}^{0}; A_{l}} = D_{m + i + 1, m + j + 1}$ for $i, j \in \{ l-m, \dots, d - m - 1 \}$. For $m = l$ we set $F^{A_{l}^{0}; A_{l}} := F^{A_{l}^{0}; A_{l}}(A_{l+1}, A_{l+2}, \dots, A_{d})$ which is equal to $F$ without the first $l$ rows and columns. Also note that for $l = d$ we have $F^{A_{m}^{0}; A_{d}} := F^{A_{m}^{0}; A_{d}}(A_{m}^{0}, A_{m+1}^{0}, \dots, A_{d - 1
}^{0})$ which is equal to $F^{0}$ without the first $m - 1$ rows and columns and without the last row and the last column.
\par
Since $A_{j} = A_{j}^{0} + A_{j}^{\varepsilon}$ and using the fact that the matrices $F$ and $F^{0}$ are symmetric then one can calculate that
\begin{align}
\label{eq:1 lem:the reduced $F$-property}
&\det(F) - (1 - \delta) \det(F^{A}) \nonumber
\\
&= \det(F^{0}) - (1 - \delta) \det(F^{A^{0}}) \nonumber
\\
&+ A_{1}^{\varepsilon} [ \det(F^{A_{1}^{0}; A_{1}}(A_{2}, A_{3}, A_{4}, \dots, A_{d}))
- (1 - \delta) \det(\diag(A_{2}, A_{3}, A_{4}, \dots, A_{d})) ] \nonumber
\\
&+ A_{2}^{\varepsilon} [ \det(F^{A_{1}^{0}; A_{2}}(A_{1}^{0}, A_{3}, A_{4}, \dots, A_{d}))
- (1 - \delta) \det(\diag(A_{1}^{0}, A_{3}, A_{4}, \dots, A_{d})) ] \nonumber
\\
&+ A_{3}^{\varepsilon} [ \det(F^{A_{1}^{0}; A_{3}}(A_{1}^{0}, A_{2}^{0}, A_{4}, \dots, A_{d}))
- (1 - \delta) \det(\diag(A_{1}^{0}, A_{2}^{0}, A_{4}, \dots, A_{d})) ] \nonumber
\\
&+ \dots + \nonumber
\\
&+ A_{d}^{\varepsilon} [ \det(F^{A_{1}^{0}; A_{d}}(A_{1}^{0}, A_{2}^{0}, A_{3}^{0}, \dots, A_{d-1}^{0}))
- (1 - \delta) \det(\diag(A_{1}^{0}, A_{2}^{0}, A_{3}^{0}, \dots, A_{d-1}^{0})) ] \,,
\end{align}
where $F^{0}$ is the $\binom{d}{d} = 1$ principal submatrix $P^{0}[ \{ 1, 2, \dots, d \} ]$ of size $d \times d$ and $F^{A_{1}^{0}; A_{d}}(A_{1}^{0}, A_{2}^{0}, A_{3}^{0},$ \dots$, A_{d-1}^{0}) = P^{0}[ \{ 1, 2, \dots, d - 1 \} ]$ one of the $\binom{d}{d-1} = d$ principal submatrices of $F^{0}$ of size $(d-1) \times (d-1)$. The remaining $d-1$ principal submatrices of size $(d-1) \times (d-1)$ can be calculated recursively as in equation (\ref{eq:1 lem:the reduced $F$-property}) for the $d-1$ terms in the R.H.S of the equation. For example we have
\begin{align}
A_{1}^{\varepsilon} [ & \det(F^{A_{1}^{0}; A_{1}}(A_{2}, A_{3}, A_{4}, \dots, A_{d}))
- (1 - \delta) \det(\diag(A_{2}, A_{3}, A_{4}, \dots, A_{d})) ]
\nonumber \\
=
A_{1}^{\varepsilon} \Big\{
&A_{2}^{\varepsilon} [ \det(F^{A_{2}^{0}; A_{2}}(A_{3}, A_{4}, A_{5}, \dots, A_{d}))
- (1 - \delta) \det(\diag(A_{3}, A_{4}, A_{5}, \dots, A_{d})) ]
\nonumber \\
+ &A_{3}^{\varepsilon} [ \det(F^{A_{2}^{0}; A_{3}}(A_{2}^{0}, A_{4}, A_{5}, \dots, A_{d}))
- (1 - \delta) \det(\diag(A_{2}^{0}, A_{4}, A_{5}, \dots, A_{d})) ]
\nonumber \\
+ &A_{4}^{\varepsilon} [ \det(F^{A_{2}^{0}; A_{4}}(A_{2}^{0}, A_{3}^{0}, A_{5}, \dots, A_{d}))
- (1 - \delta) \det(\diag(A_{2}^{0}, A_{3}^{0}, A_{5}, \dots, A_{d})) ]
\nonumber \\
+ &\dots +
\nonumber \\
+ &A_{d}^{\varepsilon} [ \det(F^{A_{2}^{0}; A_{d}}(A_{2}^{0}, A_{3}^{0}, A_{4}^{0}, \dots, A_{d-1}^{0}))
- (1 - \delta) \det(\diag(A_{2}^{0}, A_{3}^{0}, A_{4}^{0}, \dots, A_{d-1}^{0})) ]
\nonumber \\
+ &\det( P^{0}[ \{ 2, 3, \dots, d \} ] )
- (1 - \delta) \det(P^{A^{0}}[ \{ 2, 3, \dots, d \} ])
\Big\} \,.
\end{align}
Note that $F^{A_{2}^{0}; A_{d}}(A_{2}^{0}, A_{3}^{0}, A_{4}^{0}, \dots, A_{d-1}^{0}) = P^{0}[ \{ 2, 3, \dots, d-1 \} ]$ is one of the $\binom{d}{d-2}$ principal submatrices of $F^{0}$ of size $(d-2) \times (d-2)$. The remaining $\binom{d}{d-2} - 1$ principal submatrices of size $(d-2) \times (d-2)$ can be calculated recursively in the same way as above.
\par
Continuing the calculation recursively (for each of the terms) we get,
\begin{align}
&\det(F) - (1 - \delta) \det(F^{A})
\nonumber \\
= &\det(P^{0}[ \{ 1, 2, \dots, d \} ]) - (1 - \delta) \det(P^{A^{0}}[ \{ 1, 2, \dots, d \} ])
\nonumber \\
+ A_{1}^{\varepsilon} \Big\{ A_{2}^{\varepsilon} \Big\{ \dots A_{d-3}^{\varepsilon} \Big\{
&A_{d-2}^{\varepsilon} [ \det(F^{A_{d-2}^{0}; A_{d-2}}(A_{d-1}, A_{d}))
- (1 - \delta) \det(\diag(A_{d-1}, A_{d} )) ]
\nonumber \\
+ &A_{d-1}^{\varepsilon} [ \det(F^{A_{d-2}^{0}; A_{d-1}}(A_{d-2}^{0}, A_{d}))
- (1 - \delta) \det(\diag(A_{d-2}^{0}, A_{d})) ]
\nonumber \\
+ &A_{d }^{\varepsilon} [ \det(F^{A_{d-2}^{0}; A_{d}}(A_{d-2}^{0}, A_{d-1}^{0}))
- (1 - \delta) \det(\diag(A_{d-2}^{0}, A_{d-1}^{0})) ]
\nonumber \\
+ &\det( P^{0}[ \{ d-2, d-1, d \} ] )
- (1 - \delta) \det(P^{A^{0}}[ \{ d-2, d-1, d \} ])
\Big\} \dots \Big\}
\nonumber \\
+ & \dots
\end{align}
That means, we have rewritten the term $\det(F) - (1 - \delta) \det(F^{A})$ into terms of $\binom{d}{l}$ (distinct) principal submatrices $P^{0}$ of $F^{0}$ of size $l \times l $ where $l \in \{ 3, \dots, d \}$.
Moreover, we are dealing with the determinants of the $2 \times 2$ matrices as follows: for example and since $A_{d} \geq A_{d}^{0}$ we have
\begin{align}
&\det(F^{A_{d-2}^{0}; A_{d-1}}(A_{d-2}^{0}, A_{d}))
- (1 - \delta) \det(\diag(A_{d-2}^{0}, A_{d}))
\nonumber \\
&= \delta A_{d-2}^{0} A_{d} - | D_{d-2, d} |^{2}
\nonumber \\
&\geq \delta A_{d-2}^{0} A_{d}^{0} - | D_{d-2, d} |^{2}
\nonumber \\
&= \det( P^{0}[ \{ d-2, d \} ] ) - (1 - \delta) \det( P^{A^{0}}[ \{ d-2, d \} ] ) \,.
\end{align}
The same holds analogously for the other $2 \times 2$ principal submatrices by the fact that $A_{j} \geq A_{j}^{0}$ for $j=1, \dots, d$. So, since $A_{j}^{\varepsilon} \geq 0$ for $j=1, \dots, d$ and since by assumption the inequality (\ref{eq:0 lem:the reduced $F$-property}) holds, then we can estimate
\begin{align}
&\det(F) - (1 - \delta) \det(F^{A})
\nonumber \\
\geq &\det(P^{0}[ \{ 1, 2, \dots, d \} ]) - (1 - \delta) \det(P^{A^{0}}[ \{ 1, 2, \dots, d \} ])
\nonumber \\
+ A_{1}^{\varepsilon} \Big\{ A_{2}^{\varepsilon} \Big\{ \dots A_{d-3}^{\varepsilon} \Big\{
&A_{d-2}^{\varepsilon} [ \det(P^{0}[ \{A_{d-1}^{0}, A_{d}^{0} \} ])
- (1 - \delta) \det(P^{A^{0}}[ \{A_{d-1}^{0}, A_{d}^{0} \} ]) ] \allowdisplaybreaks
\nonumber \\
+ &A_{d-1}^{\varepsilon} [ \det(P^{0}[ \{A_{d-2}^{0}, A_{d}^{0} \} ])
- (1 - \delta) \det(P^{A^{0}}[ \{A_{d-2}^{0}, A_{d}^{0} \} ]) ]
\nonumber \\
+ &A_{d }^{\varepsilon} [ \det(P^{0}[ \{A_{d-2}^{0}, A_{d-1}^{0} \} ])
- (1 - \delta) \det(P^{A^{0}}[ \{A_{d-2}^{0}, A_{d-1}^{0} \} ]) ]
\nonumber \\
+ &\det( P^{0}[ \{ d-2, d-1, d \} ] )
- (1 - \delta) \det(P^{A^{0}}[ \{ d-2, d-1, d \} ])
\Big\} \dots \Big\}
\nonumber \\
+ & \dots
\nonumber \\
\geq & \, 0 .
\end{align}
That means the quantity $\det(F) - (1 - \delta) \det(F^{A})$ can be estimated from below by the determinants of principal submatrices by terms as in (\ref{eq:0 lem:the reduced $F$-property}) of $F^{0}$ and so by assumption the claim follows.
\end{proof}
Proposition \ref{prop:the reduced $F$-property} gives us an example when the $F$-property is fulfilled.
\begin{prop}
\label{prop:the reduced $F$-property}
Assume that the covariance matrix $F_{k}^{0}$ is positive definite at all times $k = 0, 1, \dots, T$ and $S^{j}$ has independent returns for each $j = 1, \dots, d$. Then the $F$-property holds.
\end{prop}
\begin{proof}
Let $d \in \mathbb{N}_{\geq 2}$.
\par
Fix $k \in \{ 0, 1, \dots, T \}$. First we introduce the notation $\bar A_{k; j}^{0} := {\mathbb{V}ar}( \rho_{k+1}^{j} )$, $\bar D_{k; i, j} := {\mathbb{C}ov}(\rho_{k+1}^{i}, \rho_{k+1}^{j})$ for $i \neq j$ where $\bar F_{k; i, j}^{0} = \bar A_{k; j}^{0}$ for $i=j$, $\bar F_{k; i, j}^{0} = \bar D_{k; i, j}$ otherwise. Our aim is to make use of Lemma \ref{lem:the reduced $F$-property}. For simplicity we omit the time $k$ and denote $F = F_{k}$.
\par
First note that since the covariance matrix $F^{0}$ is positive definite then
\begin{equation}
\det(F^{0}) > 0 \text{ and } \det(F^{A^{0}}) > 0 \,.
\end{equation}
Now using $\Delta S_{k+1}^{j} = S_{k}^{j} \rho_{k+1}^{j}$, the fact that $\rho_{k+1}^{j}$ is independent of $\mathcal{F}_{k}$ for all $j=1, \dots, d$, the properties of the determinant and the symmetry of the covariance matrix $F^{0}$ we get
\begin{align}
\det(F^{0}) &= |S^{1}_{k}|^{2} \cdots |S^{d}_{k}|^{2} \det(\bar F^{0})
> 0
\nonumber \\
\det(F^{A^{0}}) &= |S^{1}_{k}|^{2} \cdots |S^{d}_{k}|^{2} \det(\bar F^{\bar A^{0}})
> 0
\end{align}
with the obvious notation $\bar F_{k}^{\bar A^{0}} := diag(\bar A_{k; 1}^{0}, \dots, \bar A_{k; d}^{0})$.
Since $S_{k}^{j} > 0$, this implies
\begin{align}
\label{eq:1 the reduced $F$-property}
\det(F^{0}) - (1 - \delta) \det(F^{A^{0}}) \geq 0
\iff
\det(\bar F^{0}) - (1 - \delta) \det(\bar F^{A^{0}}) \geq 0
\end{align}
for $\delta \in (0, 1)$.
Furthermore, since $\bar F^{0}$ and $\bar F^{\bar A^{0}}$ are deterministic matrices with $\det(\bar F^{0}) > 0$ and $\det(\bar F^{\bar A^{0}}) > 0$, then
\begin{align}
\det(\bar F^{0}) - (1 - \delta) \det(\bar F^{\bar A^{0}})
\geq 0
\end{align}
for some $\delta \in (0, 1)$. For the $1$ principal submatrix of $F^{0}$ of size $d \times d$ which is again the matrix $F^{0}$ we want to show that
\begin{align}
\det(F^{0}) + (1 - \delta) \det (F^{A^{0}})
\geq 0 \,
\end{align}
which for independent returns and positive marginal price process is equivalent to $\det(\bar F^{0}) + (1 - \delta) \det (\bar F^{A^{0}}) \geq 0$ as shown in the equivalence relation (\ref{eq:1 the reduced $F$-property}). So it remains to show that for the all (distinct) $\binom{d}{l}$ principal submatrices $P^{0}$ of $F^{0}$ of size $l \times l $ where $l \in \{ 2, \dots, d - 1 \}$ we have that $\det(P^{0}) + (1 - \delta) \det (P^{A^{0}}) \geq 0$ for some $\delta \in (0, 1)$. Now using again the fact that $F_{k}^{0}$ is positive definite then we know that each principal submatrix $P^{0}$ is positive definite \citep[Observation 7.1.2]{horn.johnson:2012}. That means
\begin{equation}
\det(P^{0}) > 0 \text{ and } \det(P^{A^{0}}) > 0 \,.
\end{equation}
Since all principal submatrices $P^{0}$ of $F^{0}$ are covariance matrices, then by the same argumentation (and obvious notation) as above we get $\det(\bar P^{0}) - (1 - \delta) \det(\bar P^{\bar A^{0}}) \geq 0$ for some $\delta \in (0, 1)$ which for independent returns and $S_{k}^{j} > 0$ is equivalent to
\begin{align}
\det( P^{0}) + (1 - \delta) \det ( P^{A^{0}}) \geq 0 \,.
\end{align}
Finally, from Lemma \ref{lem:the reduced $F$-property} the claim follows.
\end{proof}
\begin{prop}
\label{prop:the reduced $F$-property for independent increments}
Assume that the covariance matrix $F_{k}^{0}$ at all times $k = 0, 1, \dots, T$ is positive definite and $S^{j}$ has independent increments for each $j = 1, \dots, d$. Then the $F$-property holds.
\end{prop}
\begin{proof}
Follows by analogous arguments as in Proposition \ref{prop:the reduced $F$-property}.
\end{proof}
\begin{remark}
Note that rewriting Lemma \ref{lem:the reduced $F$-property} when $\varepsilon = 0$ then the condition simply reduces to the covariance matrix being such
\begin{equation}
\det(F^{0}) - (1 - \delta) \det(F^{A^{0}})
\geq 0
\end{equation}
for some $\delta \in (0, 1)$ and principal submatrices do not need to be considered.
\end{remark}
\begin{remark}
\label{rem:the 2 dim F property}
In the $2$-dimensional case in order to ensure that $F_{k}^{0}$ is positive definite\footnote{Recall that a matrix $F$ is positive definite if and only if its leading principal minors are all positive.}, i.e. $A_{k; 1}^{0} A_{k; 2}^{0} - D_{k; 1, 2}^{2} > 0$, $A_{k; 1}^{0} > 0$, $A_{k; 2}^{0} > 0$, in the case of independent returns (or increments) we just need strict Cauchy-Schwarz inequality, which means that $S^{1}$ and $S^{2}$ must be linearly independent. Then Proposition \ref{prop:the reduced $F$-property} can be applied.
\end{remark}
\subsection{Nonnegative supply curve}
\label{sec:nonnegative supply curve}
In this section we consider the $1$-dimensional case for simplicity. An extension to the multidimensional case is straightforward. As we already mentioned the (linear) supply curve $S_{k}(x) = (1 + x \varepsilon_{k}) S_{k}$ can also take negative values when a negative transaction $x$ is such that $x\leq -1/\varepsilon$. So, a natural question to ask is how one could define a function $h: \mathbb{R} \to \mathbb{R}$ so that the supply curve process
\begin{equation}
S_{k}(x) = h(x)S_{k}
\end{equation}
is nonnegative. This can be done for example by the function
\begin{equation}
h(x)
= ( 1 + x \varepsilon_{k} ) \mathbf{1}_{ \{ x \geq -z_{k} \} }
+ ( 1 - z_{k} \varepsilon_{k} ) \mathbf{1}_{ \{ x < -z_{k} \} }
\end{equation}
defined for some deterministic positive process $z = (z_{k})_{k=0, 1, \dots, T}$ where $0 < z_{k} \leq 1/\varepsilon_{k}$ for all $k=0, 1, \dots, T$. Then $z_{k} S_k$ represents a lower bound for the price received when selling a large quantity of shares.
\par
The corresponding cost process under illiquidity $ \hat C^{b}(\varphi) = (\hat C_{k}^{b}(\varphi))_{k=0,1, \dots, T}$ of a strategy $\varphi = (X, Y)$ is then
\begin{align}
\hat C_{k}^{b}(\varphi)
:= V_{k} (\varphi) - \sum_{m=1}^{k} X_{m} \Delta S_{m}
& + \sum_{m=1}^{k} \varepsilon_{m} S_{m} | \Delta X_{m+1} |^{2} \mathbf{1}_{ \{ \Delta X_{m+1} \geq -z_{m} \} }
\nonumber \\ &
- \sum_{m=1}^{k} z_{m} \varepsilon_{m} S_{m} \Delta X_{m+1} \mathbf{1}_{ \{ \Delta X_{m+1} < -z_{m} \} }.
\end{align}
Moreover, as in Section \ref{sec:existence and recursive construction of an optimal strategy under illiquidity} and by Proposition \ref{prop:minimizing variance} at time $k$ we want to minimize the expression (w.l.o.g. $\alpha = 1$)
\begin{align}
& \condVar{ V_{k+1}(\varphi) - X'_{k+1} \Delta S_{k+1} }
\nonumber \\& \quad \quad \quad \quad
+ \condE{ \varepsilon_{k+1} S_{k+1} | X_{k+2} - X'_{k+1} |^{2} \mathbf{1}_{ \{ X_{k+2} - X'_{k+1} \geq -z_{k+1} \} } }
\nonumber \\& \quad \quad \quad \quad
- \condE{ z_{k+1} \varepsilon_{k+1} S_{k+1} (X_{k+2} - X'_{k+1}) \mathbf{1}_{ \{ X_{k+2} - X'_{k+1} < -z_{k+1} \} } }
\end{align}
over all appropriate $X'_{k+1}$. Rewriting the above expression, one needs to minimize the function $\hat f_{k}^{b} : \mathbb{R} \times \Omega \rightarrow \mathbb{R}^{+}$ defined by
\begin{align}
\hat f_{k}^{b}(c, \omega)
&= | c |^{2} \hat A_{k}^{b}(\omega)
- 2 c \hat b_{k}^{b}(\omega)
+ c \hat d_{k}^{b}(\omega)
\nonumber \\
&+ \condVar{ V_{k+1} }(\omega)
+ \condE{ \varepsilon_{k+1} S_{k+1} | X_{k+2} |^{2} \mathbf{1}_{ \{ X_{k+2} - c \geq -z_{k+1} \} } }(\omega)
\nonumber \\
& \quad \quad \quad \quad \quad \quad \quad
- \condE{ z_{k+1} \varepsilon_{k+1} S_{k+1} X_{k+2} \mathbf{1}_{ \{ X_{k+2} - c < -z_{k+1} \} } } (\omega)
\end{align}
where the following notation is used,
\begin{align}
\hat A_{k}^{b}
&= \condVar{ \Delta S_{k+1} }
+ \condE{ \varepsilon_{k+1} S_{k+1} \mathbf{1}_{ \{ X_{k+2} - c \geq -z_{k+1} \} } }
\nonumber \\
\hat b_{k}^{b}
&= \condCov{ V_{k+1}, \Delta S_{k+1} }
+ \condE{ \varepsilon_{k+1} S_{k+1} X_{k+2} \mathbf{1}_{ \{ X_{k+2} - c \geq -z_{k+1} \} } }
\nonumber \\
\hat d_{k}^{b}
&= \condE{ z_{k+1} \varepsilon_{k+1} S_{k+1} \mathbf{1}_{ \{ X_{k+2} - c < -z_{k+1} \} } } \,.
\end{align}
Furthermore, under similar arguments and assumptions as in Sections \ref{sec:existence and recursive construction of an optimal strategy under illiquidity} and \ref{sec:assumptions}, one can use the dominated convergence theorem to show that the equation $\frac{d}{dc} \hat f_{k}^{b}(c) = 0$ gives that the optimal strategy $\hat{\varphi}=(\hat{X}, \hat{Y})$ fulfills the implicit relation
\begin{align}
\hat X_{k+1}
= \frac{ \condCov{ V_{k+1}, \Delta S_{k+1} }
+ \condE{ \varepsilon_{k+1} S_{k+1} \hat X_{k+2} \mathbf{1}_{ \{ \hat X_{k+2} - \hat X_{k+1} \geq -z_{k+1} \} } }
- \frac{1}{2} Q
}
{ \condVar{ \Delta S_{k+1} }
+ \condE{ \varepsilon_{k+1} S_{k+1} \mathbf{1}_{ \{ \hat X_{k+2} - \hat X_{k+1} \geq - z_{k+1} \} } }
}
\end{align}
with
\begin{equation}
Q = \condE{ z_{k+1} \varepsilon_{k+1} S_{k+1} \mathbf{1}_{ \{ \hat X_{k+2} - \hat X_{k+1} < -z_{k+1} \} } }.
\end{equation}
\section{Application to Electricity Markets}
\label{sec:applications}
In this section we apply the previous results to hedge an Asian-style electricity option with electricity futures that are exposed to liquidity costs. These futures might have different maturities, i.e.~certain hedge instruments might terminate before maturity of the option (final time horizon $T$) and hedging in these instruments is only possible on certain subintervals of $[0,T]$. A priori this situation is not covered by our setting in the previous sections where it is assumed that hedging is possible until $T$ in all hedge instruments. In Subsection~\ref{sec:the general case}, we thus shortly sketch how hedge instruments with different maturities can be embedded in our setting from the previous sections, before we focus our example on electricity markets in Subsection~\ref{sec:the energy market case}.
\subsection{Hedge instruments with different maturities}
\label{sec:the general case}
\par
On our stochastic basis $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ with final time horizon $T$, consider now nonnegative price processes $S^j = (S^j_{k})_{k=0, 1, \dots, T_j}$ of $d$ available hedge instruments with maturity $T_j\le T$, $j=1, \dots, d$. That is, hedging in asset $j$ is only possible until time $T_j\le T$, $j=1, \dots, d$, where without loss of generality we assume $0 < T_{1} \le T_{2} \le \dots \le T_{d}\le T$. To fit this situation into our general setting, we introduce an associated $d$-dimensional price process $\tilde S = (\tilde S_{k})_{k=0, 1, \dots, T}$ by artificially keeping each asset $S^j$ constant on the remaining interval $[T_j,T]$:
\begin{equation}
\label{eq:price process general applications}
\tilde S_{k}^{j}
= S_{k}^{j} \mathbf{1}_{[0, T_{j} )}(k) + S_{T_{j}}^{j} \mathbf{1}_{[T_{j}, T ]}(k)
\end{equation}
for $j=1, \dots, d$ and $k \in \{ 0, 1, \dots, T \}$. Moreover, we consider a positive, deterministic $\mathbb{R}^{d}$-valued liquidity process $\varepsilon=(\varepsilon_{k})_{k=0, 1, \dots, T}$, which is extended by some $\varepsilon_{m}^{j} > 0$ on the intervals $m \in[T_{j},T] $ for all $j \in \{ 1, \dots, d \}$, i.e.~we assume positive liquidity costs during the extended price dynamics.
It is then clear already intuitively that an investor would not trade in asset $j$ during the interval $[T_{j},T]$ since during this time frame the asset generates zero gains while incurring positive liquidity costs. Indeed, employing the fact for $k \geq T_{l}$ we have $\Delta \tilde S_{k+1}^{l} = 0$, it is straightforward to see from Proposition \ref{prop:minimizing variance}, Property \ref{prop:2}, that in this situation a LRM-strategy must be of the form $\tilde X_{m}^{l} = 0 \text{ for } m = T_{l}+1, \dots, T \,\,, l \in \{ 1, \dots, d \}$. i.e.~the hedger liquidates his position in the $j$-th asset at time $T_{j}+1$. Thus, in our extended market a LRM-strategy $\tilde X$ automatically respects the original hedge constraints beyond maturities $T_j$, $j \in \{ 1, \dots, d \}$ and is thus also a LRM-strategy in our setting with hedge instruments with different maturities. In the following we say the asset $\tilde S^j$ is \emph{active} at time $k$ if $k\le T_j$ and \emph{in
active} at time $k$ if $k > T_j$.
The existence and computation of a LRM-strategy under a linear supply curve $\tilde S_{k}^{j}(x^{j}) = \tilde S_{k}^{j} + x^{j}\varepsilon_{k}^{j} \tilde S_{k}^{j}$ as developed in Section~\ref{sec:linear:supply} now takes the following form for hedge instruments with different maturities. Using the fact that a LRM-strategy $\tilde X$ fulfills $\tilde X_{m}^{l} = 0 \text{ for } m = T_{l}+1, \dots, T \,\,, l \in \{ 1, \dots, d \}$, the minimization at step $k \in \{ 0, 1, \dots, T-1 \}$ of the function $f_k$ in \eqref{eq:minimizing function} reduces to the minimization of the function $\tilde f_{k} : \mathbb{R}^{d-l_k} \times \Omega \rightarrow \mathbb{R}^{+}$ defined by
\begin{align}\label{ftilde}
\tilde f_{k}(c, \omega)
&= \sum_{j = l_k + 1}^{d} | c_{j} |^{2} A_{k; j}(\omega)
- 2 \sum_{j = l_k + 1}^{d} c_{j} b_{k; j}(\omega)
+ \sum_{j \neq i, j, i \geq l_k + 1}^{d} c_{j} c_{i} D_{k; j, i}(\omega)
\\
&+ \condVar{ V_{k+1} }(\omega)
+ \sum_{j = l_k + 1}^{d} \condE{
\varepsilon_{k+1}^{j} S_{k+1}^{j} | X_{k+2}^{j} |^{2} }(\omega) \nonumber
\end{align}
where the sums are only over the assets $\tilde S^j$, $j=l+1,...,d$, that are active during the $k$'th period, i.e.~$l_k:=\max\{r\in \{ 1, \dots, d \}:T_{r} < k\}$. Thus, the conditions required in Theorem~\ref{theo:existence} for existence of a LRM-strategy reduce to lower-dimensional conditions that in each period only concern the active hedge instruments. More precisely, using the notation from Section \ref{sec:linear:supply}, we define for each period $k \in \{ 0, 1, \dots, T-1 \}$ the symmetric matrix $\tilde F_{k} \in \mathbb{R}^{d-l_k \times d-l_k}$ (a principal submatrix of $F_{k}$) by $\tilde F_{k; i, j} = D_{k; i+l_k, j+l_k}$ for $i \neq j$, $\tilde F_{k; i, j} = A_{k; j+l_k}$ for $i = j$, $i, j \in \{ 1, \dots, d-l_k \}$ and $\tilde b_{k} := ( b_{k; l_k+1}, \dots, b_{k; d} )^{*} \in \mathbb{R}^{d-l_k}$. Then minimizing \eqref{ftilde} amounts to solving the linear system
\begin{equation}
\tilde F_{k} c = \tilde b_{k} \,.
\end{equation}
in $c\in \mathbb{R}^{d-l_k}$. Note that $\tilde F_{k} = \tilde F_{k}^{0} + \tilde F_{k}^{\varepsilon}$ where $\tilde F^{\varepsilon} = diag(A_{k;l+1}^{\varepsilon}, \dots, A_{k;d}^{\varepsilon})$ and $\tilde F_{k}^{0}$ is the matrix $\tilde F_{k}$ with $\varepsilon_{k+1}^{j} = 0$ for $j=l+1, \dots, d$, that is a reduced form of the covariance matrix of the price process $\tilde S$. Following the arguments in Section \ref{sec:linear:supply}, we then get the following version of Theorem~\ref{theo:existence} on the existence of a LRM-strategy in the context of hedge instruments with different maturities:
\begin{cor}
\label{cor:existence}
Consider a contingent claim $H = \bar X_{T+1}^{*} S_{T} + \bar Y_{T} \in \mathbb{L}_{T}^{2, 1}$ with $\bar X_{T+1} = 0$ and a price process of the form in equation (\ref{eq:price process general applications}). Assume that for each $k$-th period, the covariance matrix $\tilde F_{k}^{0}$ is positive definite. Furthermore assume that bounded mean-variance tradeoff, the $F$-property and the $F$-diagonal condition hold for the active assets in the $k$-th period at time $k \in \{ 0, 1, \dots, T-1 \}$. Then there exists a LRM-strategy $\hat \varphi = ( \hat X, \hat Y )$ under illiquidity with $\hat X_{T+1} = 0$, $\hat Y_{T} = H$. In particular for $k \in \{ 0, 1, \dots, T-1 \}$ we have $\hat X = (\bar 0, \tilde X)$ with $\bar 0 = (0, \dots, 0) \in \mathbb{R}^{l_{k}}$ and
\begin{equation}
\tilde X_{k+1}
= \tilde F_{k}^{-1} \tilde b_{k}
\quad \mathbb{P}-a.s.
\end{equation}
in $\mathbb{R}^{d-l_{k}}$ and for $k \in \{ 0, 1, \dots, T-1 \}$
\begin{equation}
\hat Y_{k}
= \condE{ \hat W_{k} } - \hat X_{k+1}^{*} \tilde S_{k}
\quad \ensuremath{\mathbb{P}\text{ -- a.s.}}
\end{equation}
where $\hat W_{k} = H - \sum_{m=k+1}^{T} \hat X_{m}^{*} \Delta \tilde S_{m}$.
\end{cor}
\subsection{LRM strategies in electricity markets}
\label{sec:the energy market case}
In the remaining parts of the section, we now consider the example of hedging an Asian-style electricity option with electricity futures under liquidity costs by a LRM-strategy. The price processes for electricity futures we are considering are based on a continuous-time multi-factor spot price model proposed in~\cite{benth.meyerbrandis.kallsen:2007}, which we recall in Subsection~\ref{model} before we explicitly compute and simulate LRM-strategies in an example in Subsection~\ref{simulation}.
\subsubsection{An electricity market model} \label{model}
In~\cite{benth.meyerbrandis.kallsen:2007}, the price $E(t)$ of spot electricity at time $t\in[0,T]$ is modeled by
\begin{equation}\label{spotmodel}
E(t)
=\sum_{i=1}^{n} \Lambda_i(t) Y_{i}(t)\,,
\end{equation}
where for $i=1, \dots, n$ the positive and deterministic function $\Lambda_i$ accounts for seasonality and $Y_{i}$ is the solution to an Ornstein-Uhlenbeck stochastic differential equation
\begin{equation}
dY_{i}(t)
= - \lambda_{i} Y_{i}(t) dt + \sigma_{i}(t) dL_{i}(t)
\quad , \quad
Y_{i}(0) = y_{i}\,,
\end{equation}
where $\lambda_{i} > 0$ are constants, and $\sigma_{i}(t)$ are deterministic, positive bounded functions.
Moreover, the $L_{i}$'s are independent, increasing pure jump L\'{e}vy processes with jump measures $N_{i}(dt, dz)$ which have deterministic predictable compensators of the form \linebreak $\nu_{i}(dt, dz) = dt \nu_{i}(dz)$. Note that by the increasing nature of the $L_{i}$'s the positivity of the $Y_{i}$'s and thus also of the spot price $E$ is ensured. We assume that the model~\eqref{spotmodel} is defined on a stochastic basis $( \Omega, \mathbb{F}, (\mathcal{F}_{t})_{0 \leq t \leq T}, \mathbb{P} )$ where the filtration $(\mathcal{F}_{t})_{0 \leq t \leq T}$ is generated by the $L_{i}$'s.
The available hedge instruments are electricity futures, which, by the flow character of electricity, delivers spot electricity over a delivery period $[ T_{1}^{F}, T_{2}^{F} ]$ for $T_{1}^{F}<T_{2}^{F}\le T$ rather than at a fixed point in time. That is, the pay-off of the (financially settled) futures at the end of the delivery period is
\begin{equation}
\frac{1}{T_{2}^{F} - T_{1}^{F}} \int_{T_{1}^{F}}^{T_{2}^{F}} E(u) du \,,
\end{equation}
and the life of the asset terminates at $T_{2}^{F}$. In order to compute the price dynamics of an electricity futures we assume for simplicity that $\mathbb{P}$ is already an equivalent martingale measure, such that the the price $F(t;T_{1}^{F}, T_{2}^{F})$ of the futures at time $t\le T_{2}^{F}$ as a traded asset is given by
\begin{equation} \label{futures}
F(t;T_{1}^{F}, T_{2}^{F})
= {\mathbb E} \left[ \frac{1}{T_{2}^{F} - T_{1}^{F}} \int_{T_{1}^{F}}^{T_{2}^{F}} E(u) du \middle| \mathcal{F}_{t} \right ] \,.
\end{equation}
Using the explicit solution
\begin{equation}
Y_{i}(u)
= Y_{i}(t) \ensuremath{e}^{-\lambda_{i}(u-t)}
+ \int_{t}^{u} \sigma_{i}(s) \ensuremath{e}^{-\lambda_{i}(u-s)} dL_{i}(s)
\end{equation}
for the Ornstein-Uhlenbeck components $Y_i$, $i=1,...n$, a straightforward computation of the conditional expectation in \eqref{futures} yields the following price of futures contracts in the continuous-time spot model :
\begin{prop}
\label{prop:future dynamics}
The price $F(t, T_{1}^{F}, T_{2}^{F})$ at time $t$ of an electricity futures with delivery period $[T_{1}^{F}, T_{2}^{F}]$ is given by
\begin{align}
F(t, T_{1}^{F}, T_{2}^{F})
&= \sum_{i=1}^{n} Y_{i}(t) \frac{1}{T_{2}^{F} - T_{1}^{F}}
\int_{T_{1}^{F}}^{T_{2}^{F}} \Lambda_{i}(u) \ensuremath{e}^{-\lambda_{i}(u-t)} du
\nonumber \\
&+ \frac{1}{T_{2}^{F} - T_{1}^{F}} \int_{T_{1}^{F}}^{T_{2}^{F}} \int_{t}^{u} \int_{\mathbb{R}^{+}}
\sigma_{i}(s) \Lambda_{i}(u) \ensuremath{e}^{-\lambda_{i}(u-s)} z \nu_{i}(dz) ds du
\end{align}
for $0 \leq t \leq T_{1}^{F}$, and
\begin{align}
F(t, T_{1}^{F}, T_{2}^{F})
= \frac{1}{T_{2}^{F} - T_{1}^{F}} \int_{T_{1}^{F}}^{t} E(u) du
&+ \sum_{i=1}^{n} Y_{i}(t) \frac{1}{T_{2}^{F} - T_{1}^{F}}
\int_{t}^{T_{2}^{F}} \Lambda_{i}(u) \ensuremath{e}^{-\lambda_{i}(u-t)} du
\nonumber \\
&+ \frac{1}{T_{2}^{F} - T_{1}^{F}} \int_{t}^{T_{2}^{F}} \int_{t}^{u} \int_{\mathbb{R}^{+}}
\sigma_{i}(s) \Lambda_{i}(u) \ensuremath{e}^{-\lambda_{i}(u-s)} z \nu_{i}(dz) ds du
\end{align}
for $T_{1}^{F} \leq t \leq T_{2}^{F}$.
\end{prop}
Based on this continuous-time spot and futures price model, we now construct a discrete-time electricity market model that fits into our framework by sampling the continuous-time processes at finitely many trading times $0=t_0, t_1,...,T$, i.e.~our hedge instruments $S^j$, $j=1,...d$, are given by futures price processes of the form
\begin{equation}
S^j_{k}:=F^j(t_k, T_{1}^{F^j}, T_{2}^{F^j}) \quad \text{for}\quad 0\le t_k\le T_{2}^{F^j}\le T \,.
\end{equation}
In the following we always assume that delivery period times are part of the discrete time grid, i.e.~$T_{1}^{F}, T_{2}^{F}\in \{t_0, t_1,...,T\}$. After the maturity $T_{2}^{F}$, the futures contract ceases to exist and trading is not possible anymore. During the delivery period $[T_{1}^{F},T_{2}^{F}]$, depending on the conventions of the exchange, trading is either not possible at all or very illiquid. We capture this feature by specifying high liquidity costs during $[T_{1}^{F},T_{2}^{F}]$, with the impossibility of trading as the limit case when liquidity costs tend to infinity. Before the delivery period, one typically observes on electricity markets that futures become the more liquid the shorter the remaining time to delivery period is. We capture this behavior by the following liquidity structure $\varepsilon^{j}$ for the futures $F^j$, $j=1,...d$:
\begin{align} \label{TimeLiquidity}
\varepsilon^{j}_{t}
&= a_{j} (1 - \exp(-(T_{1}^{F^{j}} - t))) + \delta_j
\quad , \quad
a_{j}
= M_{j} \frac{1}{1 - \exp(-T_{1}^{F^{j}})}
\quad \text{ for } 0 \leq t \leq T_{1}^{F^{j}}\,,
\nonumber \\
\varepsilon^{j}_{t}
&= N_{j}
\quad \text{ for } T_{1}^{F^{j}} < t \leq T_{2}^{F^{j}}\,.
\end{align}
The liquidity structure $\varepsilon^{j}$ for a future $F^j$ thus starts from a constant $M_j>0$ at time $0$ and decreases exponentially in time until the start of the delivery period to a level $\delta_j>0$. During the delivery period it then jumps to a constant (high) level $N_j>0$.
Further, in our simulation study we compare the time varying liquidity structure in \eqref{TimeLiquidity} with a constant liquidity structure given by
\begin{align} \label{ConstantLiquidity}
\varepsilon^{j}_{t}
= M_{j}
\quad \text{ for } 0 \leq t \leq T_{1}^{F^{j}}\,,
\quad \quad
\varepsilon^{j}_{t}
= N_{j}
\quad \text{ for } T_{1}^{F^{j}} < t \leq T_{2}^{F^{j}}\,.
\end{align}
for $M_j>0$ and $N_j>0$.
\subsubsection{LRM-strategies of electricity call options} \label{simulation}
In the electricity market model specified in Subsection~\ref{model}, we now intend to compute a LRM-strategy of a financially settled Asian call option written on an electricity future with delivery period $[T_{1}^{c},T_{2}^{c}]$ for $0<T_{1}^{c}< T_{2}^{c}\le T$, i.e.~the claim is given by $H = \bar Y_{T}$ with
\begin{equation}\label{calloption}
\bar Y_{T} = \left( \frac{1}{T_{2}^{c} - T_{1}^{c}} \int_{T_{1}^{c}}^{T_{2}^{c}} E(u) du - K \right)^{+}
\end{equation}
for some strike price $K$. In the following we will always assume that the option maturity is equal to the terminal time horizon: $T_{2}^{c} = T$.
\par
We will analyze and compare various specifications where the investor can hedge in two different futures $F^{1}$, $F^{2}$ with corresponding delivery periods $[ T_{1}^{F^{1}}, T_{2}^{F^{1}} ]$ and $[ T_{1}^{F^{2}}, T_{2}^{F^{2}} ]$, respectively, where we assume $T_{2}^{F^{1}} \leq T_{2}^{F^{2}} \leq T$ and $T_{1}^{F^{1}} \neq T_{1}^{F^{2}}$.\footnote{Basically one needs that either $T_{2}^{F^{1}} \neq T_{2}^{F^{2}}$ or $T_{1}^{F^{1}} \neq T_{1}^{F^{2}}$ so that the conditional Cauchy-Schwarz inequality is strict. See Remark \ref{rem:the 2 dim F property}.} In this situation, Corollary~\ref{cor:existence} ensures the existence of a LRM-strategy under liquidity costs. Indeed, from Proposition~\ref{prop:independent increments} it is clear that both the bounded mean-variance tradeoff and the $F$-diagonal condition hold for the active assets in each period by the fact that the futures have independent increments. Moreover, by Proposition \ref{prop:the reduced $F$-property for i
ndependent increments} and Remark \ref{rem:the 2 dim F property}, it remains to check if the conditional Cauchy-Schwarz-Inequality is strict, i.e. if for each $k\in\{0,...,T_{2}^{F^{1}}\}$ the active hedge instruments $F^{1}$ and $F^{2}$ fulfill
\begin{equation}
\condCov{ \Delta F_{k+1}^{1}, \Delta F_{k+1}^{2} }^{2}
<
\condVar{ \Delta F_{k+1}^{1} }
\condVar{ \Delta F_{k+1}^{2} }\,,
\end{equation}
which ensures that the inverse matrix $\tilde F_{k}^{-1}$ exists and additionally the $F$-property holds. The CS-inequality is indeed strict since $T_{1}^{F^{1}} \neq T_{1}^{F^{2}}$ and this ensures that ${\mathbb P}(F_{k+1}^{1} = a F_{k+1}^{2}) < 1$ for any constant $a \in \mathbb{R}$.\footnote{That means, both futures are linearly independent with positive probability.} So by Corollary~\ref{cor:existence}, there exists a LRM-strategy $\hat \varphi = ( \hat X, \hat Y )$ under illiquidity of the form $\hat X_{T+1} = 0$, $\hat Y_{T} = H$ and $\hat X = (\bar 0, \tilde X)$ with $\bar 0 = (0, \dots, 0) \in \mathbb{R}^{l_{k}}$ and
\begin{equation}
\tilde X_{k+1}
= \tilde F_{k}^{-1} \tilde b_{k}
\quad \ensuremath{\mathbb{P}\text{ -- a.s.}}
\end{equation}
in $\mathbb{R}^{d-l_{k}}$ for $k \in \{ 0, \dots, T-1 \}$. Note that the matrix $\tilde F_{k}^{-1}$ is $2 \times 2$-dimensional for $k \in \{ 0, \dots, T_{2}^{F^{1}} - 1 \}$ and $1$-dimensional for $k \in \{ T_{2}^{F^{1}}, \dots, T_{2}^{F^{2}} - 1 \}$.
\\ \\
To compute the optimal strategy $\tilde X$ one needs to compute conditional expectations of the form ${\mathbb E}[ Y | X ]$ for square integrable random variables $X$ and $Y$. A popular method to compute such conditional expectations numerically, which we also employ in the following, is the \textit{least-squares Monte Carlo} (LSMC) method first used in finance by \cite{longstaff.schwartz:2001} for the valuation of American options. We do not go into further details of the LSMC method, but just mention that we use indicator functions constructed via the \textit{binning} method as basis functions. We refer to \cite{fries:2007} for a nice introduction to the LSMC method.
\par
In our $2$-dimensional example we need to simulate,
\begin{align}
\tilde X_{T+1}
&= 0
\nonumber \\
\tilde X_{k+1}
&= \frac{1}{A_{k; 2}} b_{k; 2}
\quad \text{ for } k \in \{ T_{2}^{F^{1}}, \dots, T_{2}^{F^{2}} - 1 \}
\nonumber \\%
\tilde X_{k+1}
&= (\tilde X_{k+1}^{1}, \tilde X_{k+1}^{2})
\quad \text{ for } k \in \{ 0, \dots, T_{2}^{F^{1}} - 1 \}, \quad \text{ where }
\nonumber \\%
\tilde X_{k+1}^{1}
&= \frac{1}{A_{k; 1}A_{k; 2} - | D_{k; 1, 2} |^{2} } ( A_{k; 2} b_{k; 1} - D_{k; 1, 2} b_{k; 2} )
\nonumber \\
\tilde X_{k+1}^{2}
&= \frac{1}{A_{k; 1}A_{k; 2} - | D_{k; 1, 2} |^{2} } ( A_{k; 1} b_{k; 2} - D_{k; 1, 2} b_{k; 1} )\,.
\end{align}
To implement the LSMC-method one needs to ensure that all random variables in the conditional expectations are square integrable. This is guaranteed by Corollary~\ref{cor:LSMC integrability condition} below, which is mostly based on Lemma \ref{lem:the alpha, beta, gamma terms}.
For Corollary \ref{cor:LSMC integrability condition}, we use the notation of Section~\ref{sec:the general case} where $\tilde S=(\tilde S^{1}, \dots, \tilde S^{d})$ is the price process of the (extended) hedge instruments.
\begin{cor}
\label{cor:LSMC integrability condition}
Assume that the components of the marginal price process $\tilde S$ and the contingent claim $H$ are both in $\ensuremath{\mathbb{L}_{T}^{4, 1}}$ as well as $\bar X_{T+1}=0$. Under the assumptions of Corollary \ref{cor:existence} there exists a LRM-strategy $\hat \varphi = ( \hat X, \hat Y )$ under illiquidity such that for some constant $C > 0$
\begin{align}
{\mathbb E}[ ( (\tilde F_{k}^{-1} \tilde b_{k})_{j} \Delta \tilde S_{k+1}^{j} )^{4} ]
&\leq C ( {\mathbb E} | V_{k+1}(\hat \varphi) |^{4}
+ \sum_{i=1}^{d}{\mathbb E} | \hat X_{k+2}^{i} |^{4} )
\\
{\mathbb E}[ ( (\tilde F_{k}^{-1} \tilde b_{k})_{j} )^{4} ]
&\leq C ( {\mathbb E} | V_{k+1}(\hat \varphi) |^{4}
+ \sum_{i=1}^{d}{\mathbb E} | \hat X_{k+2}^{i} |^{4} )
\end{align}
for $k \in \{ 0, 1, \dots, T - 1 \}$ where $V_{k+1}(\hat \varphi) = \condEplus{ H - \sum_{m=k+2}^{T} \hat X_{m}^{*} \Delta \tilde S_{m} }$. In particular, all random variables in the conditional expectations in the terms $A_{k, j}$, $b_{k, j}$ and $D_{k, j, i}$ are square integrable for all $j=l_{k}+1, \dots, d$ and $k= 0, 1, \dots, T-1$.
\end{cor}
\begin{proof}
The existence of a LRM-strategy $\hat \varphi = ( \hat X, \hat Y )$ under illiquidity follows directly from Corollary \ref{cor:existence}. The fact that $V_{k+1}(\hat \varphi) = \condEplus{ H - \sum_{m=k+2}^{T} \hat X_{m}^{*} \Delta \tilde S_{m} }$ follows also directly from $\hat Y_{k}$ defined as in Corollary \ref{cor:existence}.
\par
By Lemma \ref{lem:the alpha, beta, gamma terms} together with Lemma \ref{lem:alpha beta terms boundedness} applied for the active assets at time $k \in \{ 0, 1, \dots, T-1 \}$, we get
\begin{align}
{\mathbb E}[ ( (\tilde F_{k}^{-1} \tilde b_{k})_{j} \Delta \tilde S_{k+1}^{j} )^{4} ]
\leq C {\mathbb E}[ ( \condVar{ V_{k+1}(\hat \varphi) }
+ \sum_{i=1}^{d} \condE{ | \hat X_{k+2}^{i} |^{2} } )^{2} ] \,.
\end{align}
Furthermore, using ${\mathbb{V}ar} [X] \leq {\mathbb E}[X^2]$ we can estimate,
\begin{align}
{\mathbb E}[ ( (\tilde F_{k}^{-1} \tilde b_{k})_{j} \Delta \tilde S_{k+1}^{j} )^{4} ]
&\leq C {\mathbb E}[ ( \condE{ | V_{k+1}(\hat \varphi) |^{2} }
+ \sum_{i=1}^{d} \condE{ | \hat X_{k+2}^{i} |^{2} } )^{2} ]
\nonumber \\
& \leq C {\mathbb E}[ \condE{ | V_{k+1}(\hat \varphi) |^{4} }
+ \sum_{i=1}^{d} \condE{ | \hat X_{k+2}^{i} |^{4} } ]
\nonumber \\
& = C ( {\mathbb E} | V_{k+1}(\hat \varphi) |^{4}
+ \sum_{i=1}^{d} {\mathbb E} | \hat X_{k+2}^{i} |^{4} )
\end{align}
where for the last inequality we have used the conditional Jensen Inequality and for the equality we have applied the tower property. Analogously we also get the second inequality of the claim.
\par
This shows that,
\begin{align}
{\mathbb E}[ ( \hat X_{k+1}^{j} \Delta \tilde S_{k+1}^{j} )^{4} ]
&\leq C ( {\mathbb E} | V_{k+1}(\hat \varphi) |^{4}
+ \sum_{i=1}^{d}{\mathbb E} | \hat X_{k+2}^{i} |^{4} )
\\
{\mathbb E}[ ( \hat X_{k+1}^{j} )^{4} ]
&\leq C ( {\mathbb E} | V_{k+1}(\hat \varphi) |^{4}
+ \sum_{i=1}^{d}{\mathbb E} | \hat X_{k+2}^{i} |^{4} )
\end{align}
for all $k=0, 1, \dots, T-1$, $j=l_{k}+1, \dots, d$. By the definition of $V_{k+1}(\hat \varphi)$ and since by assumption $H \in \ensuremath{\mathbb{L}_{T}^{4, 1}}$ and $\hat X_{T+1} = 0$, one can argue recursively that both $\hat X_{k+1}^{j} \Delta \tilde S_{k+1}^{j}$ and $\hat X_{k+1}^{j}$ are in $\ensuremath{\mathbb{L}_{T}^{4, 1}}$.
\par
Furthermore, we have for some $j \in \{ l_{k}+1, \dots, d \}$ at time $k \in \{ 0, 1, \dots, T-1 \}$ for the term
\begin{equation}
b_{k; j}^{0}
= \condCov{ V_{k+1}(\hat \varphi), S_{k+1}^{j} }
= \condE{ V_{k+1} S_{k+1}^{j} } - \condE{ V_{k+1} } \condE{ S_{k+1}^{j} }
\end{equation}
that $V_{k+1}(\hat \varphi) \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$, $S_{k+1}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$ and $V_{k+1}(\hat \varphi) S_{k+1}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$ since $V_{k+1}(\hat \varphi) \in \ensuremath{\mathbb{L}_{T}^{4, 1}}$, $\tilde S_{k+1}^{j} \in \ensuremath{\mathbb{L}_{T}^{4, 1}}$ and by the Cauchy-Schwarz inequality. For the term
\begin{equation}
b_{k; j}^{\varepsilon}
= \condE{ \varepsilon_{k+1}^{j} S_{k+1}^{j} \hat X_{k+2}^{j} }
\end{equation}
we have $S_{k+1}^{j} \hat X_{k+2}^{j} \in \ensuremath{\mathbb{L}_{T}^{2, 1}}$ since $\tilde S_{k+1}^{j} \in \ensuremath{\mathbb{L}_{T}^{4, 1}}$, $\hat X_{k+2}^{j} \in \ensuremath{\mathbb{L}_{T}^{4, 1}}$ and using the Cauchy-Schwarz inequality.
\par
So, all random variables in the conditional expectations for the term $b_{k; j}$ are square integrable. Analogously the same holds for the terms $A_{k; j}$ and $D_{k; j, i}$.
\end{proof}
We now come to the specification of the electricity market model for our simulation study. To this end, we consider the spot price model \eqref{spotmodel} with two OU factors ($n=2$) $Y_{1}$ the base regime and $Y_{2}$ the spike regime with strong upward moves followed by quick reversion to normal levels and constant seasonality function $\Lambda_{1} = \Lambda_{2} =1$. We set $Y_{1}(0) = Y_{2}(0) = 0.5$, and assume constant volatilities $\sigma^{1}=0.34, \sigma^{2}=0.01$ and mean reversion rates $\lambda_{1}=0.01, \lambda_{2}=0.1$. For the driving L\'evy processes we suppose that $L_{1}$ is a Gamma process where $L_{1}(t)$ has $\Gamma(\gamma^{1} t, \alpha^{1})$-distribution and $L_{2}$ a compound Poisson process with intensity $\gamma^{2}$ and $\exp(\alpha^{2})$-distributed jumps. We set $\gamma^{1} = \gamma^{2} = \alpha^{1} = 1$, $\alpha^{2} = 0.1$. Both OU-processes are simulated using an Euler Scheme.\footnote{Note that in order to use the Least-squares Monte Carlo metho
d for calculating conditional expectations for the simulation, we need to simulate $2$-dim. basis functions using both Markov processes $L^{1}$ and $L^{2}$.}
Moreover, we set the strike price $K=1.05$ in \eqref{calloption} and $\alpha=1$ in the performance criterium \eqref{eq:quadratic linear risk process new}, which means an equal concern between the risk from market price fluctuations and the cost of liquidity costs.
We will simulate and analyze two different settings, each with various pairs of futures with different delivery periods as available hedge instruments for the call option. In the first setting we focus on hedging the option with \emph{various combinations of futures that cover the delivery period $[T_{1}^{c},T_{2}^{c}]$ of the option}. To this end we consider three futures $F^{1}, F^{2}, F^{3}$ with delivery periods $[ T_{1}^{F^{1}}, T_{2}^{F^{1}} ], [ T_{1}^{F^{2}}, T_{2}^{F^{2}} ], [ T_{1}^{F^{3}}, T_{2}^{F^{3}} ]$, respectively, where we set $T^c_{1} = T_{1}^{F^{1}} = T_{1}^{F^{2}} = 0.0125$, $T^c_{2} = T_{2}^{F^{2}} = T_{2}^{F^{3}} = 0.1$, $T_{2}^{F^{1}} = T_{1}^{F^{3}} = 0.05$. We consider both the time varying liquidity structure \eqref{TimeLiquidity}, where we set $M_{i}=0.005$, $N_{i}=2M_{i}$, $\delta_i = 0.000001$ and the constant liquidity structure \eqref{ConstantLiquidity}, where we set $M_{i}=N_{i}=0.01$ for $i=1,2,3$.
We compute the criteria $ T_{0}(\varphi) $, $\tilde T_{0}(\varphi) $, $L_{0}(\varphi)$, and $C_{0}(\varphi)$ for LRM-strategies $\varphi=(X,Y)$, where $\tilde T_{0}(\varphi) = {\mathbb E} [(C_{T}(\varphi) - C_{0}(\varphi))^{2}]$ is the quadratic hedge criterion, $L_{0}(\varphi) = {\mathbb E}[ \sum_{m=1}^{T} \Delta X_{m+1}^{*} [ S_{m}(\Delta X_{m+1}) - S_{m}(0) ] ]$ the liquidity costs, $ T_{0}(\varphi) = \tilde T_{0}(\varphi) + L_{0}(\varphi)$ our combined LRM minimization criterion \eqref{eq:quadratic linear risk process}, and $C_{0}(\varphi) = {\mathbb E}[ H - \sum_{m=1}^{T} (X_{m})^{*} \Delta S_{m} ]$ the cost for a strategy $\varphi$ at time $0$. In Tables~\ref{tab:constant liquidity phenom 1} and \ref{tab:time varying liquidity phenom 1} the results are displayed for a LRM-strategy $\varphi^{L}=(X^{L},Y^{L})$ with time varying liquidity~\eqref{TimeLiquidity} and constant liquidity~\eqref{ConstantLiquidity}, respectively. In addition, we compute the results with the classical LRM-strategy $\varphi^{C}=(X^{C},Y^{C})$ with zero liquidity costs (i.e., $\varepsilon^{i} = 0$). Recall that the quantity $T_{0}$ is minimized by $\varphi^{L}$ and $\tilde T_{0}$ is minimized by $\varphi^{C}$. For comparison, we use the same trajectories in both cases.
The first observation that can be made is that the hedging costs and the corresponding minimization criterion indeed decrease in the number of available hedge instruments. Also, the initial cost for using the strategy $\varphi^{L}$ is more than using $\varphi^{C}$ since it will cost more to generate the optimal strategy $\varphi^{L}$ under liquidity costs.
To focus on the hedge performance with two futures that cover the delivery period of the option we consider two examples. In the first one we consider the futures $F^{1}, F^{2}$ with overlapping delivery periods while in the second one, the futures $F^{1}, F^{3}$ (see Figure \ref{fig:example1_2}) have different delivery periods. From Tables \ref{tab:constant liquidity phenom 1} and \ref{tab:time varying liquidity phenom 1} and by comparing the quantity $T_{0}(\varphi^{L})$ we see that the case with the futures $F^{1}, F^{3}$ performs better since they incur less cost. In Table \ref{tab:time varying liquidity phenom 1} with time-varying liquidity this is due to the fact that $F^{3}$ has shorter delivery period than $F^{2}$ and can be used for hedging longer in time. In Table \ref{tab:constant liquidity phenom 1} we see that in the case with constant liquidity, despite that $F^{2}$ has a delivery period perfectly coinciding with the option $H$ it is better to hedge with the tw
o hedge instruments $F^{1}$ and $F^{3}$. By looking at the quantity $\tilde T_{0}(\varphi^{C})$ one can observe that also for the classical LRM-strategy under the classical LRM-criterion the futures $F^{1}$ and $F^{3}$ perform better, simply due to the increased dimension of the hedge instruments.
\par
Recall that our quadratic criterion balances low liquidity costs against poor replication. This can be seen for example in Tables \ref{tab:constant liquidity phenom 1} and \ref{tab:time varying liquidity phenom 1}. Indeed, from our example the futures $F^{1}, F^{3}$ perform better with less cost $\tilde T_{0}(\varphi^{L})$ from market fluctuations but incurring more liquidity cost $L_{0}(\varphi^{L})$ than the futures $F^{1}, F^{2}$.
\par
Note also, that Figure~\ref{fig:example1_1} corresponding to the result for $F^{2}$ in Table \ref{tab:constant liquidity phenom 1} confirms the numerical results of \cite{agliardi.gencay:2014} and \cite{rogers.singh:2010} who find that the optimal strategy under illiquidity is less volatile than the classical one. This is perfectly intuitive since changing position drastically incurs large liquidity cost. In Figure~\ref{fig:example1_2} one can observe that before the start of the delivery periods both futures are used actively, but after entering into the delivery period of $F^{1}$ then almost only the future $F^3$ is used for hedging since $F^3$ is more liquid than $F^{1}$ and expires later.
In a second setting, we focus on the \emph{trade-off between liquidity costs and hedging performance} appearing in various hedge constellations. To this end we consider three futures $G^{1}, G^{2}, G^{3}$ with delivery periods $[ T_{1}^{G^{1}}, T_{2}^{G^{1}} ], [ T_{1}^{G^{2}}, T_{2}^{G^{2}} ], [ T_{1}^{G^{3}}, T_{2}^{G^{3}} ]$, respectively, and set $T_{1}^{c} = T_{1}^{G^{1}} = T_{1}^{G^{2}} = T_{2}^{G^{3}} = 0.05$, $T_{2}^{c} = T_{2}^{G^{2}} = 0.1$, $T_{2}^{G^{1}} = 0.075$, $T_{1}^{G^{3}} = 0.0125$. Otherwise, the model specifications remain the same as in the first setting above. We consider two examples, with one common future $G^{2}$, which has the same delivery period as the option $H$. From Tables \ref{tab:constant liquidity phenom 2} and \ref{tab:time varying liquidity phenom 2} we can observe that $G^{1}, G^{2}$ performs better than $G^{2}, G^{3}$ according to the quantity $T_{0}(\varphi^{L})$. From $\tilde T_{0}(\varphi^{C})$ we see that this is also the case in the
classical setting. This is mostly due to the fact that the future $G^{1}$ expires later than $G^{3}$ and its delivery period lies within the delivery period of the option. Note that by comparing the quantity $T_{0}(\varphi^{L})$ of both examples we observe that in Table \ref{tab:time varying liquidity phenom 2} the difference between them becomes less than in Table \ref{tab:constant liquidity phenom 2}. This is due to the fact that $G^{3}$ is more liquid than $G^{1}$ in the period $[0,0.0125]$ in this case and can be used for hedging at low liquidity cost. Therefore a correct specification of the term-structure of liquidity seems important.
In Figure \ref{fig:example2} and Figure \ref{fig:example3} we display the strategies for one trajectory in both cases. In Figure~\ref{fig:example3_2} one can actually observe that $G^3$ is the more active hedge instrument in the period $[0,0.0125]$ where it is more liquid than the future $G^2$ in the case with time dependent liquidity.
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\begin{center}
\scriptsize
\begin{tabular}{ l | l r | l r | l r | l r }
Hedging Instruments & $T_{0}(\varphi^{L})$ & $T_{0}(\varphi^{C})$ & $\tilde T_{0}(\varphi^{L})$ & $\tilde T_{0}(\varphi^{C})$ & $L_{0}(\varphi^{L})$ & $L_{0}(\varphi^{C})$ & $C_{0}(\varphi^{L})$ & $C_{0}(\varphi^{C})$ \\ \hline
$F^{2}$ & 2.19E-3 & 4.79E-2 & 2.03E-3 & 3.40E-4 & 1.56E-4 & 4.76E-2 & 1.09E-2 & 9.29E-3 \\ \hline
$F^{1}, F^{2}$ & 1.86E-3 & 3.64E-2 & 1.67E-3 & 2.92E-4 & 1.88E-4 & 3.61E-2 & 1.07E-2 & 9.19E-3 \\ \hline
$F^{1}, F^{3}$ & 1.51E-3 & 1.59E-2 & 1.31E-3 & 2.20E-4 & 2.01E-4 & 1.57E-2 & 1.05E-2 & 8.92E-3 \\ \hline
\end{tabular}
\captionof{table}{Simulation results with constant liquidity parameter.}
\label{tab:constant liquidity phenom 1}
\end{center}
\begin{center}
\scriptsize
\begin{tabular}{ l | l r | l r | l r | l r}
Hedging Instruments & $T_{0}(\varphi^{L})$ & $T_{0}(\varphi^{C})$ & $\tilde T_{0}(\varphi^{L})$ & $\tilde T_{0}(\varphi^{C})$ & $L_{0}(\varphi^{L})$ & $L_{0}(\varphi^{C})$ & $C_{0}(\varphi^{L})$ & $C_{0}(\varphi^{C})$ \\ \hline
$F^{2}$ & 1.63E-3 & 4.11E-2 & 1.49E-3 & 3.40E-4 & 1.40E-4 & 4.08E-2 & 1.05E-2 & 9.29E-3 \\ \hline
$F^{1}, F^{2}$ & 1.56E-3 & 3.58E-2 & 1.35E-3 & 2.92E-4 & 2.10E-4 & 3.55E-2 & 1.04E-2 & 9.19E-3 \\ \hline
$F^{1}, F^{3}$ & 7.09E-4 & 1.28E-2 & 4.50E-4 & 2.20E-4 & 2.59E-4 & 1.26E-2 & 9.66E-3 & 8.92E-3 \\ \hline
\end{tabular}
\captionof{table}{Simulation results with time varying liquidity parameter.}
\label{tab:time varying liquidity phenom 1}
\end{center}
\begin{center}
\scriptsize
\begin{tabular}{ l | l r | l r | l r | l r }
Hedging Instruments & $T_{0}(\varphi^{L})$ & $T_{0}(\varphi^{C})$ & $\tilde T_{0}(\varphi^{L})$ & $\tilde T_{0}(\varphi^{C})$ & $L_{0}(\varphi^{L})$ & $L_{0}(\varphi^{C})$ & $C_{0}(\varphi^{L})$ & $C_{0}(\varphi^{C})$ \\ \hline
$G^{2}$ & 3.22E-3 & 2.30E-2 & 2.99E-3 & 7.75E-4 & 2.28E-4 & 2.23E-2 & 1.60E-2 & 1.41E-2 \\ \hline
$G^{1}, G^{2}$ & 2.33E-3 & 8.03E-3 & 2.06E-3 & 5.21E-4 & 2.68E-4 & 7.51E-3 & 1.55E-2 & 1.39E-2 \\ \hline
$G^{2}, G^{3}$ & 2.95E-3 & 1.52E-2 & 2.69E-3 & 7.12E-4 & 2.55E-4 & 1.45E-2 & 1.58E-2 & 1.40E-2 \\ \hline
\end{tabular}
\captionof{table}{Simulation results with constant liquidity parameter.}
\label{tab:constant liquidity phenom 2}
\end{center}
\begin{center}
\scriptsize
\begin{tabular}{ l | l r | l r | l r | l r}
Hedging Instruments & $T_{0}(\varphi^{L})$ & $T_{0}(\varphi^{C})$ & $\tilde T_{0}(\varphi^{L})$ & $\tilde T_{0}(\varphi^{C})$ & $L_{0}(\varphi^{L})$ & $L_{0}(\varphi^{C})$ & $C_{0}(\varphi^{L})$ & $C_{0}(\varphi^{C})$ \\ \hline
$G^{2}$ & 1.66E-3 & 1.45E-2 & 1.49E-3 & 7.75E-4 & 1.69E-4 & 1.37E-2 & 1.50E-2 & 1.41E-2 \\ \hline
$G^{1}, G^{2}$ & 1.32E-3 & 4.64E-3 & 1.13E-3 & 5.21E-4 & 1.92E-4 & 4.12E-3 & 1.47E-2 & 1.39E-2 \\ \hline
$G^{2}, G^{3}$ & 1.63E-3 & 1.25E-2 & 1.39E-3 & 7.12E-4 & 2.39E-4 & 1.18E-2 & 1.49E-2 & 1.40E-2 \\ \hline
\end{tabular}
\captionof{table}{Simulation results with time varying liquidity parameter.}
\label{tab:time varying liquidity phenom 2}
\end{center}
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\begin{figure}[h]
\begin{subfigure}[c]{1.0\textwidth}
\centering
\includegraphics[width=0.7\textwidth, height=9\baselineskip]{Example1_1Path9Holdings
\subcaption{Hedging with only the Future $F^{2}$ which has the same delivery period as the claim $H$. The hedging strategy $X^{C}$ corresponds to the classical case without liquidity cost and $X^{L}$ to the case with constant liquidity structure (\ref{ConstantLiquidity}) with parameters $M_{2}=N_{2}=0.01$. Observe that the optimal LRM-strategy $X^{L}$ under illiquidity is less volatile than the classical LRM-strategy $X^{C}$. }
\label{fig:example1_1}
\end{subfigure}
\begin{subfigure}[c]{1.0\textwidth
\centering
\includegraphics[width=0.7\textwidth, height=9\baselineskip]{Example1_2Path9Holdings
\subcaption{Hedging with the two futures $F^{1}$ and $F^{3}$ with consecutive delivery periods which together cover the delivery period of the claim $H$. Both futures have a time-varying liquidity structure (\ref{TimeLiquidity}) with parameters $M_{1}=M_{3}=0.005, N_{1}=2M_{1}, N_{3}=2M_{3}$. The optimal LRM-strategy $X^{1, L}$ under illiquidity corresponds to the future $F^{1}$ and $X^{2, L}$ to the future $F^{3}$ which is more liquid and expires later than $F^{1}$. $F^{3}$ is used more actively as can be observed from the plot. In the delivery period both futures become very illiquid and thus a rapid drop in the holdings can be observed. }
\label{fig:example1_2}
\end{subfigure}
\caption{Comparison of the sample path of optimal LRM-strategies under different liquidity structures and for different hedge instruments. All plots based on the same realization of the underlying.}
\label{fig:example1}
\end{figure}
\begin{figure}[h]
\begin{subfigure}[c]{1.0\textwidth
\centering
\includegraphics[width=0.7\textwidth, height=9\baselineskip]{Example2_1Path9Holdings
\subcaption{Hedging with the two Futures $G^{1}, G^{2}$ with constant liquidity structure (\ref{ConstantLiquidity}) with parameters $M_{i}=N_{i}=0.01$ for $i=1, 2$ .}
\label{fig:example2_1}
\end{subfigure}
\begin{subfigure}[c]{1.0\textwidth
\centering
\includegraphics[width=0.7\textwidth, height=9\baselineskip]{Example2_2Path9Holdings
\subcaption{Hedging with two instruments using the Futures $G^{1}, G^{2}$ with time-varying liquidity structure (\ref{TimeLiquidity}) with parameters $M_{i}=0.005, N_{i}=2M_{i}$ for $i=1, 2$. The sudden drop in the holdings occurs when entering the delivery period where the futures are very illiquid.}
\label{fig:example2_2}
\end{subfigure}
\caption{Comparison of the sample path of optimal LRM-strategies under different liquidity structures but with the same hedge instruments. The two futures have overlapping delivery periods starting together but $G^{1}$ expires earlier. The delivery period of $G^{2}$ is the same as the one of the claim $H$. The LRM-strategies $X^{1, L}, X^{2, L}$ under illiquidity correspond to the future hedge instruments $G^{1}, G^{2}$ respectively.}
\label{fig:example2}
\end{figure}
\begin{figure}[h]
\begin{subfigure}[c]{1.0\textwidth
\centering
\includegraphics[width=0.7\textwidth, height=9\baselineskip]{Example3_1Path9Holdings
\subcaption{Hedging with the two Futures $G^{2}, G^{3}$ with constant liquidity structure (\ref{ConstantLiquidity}) with parameters $M_{i}=N_{i}=0.01$ for $i=2, 3$.}
\label{fig:example3_1}
\end{subfigure}
\begin{subfigure}[c]{1.0\textwidth
\centering
\includegraphics[width=0.7\textwidth, height=9\baselineskip]{Example3_2Path9Holdings
\subcaption{Hedging with the two Futures $G^{2}, G^{3}$ with time-varying liquidity structure (\ref{TimeLiquidity}) with parameters $M_{i}=0.005, N_{i}=2M_{i}$ for $i=2, 3$.}
\label{fig:example3_2}
\end{subfigure}
\caption{Comparison of the sample path of optimal LRM-strategies under different liquidity structures but with the same hedge instruments. The two futures have consecutive delivery periods with the one of $G^{3}$ starting earlier. The delivery period of $G^{2}$ and the claim $H$ coincide. The optimal LRM-strategy $X^{1, L}$ under illiquidity corresponds to the hedge instrument $G^{3}$ and $X^{2, L}$ to $G^{2}$.}
\label{fig:example3}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In an arbitrage-free model framework, this paper has presented a new quadratic hedging criterion that targets at minimizing the risk against random fluctuations of the underlying stock price while simultaneously incurring low liquidity costs. It extends the quadratic local-risk minimization approach of \cite{schweizer:1988} in the spirit of \cite{rogers.singh:2010} and \cite{agliardi.gencay:2014}. It is mathematically tractable enough to allow for computable formulae. Under mild conditions, the optimization problem can be solved in closed-form. Furthermore, by embedding a multi-dimensional price process with different maturities in our setting it is possible to consider as one possible application the hedging of an Asian-style option in an electricity exchange using a variety of futures. In a simulation study we analyze hedge performance and cost under various pairs of futures with different delivery periods and liquidity levels, allowing us to investigate the tradeoff between hedge performance and liquidity cost.
\section{Appendix}
\label{appendix}
\begin{proof}[Proof of Lemma \ref{lem:cost process martingale}]
The arguments follow those in the proof of Lemma 1 in \cite{lamberton.pham.schweizer:1998}.
\par Let $\varphi=(X,Y)$ be a LRM-strategy under illiquidity and fix some $k \in \{ 0, 1, \dots, T-1 \}$. Assuming that $C(\varphi)$ is not a martingale, we can choose a local perturbation $\varphi ' = (X',Y')$ of $\varphi$ at time $k$ by defining $X':=X$ and only modifying the cash holding $Y'$ at time $k$, by adding the conditional expectation of the incremental cost at time $k$ to $Y$,
\begin{equation}
Y_{k}^{'} := \condE{ C_{T}(\varphi) - C_{k}(\varphi) } + Y_{k} \,.
\end{equation}
This implies that $\condE{ C_{T}(\varphi') - C_{k}(\varphi') } = 0$ and
$\condVar{ C_{T}(\varphi') - C_{k}(\varphi') } = \condVar{C_{T}(\varphi) - C_{k}(\varphi) }$. Since ${\mathbb E}[X^2]= {\mathbb{V}ar} [X] + ({\mathbb E}[X])^{2}$ for a random variable $X$, one can conclude that using the strategy $\varphi'$ the risk process becomes less, that is,
\begin{equation}
R_{k}(\varphi')
\leq R_{k}(\varphi) \,.
\end{equation}
Since $X':=X$, the liquidity costs of $\varphi'$ and $\varphi$ equal. This implies,
\begin{equation}
T_{k}^{\alpha}(\varphi')
\leq T_{k}^{\alpha}(\varphi) \,.
\end{equation}
By the fact that $\varphi$ is a LRM-strategy under illiquidity, we must have equality on $T_{k}^{\alpha}$ which implies equality on $R_{k}$ i.e., $R_{k}(\varphi') = R_{k}(\varphi)$. So, the cost process $C(\varphi)$ must be a martingale.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:quadratic linear risk process modified}]
As in \cite{lamberton.pham.schweizer:1998} (see proof of Proposition 2), by using Lemma \ref{lem:cost process martingale} and the fact that
\begin{equation}
\condE{ C_{T}(\varphi') - C_{k}(\varphi') }
= \Delta C_{k+1}(\varphi'),
\end{equation}
which follows from the martingale property of $C(\varphi)$, one can conclude that
\begin{equation}
R_{k}(\varphi ')
= \condE{ R_{k+1}(\varphi) }
+ \condE{ (\Delta C_{k+1}(\varphi '))^{2} } \,.
\end{equation}
Furthermore since $\varphi '$ is a local perturbation of $\varphi$ at time $k$, we have
\begin{align}
& \condE{ (X'_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X'_{k+2} - X'_{k+1}) - S_{k+1}(0) ] }
\nonumber \\ & \quad
= \condE{ (X_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X_{k+2} - X'_{k+1}) - S_{k+1}(0) ] }
\end{align}
and the claim follows.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:minimizing variance}]
The proof follows the steps in the proof of Proposition 2 in \cite{lamberton.pham.schweizer:1998}.
\par Let us first show the $``\Leftarrow"$ direction of the proof. We want to show that $\varphi=(X,Y)$ is a LRM-strategy under illiquidity, according to Definition \ref{defi:local risk minimizing strategy under illiquidity}. So, fix some $k \in \{ 0, 1, \dots, T-1 \}$ and let $\varphi '=(X',Y')$ be a local perturbation of $\varphi$ at time $k$.
\par Since property \ref{prop:1} holds and $\varphi '$ a local perturbation of $\varphi$ at time $k$ then by Lemma \ref{lem:quadratic linear risk process modified} we have the equality
\begin{align}
T_{k}^{\alpha}(\varphi ')
= & \condE{ R_{k+1}(\varphi) }
+ \condE{ (\Delta C_{k+1}(\varphi '))^{2} }
\nonumber \\
&+ \alpha \condE{ (X_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X_{k+2} - X'_{k+1}) - S_{k+1}(0) ] }
\end{align}
Moreover, from the definition of the conditional variance we have
\begin{equation}
\condE{ (\Delta C_{k+1}(\varphi '))^{2} }
\geq \condVar{ \Delta C_{k+1}(\varphi ') }
\end{equation}
and so we can estimate
\begin{align}
T_{k}^{\alpha}(\varphi ')
\geq \, & \condE{ R_{k+1}(\varphi) }
+ \condVar{ \Delta C_{k+1}(\varphi ') }
\nonumber \\
& + \alpha \condE{ (X_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X_{k+2} - X'_{k+1}) - S_{k+1}(0) ] } \,.
\end{align}
Since $\varphi '$ a local perturbation of $\varphi$ at time $k$ then $X'_{k+2} = X_{k+2}$ and $Y'_{k+1} = Y_{k+1}$ and so we get
\begin{align}
\condVar{ \Delta C_{k+1}(\varphi ') }
= \condVar{ C_{k+1}(\varphi ') }
&= \condVar{ V_{k+1}(\varphi') - (X'_{k+1})^{*} \Delta S_{k+1} }
\nonumber \\
&= \condVar{ V_{k+1}(\varphi) - (X'_{k+1})^{*} \Delta S_{k+1} }
\end{align}
and we can conclude that
\begin{align}
T_{k}^{\alpha}(\varphi ')
\geq \, & \condE{ R_{k+1}(\varphi) }
+ \condVar{ V_{k+1}(\varphi) - (X'_{k+1})^{*} \Delta S_{k+1} }
\nonumber \\
& + \alpha \condE{ (X_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X_{k+2} - X'_{k+1}) - S_{k+1}(0) ] } \,.
\end{align}
Furthermore, since \ref{prop:2} holds, then
\begin{align}
\label{eq:propProofEq1}
T_{k}^{\alpha}(\varphi ')
\geq \, & \condE{ R_{k+1}(\varphi) }
+ \condVar{ V_{k+1}(\varphi) - (X_{k+1})^{*} \Delta S_{k+1} }
\nonumber \\
& + \alpha \condE{ (X_{k+2} - X_{k+1})^{*} [ S_{k+1}(X_{k+2} - X_{k+1}) - S_{k+1}(0) ] } \,.
\end{align}
On the other hand, we have by definition (see Equation (\ref{eq:quadratic linear risk process new}) )
\begin{equation}
T_{k}^{\alpha}(\varphi)
= R_{k}(\varphi)
+ \alpha \condE{ \Delta X_{k+2}^{*} [ S_{k+1}(\Delta X_{k+2}) - S_{k+1}(0) ] } \,.
\end{equation}
Since $C(\varphi)$ is a martingale, we get the representation (\ref{eq:cost process martingale}) for the risk process $R_{k}(\varphi)$. So we can conclude that
\begin{align}
\label{eq:propProofEq2}
T_{k}^{\alpha}(\varphi)
= & \condE{ R_{k+1}(\varphi) }
+ \condVar{ \Delta C_{k+1}(\varphi) }
\nonumber \\
& + \alpha \condE{ (X_{k+2} - X_{k+1})^{*} [ S_{k+1}(X_{k+2} - X_{k+1}) - S_{k+1}(0) ] }
\end{align}
Finally, since (\ref{eq:propProofEq1}) and (\ref{eq:propProofEq2}) hold then $T_{k}^{\alpha}(\varphi ') \geq T_{k}^{\alpha}(\varphi)$ and this shows the $``\Leftarrow"$ direction of the proof.
\par Now, assuming that $\varphi$ is a LRM-strategy under illiquidity i.e., $T_{k}^{\alpha}(\varphi ') \geq T_{k}^{\alpha}(\varphi)$ for any local perturbation $\varphi '$ of $\varphi$ at time $k$, we will show the $``\Rightarrow"$ direction of the proof. Property \ref{prop:1} holds from Lemma~\ref{lem:cost process martingale}. So it remains to show Property \ref{prop:2}.
\par Since $C(\varphi)$ is a martingale and $\varphi '$ a local perturbation of $\varphi$ at time $k$, then from Lemma \ref{lem:quadratic linear risk process modified} we know that equation (\ref{eq:quadratic linear risk process modified}) holds. On the other hand, since (\ref{eq:propProofEq2}) holds (from the martingale property of $C(\varphi)$) then from the fact that $T_{k}^{\alpha}(\varphi ') \geq T_{k}^{\alpha}(\varphi)$ we have
\begin{align}
& \condE{ R_{k+1}(\varphi) }
+ \condE{ ( \Delta C_{k+1}(\varphi ') )^{2} }
\nonumber \\
& \quad + \alpha \condE{ (X_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X_{k+2} - X'_{k+1}) - S_{k+1}(0) ] }
\nonumber \\
&\geq \condE{ R_{k+1}(\varphi) }
+ \condVar{ \Delta C_{k+1}(\varphi) }
\nonumber \\
& \quad + \alpha \condE{ (X_{k+2} - X_{k+1})^{*} [ S_{k+1}(X_{k+2} - X_{k+1}) - S_{k+1}(0) ] }
\end{align}
and from the definition of the conditional variance we can conclude that
\begin{align}
& \condVar{ \Delta C_{k+1}(\varphi ') }
+ ( \condE{ \Delta C_{k+1}(\varphi ') } )^{2}
\nonumber \\
& \quad + \alpha \condE{ (X_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X_{k+2} - X'_{k+1}) - S_{k+1}(0) ] }
\nonumber \\
&\geq
\condVar{ \Delta C_{k+1}(\varphi) }
+ \alpha \condE{ (X_{k+2} - X_{k+1})^{*} [ S_{k+1}(X_{k+2} - X_{k+1}) - S_{k+1}(0) ] }
\end{align}
for all $X'_{k+1}$ and $Y'_{k}$. Fixing $X'_{k+1}$ and choosing $Y'_{k}$ as in the proof of Lemma \ref{lem:cost process martingale} the inequality still holds and the liquidity costs remain unchanged. Since this choice gives us $\condE{ \Delta C_{k+1}(\varphi ') }= 0$ (as in the proof of Lemma \ref{lem:cost process martingale}) and since $\varphi '$ a local perturbation of $\varphi$ at time $k$, we get the inequality
\begin{align}
& \condVar{ V_{k+1}(\varphi) - (X'_{k+1})^{*} \Delta S_{k+1} }
\nonumber \\
& \quad \quad + \alpha \condE{ (X_{k+2} - X'_{k+1})^{*} [ S_{k+1}(X_{k+2} - X'_{k+1}) - S_{k+1}(0) ] }
\nonumber \\
\geq & \condVar{ V_{k+1}(\varphi) - (X_{k+1})^{*} \Delta S_{k+1} }
\nonumber \\
& \quad \quad + \alpha \condE{ (X_{k+2} - X_{k+1})^{*} [ S_{k+1}(X_{k+2} - X_{k+1}) - S_{k+1}(0) ] } \,.
\end{align}
This shows that Property \ref{prop:2} holds and the proof is completed.
\end{proof}
\bibliographystyle{newapa
|
{'timestamp': '2018-07-02T02:08:19', 'yymm': '1705', 'arxiv_id': '1705.06918', 'language': 'en', 'url': 'https://arxiv.org/abs/1705.06918'}
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arxiv
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Advocates working to keep needs of poor foremost in U.S. budget debate
House Budget Committee Chairman Rep. Paul Ryan (R-Wis.), unveils the Republicans’ FY 2014 budget resolution in Washington March 12. Advocates for the poor, including Catholic Charities USA, the U.S. Conference of Catholic Bishops and Network, the Catholic social justice lobby, have blanketed Congress with their concerns that the budget reflect society’s moral obligations to care for “the least of these.” (CNS photo/Gary Cameron, Reuters)
WASHINGTON (CNS) — On Capitol Hill, when it’s spring it’s time to debate the federal budget.
Republicans and Democrats set spending parameters for fiscal year 2014 by adopting two starkly different budget plans before recessing a week before Easter. What’s expected when Congress returns April 8 is a lengthy debate over whether austerity and lower taxes or modest adjustments in government spending supplemented with selected tax law changes to boost revenues is the way to go.
The debate will come as the impact of the March 1 automatic spending cuts in hundreds of government-funded programs — known as the sequester — begins to pinch social service providers.
“The longer this has gone on, the more concerned our agencies are becoming,” Candy Hill, senior vice president for social policy and government affairs at Catholic Charities USA, told Catholic News Service March 27.
That’s because local agencies are unsure how large the funding losses will be, Hill said. While the sequester set cuts at 5.2 percent, the fact that the cuts are coming midway through the fiscal year means the funding losses will be more severe — in some cases double digits, she said.
Hill provided a partial list of diocesan Catholic Charities programs that will be squeezed: refugee assistance, Head Start, homeless services and education and the Social Services Block Grant that funds feeding, child abuse prevention and substance abuse programs. Cuts in the Special Supplemental Nutrition Program for Women, Infants and Children likely will mean more families showing up at hunger centers, she added.
Overall, programs serving youth, families, the elderly, people with disabilities and migrant workers will see $29.9 billion in cuts by Sept. 30, the end of the fiscal year, reports the National Human Services Assembly, whose membership includes Catholic Charities USA.
Most Republicans would like to see even lower spending combined with reduced tax rates. A spending plan reflecting those priorities for fiscal year 2014 was drafted by Rep. Paul Ryan, R-Wis., House Budget Committee chairman. It passed 221-207 March 21.
Ryan advocated that his plan carried out for the next decade would lead to a balanced budget in 2023 as spending on non-military programs would decline, Medicaid and Medicare would be remade and the Affordable Care Act would be repealed. The plan also calls for simplifying the tax code with a 10 percent rate for lower income earners and a 25 percent rate for higher income earners and corporations.
An anaylsis by the Washington-based Tax Policy Center projected that the plan would reduce government revenues by more than $5.7 trillion over the next decade, leading to drastic cuts in domestic spending.
Ryan spokesman Kevin Seifert said the congressman was unavailable to comment on the analysis.
In the Senate, where Democrats hold the majority, a more moderate budget was approved, 50-49, early March 23. It calls for slight adjustments in spending and raising revenues by closing loopholes and altering deductions for high income Americans as the way to stabilize the country’s growing debt.
Sen. Patty Murray, D-Wash., Senate Budget Committee chairwoman, has said the spending plan reflects a balanced approach to solving the country’s budget concerns. In introducing the proposal, Murray said the budget “tackles the deficit and debt the way the American people wanted it done.”
As the congressional votes neared, advocates for the poor raised their voices in an attempt to minimize budget cuts on programs benefiting people whose voices go largely unheard on Capitol Hill. A diverse pool that included Catholic Charities, the U.S. Conference of Catholic Bishops and Network, the Catholic social justice lobby, blanketed Congress with their concerns that the budget must reflect society’s moral obligation to care for “the least of these.”
Among the most persistent voices have been the chairmen of two USCCB committees.
Bishop Stephen E. Blaire of Stockton, Calif., chairman of the Committee on Domestic Justice and Human Development, and Bishop Richard E. Pates of Des Moines, Iowa, chairman of the Committee on International Justice and Peace, cited Catholic social teaching in reiterating their concern in a March 18 letter to all members of Congress that government programs serving poor and marginalized people deserve the highest priority.
“While we lack the competence to offer a detailed critique of entire budget proposals, we do ask you to consider the human and moral dimensions of these choices,” they wrote.
Matthew Hale, associate professor of political science and public affairs at Seton Hall University in South Orange, N.J., while remaining neutral, explained that in today’s polarized political climate, there is little reason for members of Congress to budge from their staked out positions.
For Hale, Catholic social teaching and its emphasis on human dignity illustrates the balance that policymakers can work to reach.
Those concerns were discussed at a small ecumenical gathering of advocates for the poor in the Upper Senate Park near the Capitol March 20. Sponsored by Faith in Public Life, a faith-based advocacy group, the gathering focused on the miracle of the loaves and fishes.
Addressing about 20 people, Sister Simone Campbell, a Sister of Social Service and executive director of Network, challenged the idea that austerity was the right path for the country.
She compared the daily struggles of needy Americans and the debate about the country’s future fiscal path with the miracle of the loaves and fishes witnessed by the throng gathered in Galilee to hear Jesus speak. In the end, she recalled, everyone was fed.
“It was the men who felt it was a miracle. The women knew what it was. They brought the food and they were willing to share,” Sister Simone said.
The miracle can serve as an example for people today, she told Catholic News Service.
“We know as Catholics that there is enough to go around if we share,” she explained. “Jesus Christ in Scripture says, ‘Feed the hungry.’ We’re saying there is enough to go around.”
— By Dennis Sadowski Catholic News Service
Previous articlePope Francis says good priests bring joy, comfort to those in need
Next articleBritish Catholic legislators ask pope to relax priestly celibacy rule
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common_crawl
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Report Highlights New Delays in Iraq Security Handover
WASHINGTON – In another sign of U.S. struggles in Iraq, the target date for putting Iraqi authorities in charge of security in all 18 provinces has slipped yet again, to at least July.
The delay, noted in a Pentagon report to Congress on progress and problems in Iraq, highlights the difficulties in developing Iraqi police forces and the slow pace of economic and political progress in some areas.
It is the second time this year the target date for completing what is known as "Provincial Iraqi Control" has been pushed back. The Pentagon report submitted to Congress on Monday hinted at the possibility of further delays.
The intent is to give the provincial governments control over security in their area as a step toward lessening — and eventually ending — the U.S. security role.
Thus far seven of the 18 provinces have reverted to Iraqi control.
The process has gained relatively little attention in the broader debate in Washington about when and how to get the Iraqis ready to provide their own security so that U.S. forces can begin to leave. That may be in part because some details of the provincial transition process are classified secret.
An independent commission that examined the issue of provincial Iraqi control this summer concluded in a report to Congress on Sept. 4 that the process is too convoluted and an impediment to the overall U.S. goals of speeding the transition to Iraqi control and supporting sovereignty.
"Our current policy of determining when a province may or may not be controlled by its own government reinforces the popular perception of the (U.S.-led) coalition as an occupation force," according to the commission, headed by retired Marine Gen. James Jones. "This may contribute to increased violence and instability."
The commission recommended that all 18 provinces return to Iraqi control immediately. U.S. forces would continue to operate in the areas they are now, in coordination with Iraqi authorities; Iraqi control would mean U.S. troops could transition to less combat-intense roles.
In an interview Wednesday, Jones said he and the other commissioners got the strong impression from Iraqi officials they met in Baghdad this summer that they want full provincial control without further delay.
"The whole process seems to be acting as more of a brake on progress than a help," Jones said. "If the Iraqi government is willing, I think we should be putting as much on them as possible. To have a sovereign government that doesn't control all of its provinces doesn't make a lot of sense to me."
In an Associated Press interview last week, Gen. David Petraeus, the top U.S. commander in Iraq, defended the transition process. It involves a series of detailed reviews and assessments by U.S. and Iraqi officials, culminating with input by Petraeus and the most senior Iraqi government leaders.
Though slow, it is helpful in sorting out problems that stand in the way of a smooth transition, he said.
"It forces people to come to grips with those issues," Petraeus said.
In January, President Bush announced his new strategy for stabilizing Iraq and his decision to send an additional 21,500 U.S. combat troops to Baghdad and to Anbar province. He, said, at the time, that the Iraqi government "plans to take responsibility for security in all of Iraq's provinces by November."
In June the Pentagon informed Congress that the target had slipped to "no later than" next March.
In this month's report, the Pentagon said its "current projection" was that all 18 provinces would move to Iraqi control "as early as" July; that would be eight months later than Bush's original projection.
The Pentagon also hinted at further delays.
"If, for example, violence worsened significantly in any of the provinces yet to transition to (Iraqi control) the likely dates for transition of those provinces would be reevaluated," the report said. It said the main reason for the delays so far is a "lack of capability in the Iraqi police services."
The Pentagon report cited a litany of problems with the police. For example, it said as few as 40 percent of those trained by coalition troops in recent years are still on the job. Also, due to combat loss, theft, attrition and poor maintenance, a "significant portion" of U.S.-issued equipment is now unusable.
Next in line for transition to Iraqi control is Karbala, a small south-central province, by the end of this month, according to the Pentagon report to Congress. It gave no further breakdown of the schedule.
The U.S. commander in northern Iraq, Army Maj. Gen. Benjamin Mixon, had recommended that Ninevah province shift to Iraqi control in August, but that date was pushed back to at least November. The province includes Mosul, the country's third largest city.
Last year, the relatively peaceful southern provinces of Muthanna, Dhi Qar and Najaf were returned to Iraqi security control. In April, Maysan province in the southeast was the fourth to convert.
In May the Kurdish regional government assumed security responsibility for the three provinces that make up the largely peaceful Kurdish region of northern Iraq: Dahuk, Irbil and Sulaimaniyah.
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common_crawl
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Owl and the Sparrow (Annam Films, 2007)
Jam Session (Columbia, 1944)
Boy! What a Girl (Herald Productions, 1947)
Frontline: "The Confessions" (PBS, 2010)
One Step Beyond: "Make Me Not a Witch" (Worldvisio...
LennonNYC (Two Lefts Don't Make a Right Production...
When Andrew Came Home (Hearst Entertainment/Lifeti...
A Christmas Carol (CBS/Desilu, 1954)
West of Shanghai (Warner Bros., 1937)
You'll Find Out (RKO, 1940)
Lennon Naked (Blast! Films/BBC, 2010)
Deep Down (Independent Television Service/Kentucky...
Alice in Wonderland (Paramount, 1933)
Murder on a Honeymoon (RKO, 1935)
Doctor Bull (Fox. 1933)
How the Beatles Rocked the Kremlin (BBC, 2009)
Murder on the Blackboard (RKO, 1934)
The Count of Monte Cristo (Reliance/United Artists...
Cracked Nuts (RKO, 1931)
One Angry Juror (Entertainment One/Lifetime, 2010)...
The Hoodlum (Pickford/First National, 1919)
The Penguin Pool Murder (RKO, 1932)
The Singing Fool (Warner Bros., 1928)
The Devil Is a Sissy (MGM, 1936)
Dressed to Kill (Universal, 1946)
A Bucket of Blood (Alta Vista/American Internation...
Joe Smith, American (MGM, 1942)
The Phantom of the Opera (Universal, 1925)
The Velvet Touch (Independent Artists/RKO, 1948)
House of Wax (Warner Bros., 1953)
Deep Down (Independent Television Service/Kentucky Educational Television, 2009)
by Mark Gabrish Conlan * Copyright (c) 2010 by Mark Gabrish Conlan * All rights reserved
The film was Deep Down, a 2009 documentary about the village of Maytown, Kentucky, located in the middle of the Appalachians, and the Miller Bros. Coal Company's attempt to get permits to do a mountaintop removal mine on top of the hills overlooking Willow Creek Holler. (I'd heard the word "holler" in this context before -- notably in the song "Coal Miner's Daughter," in which Loretta Lynn proudly proclaims herself as having been born in "Butcher Holler" -- but I didn't realize what it meant: it's simply Appalachia-speak for "canyon.")
The film was directed by Sally Rubin and Jen Gilomen for the PBS Independent Lens series, and from the publicity surrounding it I expected it to be considerably nastier and more confrontational than it turned out; instead it's a kind of lyrical poem to the beauty of the Appalachians and the people who live there and a serious, refreshingly non-propagandistic meditation on the nature of capitalism and how it affects people who have always lived close to the land and have been largely dependent on coal for generations. The sympathies of the filmmakers are clearly with Barbara May, who organizes her fellow Maytown residents (was the town named after her family? It's certainly possible) to oppose the mine by asking the Kentucky land management board for a petition declaring the Willow Creek area "lands unsuitable for mining," but Rubin and Gilomen are honest enough to show the "other side" as well.
Their other central character is Terry Ratliff, who lives in a house he built himself on the other side of the holler from Barbara May and who is so defiantly independent he boasts that there isn't a single level or plumb wall or beam in his self-constructed home; indeed, he tells us that he resents being called a "carpenter" since he's deliberately avoided making anything level, something that obsesses any real professional carpenter. He takes the filmmakers to a hilltop, radiantly beautiful in the orange of an Appalachian sunset and covered with ample growths of native plants, and announces that this was a mountain that was flattened 30 years before by a mining company and, as they (and we) can see, it grew back just fine.
Terry seems inclined to lease his land to the mining company at first, but gradually his doubts grow as he ponders the horror stories he's heard from other people in other communities who signed coal-company leases -- and then found themselves either paid much less than they were promised by the salespeople or not paid anything at all (apparently it's a common dodge in the area for a coal company to declare bankruptcy just as the mine is played out, and the new company that takes over to announce that they're not bound by the debts of the predecessors and therefore the people have given their land away for nothing). Ultimately he doesn't lease his land, and we se a sequence of his daughter Carly marrying a young man named Steve (we never see Carly's mother or learn what happened to her), whom we're told is as ornery and independent as his new father-in-law.
At first I was disappointed that Deep Down wasn't more confrontational -- the only time we see anyone who works for Miller Bros. (which we learn is a subsidiary of a much larger company that takes over its assets after it declares bankruptcy when their request for a permit to mine by mountaintop removal is denied) is at the hearing over the land unsuitable for mining petition, and we also see some fascinating sequences in which, this being the middle of the "Bible Belt," both pro- and anti-mining speakers quote the Bible and claim that God is on their side -- but as it wound on I found myself much more sympathetic to the softer, warmer, more lyrical approach Rubin and Gilomen actually used and quite impressed by the film.
Even the use of bluegrass music as background, which I found almost abominably cliched at first, had a basis in fact: it turned out both May and Ratliff were members of a bluegrass music school in the area and the scenes in which people on opposite sides of the mining controversy nonetheless sit down together to play guitars, banjos and fiddles in the traditional style of their forebears just add to the overall haunting quality of this surprisingly lyrical, understated film that, like the pioneering cinema verite movies of the early 1960's, doesn't use a third-person narration (when Rubin and Gilomen need to provide us with exposition to help us understand what is going on, they use one of their interviews with local people to supply it) and therefore doesn't impose a blatant editorial "we" on the material.
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common_crawl
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Obama to Israel — Time Is Running Out
Mar 2, 2014 2:00 PM ET
By Jeffrey Goldberg
When Israeli Prime Minister Benjamin Netanyahu visits the White House tomorrow, President Barack Obama will tell him that his country could face a bleak future — one of international isolation and demographic disaster — if he refuses to endorse a U.S.-drafted framework agreement for peace with the Palestinians. Obama will warn Netanyahu that time is running out for Israel as a Jewish-majority democracy. And the president will make the case that Netanyahu, alone among Israelis, has the strength and political credibility to lead his people away from the precipice.
In an hourlong interview Thursday in the Oval Office, Obama, borrowing from the Jewish sage Rabbi Hillel, told me that his message to Netanyahu will be this: “If not now, when? And if not you, Mr. Prime Minister, then who?” He then took a sharper tone, saying that if Netanyahu “does not believe that a peace deal with the Palestinians is the right thing to do for Israel, then he needs to articulate an alternative approach.” He added, “It’s hard to come up with one that’s plausible.”
Unlike Netanyahu, Obama will not address the annual convention of the American Israel Public Affairs Committee, a pro-Israel lobbying group, this week — the administration is upset with Aipac for, in its view, trying to subvert American-led nuclear negotiations with Iran. In our interview, the president, while broadly supportive of Israel and a close U.S.-Israel relationship, made statements that would be met at an Aipac convention with cold silence.
Obama was blunter about Israel’s future than I’ve ever heard him. His language was striking, but of a piece with observations made in recent months by his secretary of state, John Kerry, who until this interview, had taken the lead in pressuring both Netanyahu and the Palestinian leader, Mahmoud Abbas, to agree to a framework deal. Obama made it clear that he views Abbas as the most politically moderate leader the Palestinians may ever have. It seemed obvious to me that the president believes that the next move is Netanyahu’s.
“There comes a point where you can’t manage this anymore, and then you start having to make very difficult choices,” Obama said. “Do you resign yourself to what amounts to a permanent occupation of the West Bank? Is that the character of Israel as a state for a long period of time? Do you perpetuate, over the course of a decade or two decades, more and more restrictive policies in terms of Palestinian movement? Do you place restrictions on Arab-Israelis in ways that run counter to Israel’s traditions?”
During the interview, which took place a day before the Russian military incursion into Ukraine, Obama argued that American adversaries, such as Iran, Syria and Russia itself, still believe that he is capable of using force to advance American interests, despite his reluctance to strike Syria last year after President Bashar al-Assad crossed Obama’s chemical-weapons red line.
“We’ve now seen 15 to 20 percent of those chemical weapons on their way out of Syria with a very concrete schedule to get rid of the rest,” Obama told me. “That would not have happened had the Iranians said, ‘Obama’s bluffing, he’s not actually really willing to take a strike.’ If the Russians had said, ‘Ehh, don’t worry about it, all those submarines that are floating around your coastline, that’s all just for show.’ Of course they took it seriously! That’s why they engaged in the policy they did.”
I returned to this particularly sensitive subject. “Just to be clear,” I asked, “You don’t believe the Iranian leadership now thinks that your ‘all options are on the table’ threat as it relates to their nuclear program — you don’t think that they have stopped taking that seriously?”
Obama answered: “I know they take it seriously.”
How do you know? I asked. “We have a high degree of confidence that when they look at 35,000 U.S. military personnel in the region that are engaged in constant training exercises under the direction of a president who already has shown himself willing to take military action in the past, that they should take my statements seriously,” he replied. “And the American people should as well, and the Israelis should as well, and the Saudis should as well.”
I asked the president if, in retrospect, he should have provided more help to Syria’s rebels earlier in their struggle. “I think those who believe that two years ago, or three years ago, there was some swift resolution to this thing had we acted more forcefully, fundamentally misunderstand the nature of the conflict in Syria and the conditions on the ground there,” Obama said. “When you have a professional army that is well-armed and sponsored by two large states who have huge stakes in this, and they are fighting against a farmer, a carpenter, an engineer who started out as protesters and suddenly now see themselves in the midst of a civil conflict — the notion that we could have, in a clean way that didn’t commit U.S. military forces, changed the equation on the ground there was never true.”
He portrayed his reluctance to involve the U.S. in the Syrian civil war as a direct consequence of what he sees as America’s overly militarized engagement in the Muslim world: “There was the possibility that we would have made the situation worse rather than better on the ground, precisely because of U.S. involvement, which would have meant that we would have had the third, or, if you count Libya, the fourth war in a Muslim country in the span of a decade.”
Obama was adamant that he was correct to fight a congressional effort to impose more time-delayed sanctions on Iran just as nuclear negotiations were commencing: “There’s never been a negotiation in which at some point there isn’t some pause, some mechanism to indicate possible good faith,” he said. “Even in the old Westerns or gangster movies, right, everyone puts their gun down just for a second. You sit down, you have a conversation; if the conversation doesn’t go well, you leave the room and everybody knows what’s going to happen and everybody gets ready. But you don’t start shooting in the middle of the room during the course of negotiations.” He said he remains committed to keeping Iran from obtaining nuclear weapons and seemed unworried by reports that Iran’s economy is improving.
On the subject of Middle East peace, Obama told me that the U.S.’s friendship with Israel is undying, but he also issued what I took to be a veiled threat: The U.S., though willing to defend an isolated Israel at the United Nations and in other international bodies, might soon be unable to do so effectively.
“If you see no peace deal and continued aggressive settlement construction — and we have seen more aggressive settlement construction over the last couple years than we’ve seen in a very long time,” Obama said. “If Palestinians come to believe that the possibility of a contiguous sovereign Palestinian state is no longer within reach, then our ability to manage the international fallout is going to be limited.”
We also spent a good deal of time talking about the unease the U.S.’s Sunni Arab allies feel about his approach to Iran, their traditional adversary. I asked the president, “What is more dangerous: Sunni extremism or Shia extremism?”
I found his answer revelatory. He did not address the issue of Sunni extremism. Instead he argued in essence that the Shiite Iranian regime is susceptible to logic, appeals to self-interest and incentives.
“I’m not big on extremism generally,” Obama said. “I don’t think you’ll get me to choose on those two issues. What I’ll say is that if you look at Iranian behavior, they are strategic, and they’re not impulsive. They have a worldview, and they see their interests, and they respond to costs and benefits. And that isn’t to say that they aren’t a theocracy that embraces all kinds of ideas that I find abhorrent, but they’re not North Korea. They are a large, powerful country that sees itself as an important player on the world stage, and I do not think has a suicide wish, and can respond to incentives.”
This view puts him at odds with Netanyahu’s understanding of Iran. In an interview after he won the premiership, the Israeli leader described the Iranian leadership to me as “a messianic apocalyptic cult.”
I asked Obama if he understood why his policies make the leaders of Saudi Arabia and other Arab countries nervous: “I think that there are shifts that are taking place in the region that have caught a lot of them off guard,” he said. “I think change is always scary.”
Below is a complete transcript of our conversation. I’ve condensed my questions. The president’s answers are reproduced in full.
President Barack Obama participates in an interview with Jeff Goldberg in the Oval Office, Feb. 27, 2014. (Official White House Photo by Pete Souza)
israel, OBAMA, USA
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common_crawl
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\section{Introduction}
Shortly after Bohr and Mottelson's development of the collective model~\cite{BM1}, Morinaga showed that the spectroscopy of a number of nuclei in the $p$-shell could be interpreted
in terms of rotational bands~\cite{Morinaga56}. Perhaps one of the most obvious examples is that of $^{8}$Be, as evident from the ground state rotational band and its enhanced $B(E2)$ transition probability~\cite{nndc}. The strong $\alpha$ clustering in $^{8}$Be naturally suggests that deformation degrees of freedom play a role in the structure of the Be isotopes, a topic that has been extensively discussed in the literature (see Ref.~\cite{VonO} for a review).
Turning from a collective model to a Shell Model picture, very nearby to $^{8}$Be, the lightest example of a so-called Island of Inversion~\cite{Poves87,Warburton90} is that at $N$ = 8, where the removal of $p_{3/2}$ protons from $^{14}$C results in a quenching of the $N$=8 shell gap~\cite{Talmi, Sor08, Heyde1, Otsuka20}. This is evinced in many experimental observations, including the sudden drop of the $E(2^+)$ energy in $^{12}$Be relative to the neighboring even-even isotopes, and the change of the ground state of $^{11}$Be from the expected 1/2$^-$ to the observed positive parity 1/2$^{+}$ state.
Connecting the collective and single-particle descriptions, Bohr and Mottelson~\cite{BM} actually proposed that the shell inversion could be explained as a result of the convergence of the up-sloping [101]$\tfrac{1}{2}$ and down-sloping [220]$\tfrac{1}{2}$ Nilsson~\cite{Sven, Rag} levels with deformation as seen in Fig.~\ref{fig:fig1}. Building on these arguments, Hamamoto and Shimoura~\cite{Ikuko} explained energy levels and available electromagnetic data on $^{11}$Be and $^{12}$Be in terms of single-particle motion in a deformed potential. It is remarkable that the concept of a deformed rotating structure appears to be applicable, even when the total number of nucleons is small as in the case of light nuclei. In fact, level energies and electromagnetic properties that follow the characteristic rotational patterns emerge for $p$-shell nuclei in {\sl ab-initio} no-core configuration interaction calculations~\cite{Mark}.
\begin{figure}[ht!]
\centering\includegraphics[trim=40 0 0 60,clip,width=7cm,angle=90]{nilsson.pdf}
\caption{(Color online) Nilsson levels relevant for the structure of negative parity neutron states in $^{11}$Be and $^{11}$B (solid lines). Also shown (dashed lines) are the levels originating from the $d_{5/2}$ spherical orbital. The shaded band indicates the anticipated range of $\epsilon_2$ deformation for these nuclei and the horizontal lines the approximate Fermi levels of the odd neutron (blue) and the odd proton (red). Energies are in units of the harmonic oscillator frequency, $\hbar\omega_0$. }
\label{fig:fig1}
\end{figure}
In a series of articles we have recently applied the collective model to understand the structure of nuclei in the $N$=8 Island of Inversion and spectroscopic factors obtained from direct nucleon addition and removal reactions~\cite{aom1,aom2,aom3}. The mean-field description seems to capture the main physics ingredients and provides a satisfactory explanation of spectroscopic data, in a simple and intuitive manner.
Here we extend this approach to discuss the results of a recent study of the $^{12}$B($d$,$^3$He)$^{11}$Be reaction~\cite{chen} in terms of the Particle Rotor Model (PRM)~\cite{Rag,Larsson,Semmes}. Estimates of spectroscopic factors in the strong coupling limit, given in Ref.~\cite{chen}, underestimated the experimental data and pointed out that Coriolis effects in the structure of the $1^+$ ground state in $^{12}$B should be taken into account, for which the PRM framework is the one of choice. We present this analysis here.
\begin{figure}
\centering
\includegraphics[width=6cm]{LevelSchemePRM.pdf}
\caption{ Left: the experimental level scheme of $^{11}$B from Ref.~\cite{nndc}. Right: Results of the PRM calculations. Energies are in keV.}
\label{fig:fig2}
\end{figure}
\section{The ground state of $^{12}$B}
In order to assess the structure of the ground state of $^{12}$B, we consider first the odd-neutron and odd-proton low-lying negative parity states in $^{11}$Be and $^{11}$B respectively. Considering $^{10}$Be as a core, an inspection of the Nilsson diagram~\cite{Sven} in Fig.~\ref{fig:fig1} suggests that for $N = 7$ the last neutron is expected to occupy the [101]$\tfrac{1}{2}$ level, and for $Z = 5$ the last proton will occupy the [101]$\tfrac{3}{2}$. We have used the standard parameters, $\kappa=0.12$ and $\mu=0.0$~\cite{Sheline}, and adopt a deformation, $\epsilon_2 \approx $ 0.45 ($\beta_2 \approx 0.6$), where the crossing of the [220]$\tfrac{1}{2}$ and [101]$\tfrac{1}{2}$ levels is expected to occur, explaining the inversion of the $1/2^{+}$ and the $1/2^{-}$ states in $^{11}$Be.
We have previously discussed the positive-parity states in $^{11}$Be as arising from the strongly coupled [220]$\tfrac{1}{2}$ state~\cite{aom3}, an assignment supported by the calculated gyromagnetic factor $g_K=-2.79$, with decoupling and magnetic-decoupling parameters for the ground state of $a=1.93$ and $b=-1.27$ respectively. The low-lying negative parity states, namely the $1/2^-_1$, $3/2^-_1$ and $5/2^-_1$ states can be assigned to a $K=1/2^-$ strongly coupled band built on the neutron [101]$\tfrac{1}{2}$ level, with a decoupling parameter, $a = 0.5$ in line with the Nilsson predictions. Further, the $3/2^-_2$ originates from a neutron hole in the [101]$\tfrac{3}{2}$ level.
The case of $^{11}$B is somewhat different and more complex, requiring an explicit consideration of Coriolis coupling. An attempt to fit the energies of the {\sl yrast} negative parity states, $3/2^-_1$, $5/2^-_1$, $7/2^-_1$ ..., to leading order:
\begin{equation} \label{eq:1}
E(I) = E_K + AI(I+1)+ BI^2(I+1)^2+...
\end{equation}
requires an additional term~\cite{BM} arising from the Coriolis interaction that induces $\Delta K=\pm 2K$ mixing:
\begin{equation} \label{eq:2}
\Delta E_{rot} = (-1)^{I+K} A_{2K} \frac{(I+K)!}{(I-K)!},
\end{equation}
giving $A=$ 978 keV, $B=$ -17 keV and $A_3 \approx$ 20 keV.
Therefore we carried out a PRM calculation~\cite{Semmes} -- the results, shown in Fig.~\ref{fig:fig2}, are in good agreement with the experimental level scheme. Here we briefly discuss the physical inputs to the PRM calculation. We include the three orbits in Fig.~\ref{fig:fig1} with the Fermi level, $\lambda$, and the pairing gap, $\Delta$, obtained from a BCS calculation using a coupling constant, $G=1.9$ MeV. The solution gives $\lambda=$45.73 MeV and $\Delta=$ 3.3 MeV. The adjusted rotational constant of the core\footnote{In the PRM, we do not include a $BI^2(I+1)^2$ term in the rotational energy but set the $E(2^+)_{core}=A2(2+1)+B(2(2+1))^2$ from Eq.~\ref{eq:1}.} corresponds to a moment of inertia, $ \mathscr{I}$= 0.57 $\hbar^2$/MeV, approximately 60\% of the rigid body value and consistent with the Migdal estimate~\cite{Migdal} for A=11 and the deformation and pairing parameters above. A fit of Eq. (2) to the PRM results gives $A_3 \approx$ 25 keV.\\
The Coriolis $K$-mixing in $^{11}$B gives rise to wave-functions of the form
\begin{equation}
\psi_I = \sum_{K} \mathcal{A}_{IK} | I K \rangle
\label{eq:eq3}
\end{equation}
in the strong-coupled basis spanned by the intrinsic proton states [110]$\tfrac{1}{2}$, [101]$\tfrac{3}{2}$, and [101]$\tfrac{1}{2}$. The percent contributions (squared amplitudes) of each intrinsic proton state for the states shown in Fig.~\ref{fig:fig2} are given in Table~\ref{tab:tab1}. The results of the PRM calculations indicate a moderate $K$-mixing for the $3/2_1^-$, $5/2_1^-$ and $7/2_1^-$, and the $3/2_2 ^-$ states with dominant components of the [101]$\tfrac{3}{2}$ and [110]$\tfrac{1}{2}$ Nilsson levels in each case. The $1/2_1^-$ is essentially a pure [101]$\tfrac{1}{2}$ state. Note that the form of the Coriolis matrix elements favors the mixing of the Nilsson levels with $p_{3/2}$ parentage. It is also worthwhile noting that due to the fact that the Fermi level of the odd proton $^{11}$B is lower than the odd neutron in $^{11}$Be, there is no parity inversion in $^{11}$B and the lowest positive parity state, $1/2^+$, lies at $\approx$ 4.6 MeV relative to the $1/2^-$.
In addition to the reproduction of the energy levels, the calculated magnetic moment of the ground state is $\mu_{3/2^-}=$ 2.66 $\mu_N$ to be compared to the experimental value of 2.6886489(10) $\mu_N$~\cite{nndc2}.
There is, however, an intriguing discrepancy with the measured $Q_{3/2^-}= 0.04065(26)$eb~\cite{nndc2,11BQ}, which is consistent with the leading order collective model estimate of 0.04 eb, but the PRM result of 0.028 eb
is smaller due to the mixing of the two $I^\pi=3/2^-$ states with $K^\pi=1/2^-$ and $K^\pi=3/2^-$ that have $Q$'s of opposite signs. While one may be tempted to explain this with a larger deformation, $\epsilon_2 \approx $ 0.60~(as in Ref.~\cite{Ikuko}), it would be at the expense of losing the agreement in the energy levels. This is also the case for the mirror nucleus, $^{11}$C. We do not have an explanation for this discrepancy except to speculate that, perhaps, the deformation is decreasing with spin and the PRM reflects an average.
The Titled-Axis Cranking model~\cite{Stefan} results for the Be isotopes~\cite{Qi} may support this kind of argument.
It is also interesting to point out that a recent {\sl ab initio} no-core shell model study~\cite{Petr} of $^{10–14}$B isotopes with realistic $NN$ forces predicts $Q_{3/2^-}$ in the range 0.027 - 0.031 eb (depending on the interaction used), very close to our estimate. \\
\begin{table}
\caption{Coriolis mixing amplitudes in the PRM calculations for $^{11}$B.}
\bigskip
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c|c|ccc}
\hline\hline
$I^\pi$& Energy& & $\mathcal{A}_{IK}^2$ [ \% ] \\
& [MeV] & [110]$\tfrac{1}{2}$ & [101]$\tfrac{3}{2}$ & [101]$\tfrac{1}{2}$ \\
\hline
$\frac{3}{2}^{-}$& 0.00 & 18 & 80 & 2\\
$\frac{1}{2}^{-}$& 2.12 & 4 & 0 & 96\\
$\frac{5}{2}^{-}$& 4.21 & 6 & 90 & 4\\
$\frac{3}{2}^{-}$& 5.34 & 81 & 19 &0\\
$\frac{7}{2}^{-}$& 6.66 & 45 & 52 & 3\\
\hline\hline
\end{tabular}
\label{tab:tab1}
\end{table}
In contrast to heavy nuclei, light nuclei are more vulnerable to changes in deformation just by the addition of one particle. Relevant to our discussions is the fact that $^{12}$C is oblate~\cite{12CQ0} and one question that comes to mind is why not consider $^{11}B$ as a hole coupled to an oblate core? In fact, in an early study~\cite{Clegg}, the energy level scheme of $^{11}B$ was described in the collective model with a deformation $\epsilon_2 \approx -0.4$ corresponding to the left side of Nilsson levels shown in Fig.~\ref{fig:fig1}. We have carried out PRM calculations for oblate deformations and obtained an agreement similar to that in Fig.~\ref{fig:fig2} for $\epsilon_2 \approx -0.35$
and $ \mathscr{I}$= 0.50 $\hbar^2$/MeV. However, this solution gives $Q_{3/2^-} = -0.0063$ eb in clear disagreement with experiment. In looking at the positive parity states, we find that the lowest excitation corresponds to a $5/2^+$ at
$\approx 10$ MeV from the $3/2^-$ ground state, also inconsistent with the experimental level scheme. \\
Based on the discussion above, we adopt the prolate results to assess the structure of the ground state
in $^{12}$B, as a neutron and a proton coupled to the $^{10}$Be core. The ground state will result from the coupling of the structures discussed previously, namely a neutron 1/2$^{-}$ based on the [101]$\tfrac{1}{2}$ intrinsic neutron level, and a 3/2$^{-}$ proton state as described in Table~\ref{tab:tab1} , which will give rise to 2$^+$ and 1$^+$ states with parallel or anti-parallel coupling respectively.
Following the empirical Gallagher-Moszkowski rule~\cite{GM58}, the lower member of the doublet corresponds to the $1^+$
built from the $K^\pi =0^+, 1^+$ components in agreement with the experimental observations. Further qualitative support comes from the lowest negative parity states, $2^-$ and $1^-$ expected from the coupling to the neutron [220]$\tfrac{1}{2}$. In the oblate side, a $1^-$ to $4^-$ spin multiplet should result from the coupling to the [202]$\tfrac{5}{2}$ level.
Additionally, the measured~\cite{nndc2} magnetic moment, $\mu_{1^-}=$ 1.00306(15) $\mu_N$, and quadrupole moment, $Q_{1^+}=0.0134(14)$ eb, that compare well with leading order estimates, $\mu_{1^-}=$ 0.97 $\mu_N$, and $Q_{1^+}=0.017$ eb.
With the ingredients above we proceed to calculate the $p$-wave proton removal strengths, in terms of the Coriolis mixing amplitudes for the $^{12}$B ground state.
\section{Spectroscopic Factors}
We apply the formalism reviewed in Ref.~\cite{Elbek} to a proton-pickup reaction, such as $(d,^3$He). In the strong coupling limit, the spectroscopic factors ($S_{i,f}$) from an initial ground state $|I_iK_i\rangle$ to a final state $| I_f K_f \rangle$ can be written in terms of the Nilsson amplitudes~\cite{Elbek}:
\begin{equation}
\begin{split}
\theta_{i, f }(j\ell,K)& = \langle I_{i}j K \Omega_\pi | I_{f} 0\rangle C_{j,\ell} \langle\phi_f|\phi_i\rangle \\
S_{i, f }& = \theta_{i, f }^2(j\ell,K)
\end{split}
\label{eq:eq4}
\end{equation}
where $\langle I_{i}j K \Omega_\pi | I_{f} 0\rangle$ is a Clebsch-Gordan coefficient, $C_{j,\ell}$ is the Nilsson wavefunction amplitude, and $\langle\phi_f|\phi_i\rangle$ is the core overlap between the initial and final states, which we assume to be 1.
Due to the effects of Coriolis coupling discussed previously Eq.~\ref{eq:eq4} is generalized to~\cite{Casten71}:
\begin{equation}
S_{i, f} (j\ell) = \big( \sum_{K} \mathcal{A}_{IK} \theta_{i, f}(j\ell,K) \big)^2
\label{eq:eq5}
\end{equation}
Following from the results in Table~\ref{tab:tab1}, where the 3/2$^-$ ground state is dominated by two contributing Nilsson orbitals, we only consider the [110]$\tfrac{1}{2}$ and [101]$\tfrac{3}{2}$ proton levels which, in the spherical $|j,\ell\rangle$ basis, have the wavefunctions:
\begin{equation}
|[110]\tfrac{1}{2}\rangle= -0.34 |p_{1/2}\rangle+ 0.94|p_{3/2}\rangle
\label{eq:eq6}
\end{equation}
\begin{equation}
|[101]\tfrac{3}{2}\rangle= |p_{3/2}\rangle
\label{eq:eq7}
\end{equation}
When applied to the case of $^{12}$B, the PRM hamiltonian for the $1^+$ ground state is a 3x3 matrix in the basis:
\begin{align}
|1\rangle =& |\nu[101]\tfrac{1}{2}\otimes \pi[110]\tfrac{1}{2}\rangle_{K=0} \nonumber \\
|2\rangle =& |\nu[101]\tfrac{1}{2}\otimes \pi[110]\tfrac{1}{2}\rangle_{K=1} \\
| 3\rangle =& |\nu[101]\tfrac{1}{2}\otimes \pi[101]\tfrac{3}{2}\rangle_{K=1} \nonumber
\end{align}
giving a wavefunction of the form:
\begin{align}
|^{12}\textrm{B},1^+ \rangle_\textrm{g.s.} =~&\mathscr{A}_{1}|1\rangle + \mathscr{A}_{2}| 2\rangle
+ \mathscr{A}_{3}| 3 \rangle
\label{eq:eq9}
\end{align}
The amplitudes, $\mathscr{A}_{1-3}$
were fit using a least-squares minimization to the experimental spectroscopic factor data, yielding $\mathscr{A}_1$ = -0.60(3), $\mathscr{A}_2$ = 0.70(3) and $\mathscr{A}_3$ = 0.40(4) given by the normalization condition $\mathscr{A}_1^2 + \mathscr{A}_2^2$ +$\mathscr{A}_3^2$=1. The derived amplitudes confirm that Coriolis mixing is required to explain the experimental data, reflecting the PRM results for the $3/2^-$ band in $^{11}$B, since the [101]$\tfrac{1}{2}$ neutron can be seen to act as a spectator. The Coriolis effects appear to be somewhat larger in $^{12}$B, which may suggest a core with smaller deformation and a reduced momenta of inertia, both favoring the increased mixing.
We note that within our framework, pickup to the $3/2_2^-$ is not possible since a neutron hole
in the [101]$\tfrac{3}{2}$ level is not present in the ground state of $^{12}$B. In any case, a contribution to the $3/2_2^- + 5/2_1^-$ doublet due to $K-$mixing in $^{11}$Be is expected to be quite small.
The calculated relative $S_{if}$ in the strong coupling limit and the PRM are compared to the experimental data in Table~\ref{tab:tab2}, which also includes those for the $^{12}$C($p,2p)^{11}$B reaction~\cite{Panin16}. As already mentioned, the spectroscopic factors of the $3/2_1^-$ and $5/2_1^-$ states were underestimated in the strong coupling limit and the inclusion of Coriolis coupling appears to solve the discrepancy, bringing the collective model predictions in line with those of the shell-model and the {\sl ab-initio} Variational Monte Carlo results discussed in Ref.~\cite{chen}. It would be of interest if the study of Ref.~\cite{Petr} could be extended to obtain spectroscopic factors for the reactions in Table~\ref{tab:tab2}.
\begin{table}[ht]
\caption{Comparison between the experimental $\ell=1$ proton removal spectroscopic factors, the Nilsson strong coupling limit and the PRM results. Note that these are relative to the transitions normalized to 1 and, as such, quenching effects largely cancel.}
\bigskip
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{c|c|c|ccc}
\hline\hline
Initial & Final & Energy & & $S_{i,f}$&\\
State & State & [MeV] & Exp & Strong Coupling &Coriolis\footnote{Note that for $^{12}$B,
having two data points and two unknowns, the minimization solution reproduces the data exactly.}\footnote{For $^{12}$C these are based on the $^{11}$B amplitudes in Table.~\ref{tab:tab2} } \\
\hline
$^{12}$B & $^{11}$Be& &\\
1$^{-}_1$ & $\frac{1}{2}^{-}$& 0.32&1 &1&1\\
& $\frac{3}{2}^{-}$& 2.35 &2.6(10)&0.8&2.6\\
& $\frac{5}{2}^{-}$ &3.89 &1.7(6) &0.2&1.7\\
\hline
$^{12}$C & $^{11}$B& & & \\
$ 0^+_1$ & $\frac{3}{2}^{-}$ & 0.00&1&1&1\\
& $\frac{1}{2}^{-}$ & 2.12&0.12(2)&0.9&0.1\\
& $\frac{3}{2}^{-}$ & 5.02&0.10(2)&0.9&0.3\\
\hline\hline
\end{tabular}
\label{tab:tab2}
\end{table}
\section{Conclusion}
We have analyzed spectroscopic factors extracted from the $^{12}$B($d$,$^3$He)$^{11}$Be proton-removal reaction in the framework of the collective model.
The PRM quantitatively explains the available structure data in $^{11}$Be and $^{11}$B and provides clear evidence of Coriolis coupling in the ground state of $^{12}$B.
The resulting $K$-mixing in the wave-function is key to understand the experimental (relative) spectroscopic factors, which are underestimated in the strong coupling limit.
The amplitudes, empirically adjusted to reproduce the data, are in agreement with the PRM expectations. An application of our phenomenological description to the structure of $^{12}$B (Eqs.~8 and \ref{eq:eq9}) with the two-particle plus rotor model would be an interesting extension to explore.
\begin{acknowledgments}
We would like to thank Profs. Ikuko Hamamoto and Stefan Frauendorf for enlightening discussions on the Nilsson and Particle Rotor Models. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Contracts No. DE-AC02-05CH11231 (LBNL)
and DE-AC02-06CH11357 (ANL).
\end{acknowledgments}
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{'timestamp': '2020-12-10T02:02:38', 'yymm': '2012', 'arxiv_id': '2012.04722', 'language': 'en', 'url': 'https://arxiv.org/abs/2012.04722'}
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"Board of Distinction\nSchool Funding System\nThe Federal Way School Board Members are your official(...TRUNCATED)
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"Design Book List: October 2015\nOctober 6, 2015 by Kathryn Heine Books\nThere are plenty of new des(...TRUNCATED)
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"Paul Weinberg\nPaul Weinberg has worked in journalism most of his life, primarily as a freelance wr(...TRUNCATED)
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common_crawl
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"How to Feel Good About Your Body (It helps to like other women!)\nFound on Psychology Today\nWritte(...TRUNCATED)
| "{\"pred_label\": \"__label__cc\", \"pred_label_prob\": 0.507014811038971, \"wiki_prob\": 0.49298518(...TRUNCATED)
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common_crawl
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in Data Studio
RedPajama-1B-Weighted
A canonical 1 Billion token weighted subset of the RedPajama-Data-1T dataset.
Dataset Description
This is a strict, downsampled version of the RedPajama-10B-Weighted dataset. It maintains the exact domain distributions of the full 1T dataset, resized to a lightweight 1 Billion token footprint.
This dataset is ideal for:
- Rapid Prototyping: Train small models or debug pipelines in minutes rather than days.
- Reference Baselines: Use a standard, well-defined subset for comparative benchmarks.
- Educational Use: Explore the properties of large-scale pretraining data on consumer hardware.
Dataset Details
- Total Tokens: ~1,000,000,000 (1 Billion)
- Parent Source: krisbailey/RedPajama-10B-Weighted
- Original Source: togethercomputer/RedPajama-Data-1T
- Language: English
- Format: Apache Parquet
- Producer: Kris Bailey
Motivation
While the 10B subset is manageable, sometimes you need something even faster. A 1 Billion token dataset is the "Goldilocks" size for many initial experiments—large enough to train a statistically significant small language model (e.g., TinyLlama size) but small enough to download and process on a laptop.
We created this by strictly downsampling the 10B dataset to ensure that the distribution remains consistent with the larger parent datasets.
Dataset Creation Process
1. Source Selection
We utilized the RedPajama-10B-Weighted dataset as the source. This parent dataset was already constructed via weighted interleaving of the original RedPajama corpus.
2. Global Shuffling
The 10B dataset was globally shuffled (Seed: 43) to ensure that selecting the first $N$ tokens results in a random, representative sample, rather than a temporal slice.
3. Truncation
We selected the first 1 Billion tokens from the shuffled stream.
4. Verification
We verified that the final subset retains the correct proportional mix of CommonCrawl, C4, GitHub, etc., matching the target distribution.
Composition
| Subset | Weight | Approx. Tokens |
|---|---|---|
| CommonCrawl | 74.16% | ~741.6 M |
| C4 | 14.78% | ~147.8 M |
| GitHub | 4.98% | ~49.8 M |
| ArXiv | 2.36% | ~23.6 M |
| Wikipedia | 2.03% | ~20.3 M |
| StackExchange | 1.69% | ~16.9 M |
Usage
from datasets import load_dataset
# Load the 1B weighted subset
ds = load_dataset("krisbailey/RedPajama-1B-Weighted", split="train")
print(ds)
Citation
If you use this dataset, please cite the original RedPajama work:
@software{together2023redpajama,
author = {Together Computer},
title = {RedPajama: An Open Source Recipe to Reproduce LLaMA training dataset},
month = April,
year = 2023,
url = {https://github.com/togethercomputer/RedPajama-Data}
}
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