Dataset Preview
The full dataset viewer is not available (click to read why). Only showing a preview of the rows.
The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 41
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 33480)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 41
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
datasets.exceptions.DatasetGenerationError: An error occurred while generating the datasetNeed help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
text
string | meta
dict |
|---|---|
\section{Introduction \label{sec:intro}}
Bitcoin, the first-ever created cryptocurrency, stems from a proposal to bypass the established financial system to make peer-to-peer payments. Such initiative, born in the aftermath of the financial crisis of 2007-2008, was funneled using an anonymously posted white paper \citep{Nakamoto}. The initial popularity of Bitcoin came from the libertarian point of view, advocating for a competition of this peer-to-peer system with fiat money to overcome national boundaries. Despite its impracticability as a means of payment\footnote{According to \cite{coinmap}, there are only 16614 venues (cafeterias, groceries, ATMs, etc.) in the world that accept Bitcoin as means of payment.}, market participants soon acknowledged it as a new investment instrument. Popularity has been gaining momentum and stimulating the creation of new alternative cryptocurrencies (altcoins). Litecoin was created in 2011, Ripple in 2013, Dash, NEO, and Monero in 2014, to name a few. As of July 2020, there are more than 5000 cryptocurrencies, with a total market capitalization of \$B 344. Even though Bitcoin is still the leading player, accounting for 60\% of the market capitalization, it has been losing market participation, especially since June 2017 \citep{coinmarketcap}.
Cryptocurrencies have been consolidated as an alternative investment to traditional assets. Considering the increase in the number of coins, some cryptocurrency development companies begin to design cryptocurrency indices to monitor the evolution of the market. Also, some investors have started considering investing in portfolios predominantly constituted by different cryptocurrencies. In light of these recent market developments, this paper aims at examining the dynamic market linkages in cryptocurrencies for both returns and volatility. We believe our findings have important implications for active diversification strategies in portfolios consisting of several cryptocurrencies, and for prudential regulation regarding the stability of the market.
The paper tackles these questions by studying the evolution of return and volatility linkages across cryptocurrencies since 2015. The contribution to the existing literature is multiple. First, we employ an extended sample that includes recent data up to July 2020, thus covering recent events that might have largely affected these markets, such as the COVID-19 pandemic. Second, we adopt different econometric methodologies to cross-check the relevance of our findings. Third, we assess the linkages across frequency-ranges, which allow us to distinguish whether the transmitted shocks across cryptocurrencies have short or long-run effects. This distinction is crucial to interpret connectedness in terms of systemic risk because market participants have different preferences over trading horizons.\footnote{Consider, for example, that cryptocurrencies were tight at high frequencies only. In such a situation, transmitted shocks are not persistent, having short-term effects only. As a result, interdependencies would not be much of an issue for an agent looking for long-run profits but would matter for a short-term trader.}
The rest of the paper is organized as follows: section \ref{sec:literature} conducts a brief literature review; section \ref{sec:methods} outlines the methodology used in the paper; section \ref{sec:data} describes data and discusses the main results; finally, section \ref{sec:conclusions} summarize the main conclusions.
\section{Literature review \label{sec:literature}}
The early financial literature on cryptocurrencies mainly focuses on the assessment of the informational efficiency of Bitcoin. \cite{Urquhart2016} employs autocorrelation, runs, and variance ratio tests over 2010 -2016 and finds that although Bitcoin showed signs of inefficiency in the initial period of 2010-2013, it evolved towards a more efficient market later on. Shortly afterward, \cite{Bariviera2017} shows that although returns have become more efficient over time, volatility still exhibits substantial persistence. \cite{TIWARI2018106} later confirms these results.
A different stream of the literature studies the relationships across cryptocurrencies, or between cryptocurrencies and traditional assets. \cite{CORBET201828}, for example, provides evidence of a relative detachment of Bitcoin, Ripple, and Litecoin, from stocks, government bonds, and gold indices, thus offering some diversification benefits for investors in the short term. In a similar vein, \cite{Aslanidis2019} find a positive but time-varying conditional correlation among cryptocurrencies (Bitcoin, Ripple, Dash, Monero), and confirm their negligible relationship with traditional assets. Additionally, \cite{VIDALTOMAS-Herding2019} finds evidence of herding behavior during down markets, and that the smallest coins follow the path of the larger ones (not only that of Bitcoin). In a more recent paper \cite{BOURI2020101497} reports that the average return equicorrelation between cryptocurrencies is upward trending, which suggests that market linkages are increasing over time. \cite{Kurka2019} argues that although Bitcoin seems isolated from other financial assets over the entire period, market linkages arise when sub-periods are carefully examined.
Perhaps, the most closely related works are \cite{YI201898} and \cite{JI2019257}. These two studies are based on the connectedness methodology of \cite{Diebold2009,Diebold2012,Diebold2014} and quantify the interdependencies across cryptocurrencies using data up to 2018. In particular, \cite{YI201898} analyzes return and volatility connectedness between six leading cryptocurrencies, stressing the importance of Bitcoin and Litecoin as sources of uncertainty. \cite{JI2019257} focuses only on volatility linkages using a large set of coins.The authors document a period of increasing interdependencies across volatilities from mid- 2016, emphasizing the role of cryptocurrencies other than Bitcoin in emitting uncertainty.
We complement these two interesting studies in two ways. First, we study both return and volatility linkages adopting a more general set of methodologies and a broader time coverage (up to July 2020), thus capturing the most recent events. In addition, we quantify the market linkages across frequency ranges, determining the specific frequencies at which cryptocurrencies are more tightly connected. The frequency-domain analysis allows us to document new stylized facts about the cyclical properties of the transmission mechanism, which are essential to make an overall assessment of systemic risk.
For the sake of brevity, we refer to two recent surveys covering most aspects of cryptocurrencies research topics \citep{Corbet2019,MeredizBariviera2019}.
\section{Methodology \label{sec:methods}}
We conduct the empirical analysis using three different methodologies to assess cryptocurrency market linkages. Our first approach, Principal Component Analysis (PCA), is a statistical method that converts a set of correlated variables into a set of uncorrelated components through an orthogonal transformation. PCA aims to reduce the dimension of the data retaining as much variance (information) as possible. See, e.g., \cite{Wei} for further details about PCA methodology.
Our second approach to assess cryptocurrency market linkages is based on the estimation of cross-sectional dependence. Specifically, we first obtain the pair-wise cross-sectional correlations of the cryptocurrencies, $\hat{\rho}_{ij}$. Then, we calculate the average correlations across all pairs as $\overbar{\hat{\rho}}=(2/N(N-1))\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\hat{\rho}_{ij}$, and the associated cross-sectional dependence statistic of \cite{Pesaran2015} as $CD=\left[TN(N-1)/2\right]^{1/2}\overbar{\hat{\rho}}.$ \cite{Pesaran2015} establishes that the implicit null hypothesis of the CD test is that of weak cross sectional dependence versus the alternative hypothesis of strong cross sectional dependence\footnote{For further developments on cross-sectional dependence, we refer to the Special Issue edited by \cite{BaltagiPesaran}}.
Our third approach consists of constructing quantitative measures of market interdependence (or connectedness) based on the vector autorregression (VAR) framework of \cite{Diebold2009,Diebold2012,Diebold2014}. This methodology has also been used by other papers in the literature \citep{YI201898, JI2019257}. Beside this, we also follow the approach of \cite{BarunikKrehlik2018}, who extend the traditional Diebold-Yilmaz framework to the frequency-domain. The advantage of the frequency-domain is twofold. First, the frequency-domain analysis allows us to distinguish whether shocks across cryptocurrencies have long- or short-term effects. Second, one can recover standard, (time-domain) indices by aggregating frequency-domain connectedness measures over all frequencies. Thus, the approach in \cite{BarunikKrehlik2018} allows for a simultaneous assessment over time and across frequencies. \cite{CORBET201828} for example, also rely on this approach when assessing connectedness between cryptocurrencies and traditional assets. We briefly discuss the major features of the frequency-domain measures in the supplementary material. However, for the standard time-domain indices, we refer to \cite{Diebold2012, Diebold2014}.
\section{Data and results \label{sec:data}}
The empirical analysis employs daily data for seventeen major cryptocurrencies obtained from \url{https://coinmarketcap.com}. Since these cryptocurrencies were not launched at the same time, to capture more information we expand our sample adding more coins as we move through in time. Sample 1 includes seven important cryptocurrencies traded since (at least) August 2015. Sample 2 adds three more cryptocurrencies with data starting in October 2016. Finally, Sample 3 adds seven more coins, and data coverage starts in October 2017. Returns are computed as the log-price differences. We follow \cite{Diebold2012}, and estimate daily range-based return volatilities from open, close, high, and low daily prices, as in \cite{Garman}. Given that realized volatilities are right-skewed but approximately Gaussian after taking logs \citep{Anders}, we consider logarithmically transformed volatilities as time series for our estimations, as in \cite{Diebold2016} or \cite{Dem}.
\begin{table}[!htbp]
\centering
\caption{Factor analysis computed with different number of coins in each dataset and in non-overlapping 1-year windows.}
\resizebox{.99\textwidth}{!}{
\begin{tabular}{clrrrrrrrrrrrrrrrrrrrrrr}
\cmidrule{3-12} \multicolumn{2}{l}{\textbf{Sample 1 (7 coins)}} & \multicolumn{5}{c|}{Returns} & \multicolumn{5}{c}{Volatilities} \\
& & 08/08/2015 & 08/08/2016 & 08/08/2017 & 08/08/2018 & \multicolumn{1}{r|}{08/08/2019} & 08/08/2015 & 08/08/2016 & 08/08/2017 & 08/08/2018 & 08/08/2019 \\
& & 07/08/2016 & 07/08/2017 & 07/08/2018 & 07/08/2019 & \multicolumn{1}{r|}{17/07/2020} & 07/08/2016 & 07/08/2017 & 07/08/2018 & 07/08/2019 & 17/07/2020 \\
\cmidrule{3-12} \multicolumn{1}{c}{\% Variance explained by first PC} & \ & 36\% & 37\% & 62\% & 80\% & 83\% & 41\% & 54\% & 70\% & 73\% & 71\% \\
\multirow{7}[1]{*}{Squared component loading} & BTC & 76\% & 49\% & 60\% & 79\% & 86\% & 61\% & 63\% & 76\% & 76\% & 67\% \\
& DASH & 21\% & 39\% & 59\% & 78\% & 65\% & 36\% & 49\% & 69\% & 74\% & 69\% \\
& ETH & 7\% & 35\% & 80\% & 88\% & 93\% & 27\% & 66\% & 76\% & 85\% & 83\% \\
& LTC & 68\% & 36\% & 68\% & 79\% & 91\% & 43\% & 67\% & 75\% & 74\% & 83\% \\
& XLM & 27\% & 36\% & 49\% & 79\% & 74\% & 40\% & 55\% & 58\% & 64\% & 48\% \\
& XMR & 31\% & 41\% & 70\% & 83\% & 85\% & 39\% & 13\% & 71\% & 76\% & 72\% \\
& XRP & 19\% & 19\% & 50\% & 70\% & 87\% & 37\% & 61\% & 67\% & 63\% & 74\% \\
\midrule
& & & & & & & & & & & \\
\cmidrule{4-7}\cmidrule{9-12} \multicolumn{2}{l}{\textbf{Sample 2 (10 coins)}} & & \multicolumn{4}{c}{Returns} & & \multicolumn{4}{c}{Volatilities} \\
& & & 30/10/2016 & 30/10/2017 & 30/10/2018 & 30/10/2019 & & 30/10/2016 & 30/10/2017 & 30/10/2018 & 30/10/2019 \\
& & & 29/10/2017 & 29/10/2018 & 29/10/2019 & 17/07/2020 & & 29/10/2017 & 29/10/2018 & 29/10/2019 & 17/07/2020 \\
\cmidrule{4-7}\cmidrule{9-12} \multicolumn{1}{c}{\%Variance explained by first PC} & \ & & 37\% & 65\% & 79\% & 82\% & & 48\% & 75\% & 72\% & 72\% \\
\multirow{10}[0]{*}{Squared component loading} & BTC & & 52\% & 56\% & 79\% & 87\% & & 55\% & 79\% & 76\% & 64\% \\
& DASH & & 44\% & 62\% & 80\% & 66\% & & 50\% & 76\% & 72\% & 75\% \\
& ETC & & 52\% & 64\% & 72\% & 79\% & & 54\% & 73\% & 74\% & 73\% \\
& ETH & & 50\% & 80\% & 89\% & 92\% & & 67\% & 81\% & 83\% & 81\% \\
& LTC & & 46\% & 67\% & 78\% & 91\% & & 57\% & 80\% & 72\% & 85\% \\
& NEO & & 26\% & 66\% & 80\% & 84\% & & 24\% & 78\% & 78\% & 65\% \\
& XLM & & 27\% & 58\% & 76\% & 77\% & & 49\% & 70\% & 55\% & 55\% \\
& XMR & & 48\% & 74\% & 82\% & 86\% & & 45\% & 78\% & 68\% & 75\% \\
& XRP & & 13\% & 52\% & 78\% & 87\% & & 54\% & 61\% & 68\% & 76\% \\
& ZEC & & 16\% & 74\% & 79\% & 72\% & & 25\% & 69\% & 71\% & 70\% \\
& & & & & & & & & & & \\
\cmidrule{5-7}\cmidrule{10-12} \multicolumn{2}{l}{\textbf{Sample 3 (17 coins)}} & & & \multicolumn{3}{c}{Returns} & & & \multicolumn{3}{c}{Volatilities} \\
& & & & 03/10/2017 & 03/10/2018 & 03/10/2019 & & & 03/10/2017 & 03/10/2018 & 03/10/2019 \\
& & & & 02/10/2018 & 02/10/2019 & 17/07/2020 & & & 02/10/2018 & 02/10/2019 & 17/07/2020 \\
\cmidrule{5-7}\cmidrule{10-12} \multicolumn{1}{c}{\% Variance explained by first PC} & \ & & & 54\% & 74\% & 81\% & & & 54\% & 74\% & 81\% \\
\multirow{17}[0]{*}{Squared component loading} & ADA & & & 47\% & 85\% & 84\% & & & 47\% & 85\% & 84\% \\
& BCH & & & 47\% & 69\% & 88\% & & & 47\% & 69\% & 88\% \\
& BNB & & & 38\% & 54\% & 89\% & & & 38\% & 54\% & 89\% \\
& BTC & & & 54\% & 76\% & 87\% & & & 54\% & 76\% & 87\% \\
& DASH & & & 59\% & 78\% & 63\% & & & 59\% & 78\% & 63\% \\
& EOS & & & 54\% & 80\% & 90\% & & & 54\% & 80\% & 90\% \\
& ETC & & & 63\% & 72\% & 78\% & & & 63\% & 72\% & 78\% \\
& ETH & & & 78\% & 89\% & 93\% & & & 78\% & 89\% & 93\% \\
& LTC & & & 62\% & 79\% & 92\% & & & 62\% & 79\% & 92\% \\
& MIOTA & & & 58\% & 71\% & 75\% & & & 58\% & 71\% & 75\% \\
& NEO & & & 63\% & 84\% & 78\% & & & 63\% & 84\% & 78\% \\
& TRX & & & 34\% & 68\% & 87\% & & & 34\% & 68\% & 87\% \\
& XLM & & & 44\% & 76\% & 76\% & & & 44\% & 76\% & 76\% \\
& XMR & & & 71\% & 80\% & 84\% & & & 71\% & 80\% & 84\% \\
& XRP & & & 49\% & 77\% & 86\% & & & 49\% & 77\% & 86\% \\
& XTZ & & & 23\% & 45\% & 64\% & & & 23\% & 45\% & 64\% \\
& ZEC & & & 70\% & 77\% & 69\% & & & 70\% & 77\% & 69\% \\
\end{tabular}%
}
\label{tab:PCA}%
\end{table}%
We first conduct PCA for returns and volatilities, separately. Following the literature, we standardize the data before applying PCA to prevent undue influence of a variable. We provide comparative analysis, dividing the data into non-overlapping one-year samples. The percentage of the variance explained by the first principal component is reported in Table \ref{tab:PCA}. The table also provides the squared component loadings, which are just the squared correlations between the first principal component and each of the variables. Analogous to Pearson's R-squared, the squared component loading measures the percentage of the variance in that variable explained by the principal component.
Results in Table \ref{tab:PCA} illustrate the increasing commonality in return variances in the cryptocurrency market. The first principal component (PC) explains 36\% of the overall variance in the first year of Sample 1. It is important to point out that in this period, the first PC represents around 70\% of Bitcoin or Litecoin variance. However, for other coins such as Ethereum, the first PC would capture only 7\% of variance. This picture changes dramatically over time. In the last yearly sample (August 2019 - July 2020) the first principal component explains more than 80\% of the overall data variance and at least 65\% of the variance of any individual coin. Thus, the PCA of returns reflects a clear path toward closer cryptocurrency linkages, where shocks are rapidly transmitted across the market. Overall, we conclude that one PC is now sufficient to represent the dynamics of the entire cryptocurrency market.
Table \ref{tab:PCA} also reports PCA for the volatility series. The results are similar to those obtained for returns. Volatility linkages increase substantially over time, albeit towards the end of the sample period reaching slightly lower levels of interdependency than returns.
\begin{figure}[!htbp]
\centering
\includegraphics[width = 0.6\textwidth]{Pesaran2.png}
\caption{Pesaran cross-sectional dependence, average correlations $\overbar{\hat{\rho}}$, computed using different number of coins in each sample and in non-overlapping 1-year windows. Coins and windows dates are the same as in Table \ref{tab:PCA}}
\label{fig:Pesaran}
\end{figure}
Next, we conduct the \cite{Pesaran2015} test for cross-sectional dependence. Our results strongly reject the null hypothesis of weak in favor of strong cross-sectional dependence in all cases\footnote{Detailed results of CD test are not reported in the paper, but they are available upon request.}. The average correlation across all pairs (for returns and volatilities, separately) is plotted in Figure \ref{fig:Pesaran}. The increasing cross-sectional correlations over time are consistent with previous findings obtained using PCA. Again, towards the end of the sample the increase in correlation is slightly stronger in returns than in volatilities. Thus, our results support previous findings of steadily increasing cross-cryptocurrency market linkages.
Finally, to quantify the strength of these market linkages we carry out a more extensive linkage analysis using the methodologies developed in \cite{Diebold2009,Diebold2012,Diebold2014} and \cite{BarunikKrehlik2018}. For each dataset, we analyze the dynamic evolution of connectedness estimating VARs for returns and for volatilities in a rolling window fashion.\footnote{The results in the paper are based on a vector autoregression of order four and a rolling window length of 365 days (one year)} As in \cite{BarunikKrehlik2018}, the usual (time-domain) connectedness indices are computed by aggregating frequency connectedness over all ranges, but the results are identical to those obtained from finite-horizon formulas with the standard ten period ahead horizon used, for example, in \cite{Diebold2012}.
\begin{figure}[!htbp]
\centering
\subfloat[Returns]{\includegraphics[width = 0.6\textwidth]{TotalConnectedness-return.png} \label{fig:TotalConnecteness-A}}\\
\subfloat[Volatilities]{\includegraphics[width=0.6\textwidth]{TotalConnectedness-vol.png}\label{fig:TotalConnecteness-B}}\\
\subfloat{\includegraphics[width = 0.6\textwidth]{legend-total.PNG}}\\
\caption{Total connectedness in returns and volatilties, using rolling windows.}
\label{fig:TotalConnecteness}
\end{figure}
Figure \ref{fig:TotalConnecteness} depicts the dynamic evolution of the total connectedness indices for returns (upper panel) and for volatilities (bottom panel) using the three samples of coins. Total connectedness (also called spillover index) measures, on average, the percentage of the variance explained by shocks transmitted across coins. As Figure \ref{fig:TotalConnecteness} shows, total connectedness exhibits a substantial upward trend for both returns and volatilities. Specifically, the index for returns rises from nearly 25\% to 80\% over 2016-2020 while for volatilities from roughly 30\% to 76\% during the same period. Thus, more than three quarters of the return and volatility variances are currently explained by shocks transmitted across cryptocurrencies. Notice that, despite the different starting dates, the magnitude and pattern of total connectedness are very similar across the three samples, that is, our findings are robust and therefore independent of the number of coins considered.
Since 2019 total connectedness stabilizes around 0.8, leaving little room for further increases. Importantly, we observe a sudden upward jump around March 2020 in all plots, which is contemporaneous to stock market crashes worldwide arising from the COVID-19 pandemic. This result indicates that such a global shock has helped to knit even tighter relationships among cryptocurrencies. This finding broadens the results by \cite{Goodell2020} regarding the influence of COVID-19 on Bitcoin prices.
\begin{table}[!htbp]
\centering
\caption{ Return connectedness table for Sample 1 (7 coins)}
\resizebox{.7\textwidth}{!}{
\begin{tabular}{lrrrrrrrc}
\multicolumn{9}{c}{Period 08/08/ 2015 - 07/08/2016. Total connectedness: 0.32} \\
\toprule & \multicolumn{1}{l}{BTC} & \multicolumn{1}{l}{DASH} & \multicolumn{1}{l}{ETH} & \multicolumn{1}{l}{LTC} & \multicolumn{1}{l}{XLM} & \multicolumn{1}{l}{XMR} & \multicolumn{1}{l}{XRP} & \multicolumn{1}{l}{Total from} \\
\cmidrule{2-9} BTC & 0.47 & 0.05 & 0.01 & 0.30 & 0.05 & 0.09 & 0.04 & \multicolumn{1}{r}{0.53} \\
DASH & 0.08 & 0.79 & 0.01 & 0.06 & 0.02 & 0.02 & 0.02 & \multicolumn{1}{r}{0.21} \\
ETH & 0.02 & 0.02 & 0.88 & 0.01 & 0.02 & 0.03 & 0.01 & \multicolumn{1}{r}{0.12} \\
LTC & 0.32 & 0.04 & 0.01 & 0.52 & 0.05 & 0.04 & 0.04 & \multicolumn{1}{r}{0.48} \\
XLM & 0.09 & 0.02 & 0.02 & 0.08 & 0.67 & 0.03 & 0.09 & \multicolumn{1}{r}{0.33} \\
XMR & 0.13 & 0.03 & 0.03 & 0.07 & 0.03 & 0.70 & 0.02 & \multicolumn{1}{r}{0.30} \\
XRP & 0.05 & 0.02 & 0.02 & 0.05 & 0.11 & 0.02 & 0.73 & \multicolumn{1}{r}{0.27} \\
\cmidrule{1-8} Total to & 0.69 & 0.17 & 0.10 & 0.57 & 0.27 & 0.23 & 0.21 & \\
\bottomrule & & & & & & & & \\
\multicolumn{9}{c}{Period 19/07/2019 - 18/07/2020. Total connectedness: 0.79} \\
\toprule & \multicolumn{1}{l}{BTC} & \multicolumn{1}{l}{DASH} & \multicolumn{1}{l}{ETH} & \multicolumn{1}{l}{LTC} & \multicolumn{1}{l}{XLM} & \multicolumn{1}{l}{XMR} & \multicolumn{1}{l}{XRP} & \multicolumn{1}{l}{Total from} \\
\cmidrule{2-9} BTC & 0.20 & 0.09 & 0.17 & 0.15 & 0.10 & 0.15 & 0.13 & \multicolumn{1}{r}{0.80} \\
DASH & 0.12 & 0.26 & 0.14 & 0.14 & 0.10 & 0.13 & 0.12 & \multicolumn{1}{r}{0.74} \\
ETH & 0.15 & 0.10 & 0.19 & 0.16 & 0.11 & 0.14 & 0.15 & \multicolumn{1}{r}{0.81} \\
LTC & 0.15 & 0.10 & 0.16 & 0.19 & 0.12 & 0.14 & 0.15 & \multicolumn{1}{r}{0.81} \\
XLM & 0.12 & 0.08 & 0.14 & 0.14 & 0.22 & 0.13 & 0.16 & \multicolumn{1}{r}{0.78} \\
XMR & 0.15 & 0.10 & 0.15 & 0.14 & 0.11 & 0.21 & 0.13 & \multicolumn{1}{r}{0.79} \\
XRP & 0.13 & 0.09 & 0.16 & 0.15 & 0.14 & 0.13 & 0.20 & \multicolumn{1}{r}{0.80} \\
\cmidrule{1-8} Total to & 0.82 & 0.57 & 0.92 & 0.89 & 0.68 & 0.82 & 0.84 & \\
\bottomrule
\end{tabular}%
}
\label{tab:pairwise}%
\end{table}%
The directional connectedness indices offer further insight into the evolution of the cryptocurrency market linkages. In Table \ref{tab:pairwise} we report the complete connectedness tables for returns (also known as spillover table) estimated using the first and last years of the seven coins sample (Sample 1). Results in Table \ref{tab:pairwise} provide strong evidence of an across the board increase in return connectedness already depicted in Figure \ref{fig:TotalConnecteness}. Moreover, a comparison of the pairwise and the total FROM and TO connectedness indices shows that the relative importance of the different cryptocurrencies have also changed dramatically over time. During the first year, spillovers are mostly driven by Bitcoin and Litecoin, while other coins seem less interconnected. This result is consistent with the findings in \cite{YI201898} using data up to February 2018. However, the situation changes remarkably in recent years, since all coins are now about equally important in transmitting/receiving return spillovers (shocks across markets represent between 74\% to 80\% of the return variances of coins).
The directional connectedness analysis of the volatility linkages is similar to that of returns. See supplementary material for detailed results.\footnote{The supplement also provides connectedness tables obtained with the seventeen coins sample (Sample 3)}
Overall, our results show that cryptocurrencies have become increasingly interconnected in both return and volatility over the recent period, emerging as a compactly integrated market. These results complement the previous findings on volatility in \cite{JI2019257} and highlight that the cryptocurrency market is becoming significantly more vulnerable to within shock transmissions.
\begin{figure}[!htbp]
\centering
\includegraphics[width = 0.4\textwidth]{h-l-coneectedness-return.png}
\includegraphics[width = 0.385\textwidth]{h-l-coneectedness-vol.png}\\
\includegraphics[width = 0.6\textwidth]{legend-freq.PNG}
\caption{Connectedness across frequency ranges for Sample 1 of seven coins (Returns: left panels; Volatilities: right panels). High frequencies: 1-7 day period . Low frequencies: period longer than 7 days.}
\label{fig:h-lConnecteness-1}
\end{figure}
We further adopt the \cite{BarunikKrehlik2018} approach to evaluate connectedness in the frequency domain (high vs. low frequencies). The frequency domain approach would help us disentangle the specific frequencies that have most contributed to the observed rise in connectedness. The high-frequency range includes frequencies with periods from one to seven days (one week), while the low-frequency range frequencies with periods longer than one week. To the best of our knowledge, the frequency domain method is not performed in the other studies in the literature focusing on the existing relationships across cryptocurrencies, such as \cite{YI201898} and \cite{JI2019257}.
Figure \ref{fig:h-lConnecteness-1} plots connectedness measures by frequency ranges obtained using the seven-coin sample (Sample 1)-- corresponding figures for the other samples with more coins can be found in the Supplement. The upper panel of \ref{fig:h-lConnecteness-1} depicts the decomposition of the total connectedness index in \ref{fig:TotalConnecteness} into two frequency connectedness components: the high frequency and low frequency components. Notice that connectedness at the two frequency ranges add up the total connectedness index. The middle panel plots the connectedness created within the specific ranges. Finally, the bottom panel plots the weights used to transform within connectedness into the frequency connectedness components, which measures the relative importance of high and low fluctuations on total variance.
Regarding returns (left panels), most of the connectedness between returns is created at high frequencies. The high-frequency component accounting for most of the observed increase in the total connectedness index. This implies that shocks across cryptocurrencies have mostly temporary effects on returns, dissipating fast in the short-run. This result, however, does not imply that returns are not connected at low frequencies. In fact, as the figures in the second and third panel show, returns are about equally connected at high and low frequencies, but fluctuations at high frequencies turn out to be more important for returns. Besides, it is suggestive that in March 2020, the relative importance of high-frequency fluctuations experiences a sudden increase, which indicates that the COVID-19 pandemic has increased the importance of immediacy.
The relative small importance of low-frequency connection supports previous findings regarding long-range memory (e.g., \cite{Urquhart2016, Bariviera2017, TIWARI2018106}).
The results for volatility differ substantially from those of returns. As the right panels of Figure \ref{fig:h-lConnecteness-1} show, most of the volatility connectedness is created at low frequencies. This indicates that volatility shocks across cryptocurrencies have persistent effects. However, we also observe an increasing contribution of the high-frequency component over time, which indicates that that information is currently transmitted faster across cryptocurrencies than it was before. As the middle and lower panels of the figure show, most of this increase can be explained by a strong decline in the relative importance of low-frequency fluctuations on the volatility variances. This decline may signal that nowadays agents are better able to offset any long-run effects of shocks by switching to other assets. In any case, these are good news for long-run investors, as their exposure to systemic risk over the long-term has significantly decreased.
\section{Conclusions \label{sec:conclusions}}
This paper broadens previous studies on cryptocurrency market linkages. We tackle this issue by an ensemble of methodologies to examine return and volatility linkages across the major coins over the last five years. To account for the fact that new coins are being introduced in the market, we conduct our analysis using extended samples with an increasing number of coins.
Irrespective of the methodology adopted, we document that the cryptocurrency market has experienced a strong overall increase in market linkages (return and volatility). As of July 2020, only few coin-specific shocks are not transmitted to the rest of the coins (less than 20\%).
The insights provided by the frequency-domain approach have provided new stylized facts on the shock transmission mechanism across cryptocurrencies. The paper uncovers that the transmitted shocks have mostly short-term effects on returns. This result is in line with the view that the cryptocurrency market makes significant steps towards becoming efficient. Although a significant part of volatility connectedness is still created at low-frequencies, we show that volatility transmission at high frequencies has currently become considerably more important.
Our results have several practical implications.
First, there are now limited diversification benefits in the cryptocurrency market, with active portfolio re-balancing becoming mostly irrelevant. Second, cryptocurrency indices hardly add any information about market evolution beyond that conveyed by any individual cryptocurrency. Third, from a regulatory perspective, if cryptocurrencies were to become legal tender at some point in time, policymakers should evaluate the potentially disruptive effects of such a highly interconnected market. Nevertheless, the observed decline of the relative importance of low-frequency transmission is favoring long-term investors in terms of smaller exposure to systemic risk.
\section*{SUPLEMENTARY MATERIAL}
|
{
"timestamp": "2020-10-01T02:14:20",
"yymm": "2009",
"arxiv_id": "2009.14561",
"language": "en",
"url": "https://arxiv.org/abs/2009.14561"
}
|
\section{Introduction}
Financial market models are usually prone to statistical estimation errors, incomplete information and other reasons for model misspecifications. Especially the drift of asset prices is notoriously difficult to estimate from historical data. Drift processes tend to fluctuate randomly over time, and even for estimating a constant drift with a reasonable degree of precision one needs very long time series, an observation already made by Merton~\cite{merton_1980}. At the same time, trading strategies in portfolio optimization problems depend crucially on the drift. Strategies that are determined based on a misspecified model can therefore perform rather badly in the true financial market setting, see Chopra and Ziemba~\cite{chopra_ziemba_1993} and Kan and Zhou~\cite{kan_zhou_2007}.
There are two main approaches to deal with these problems. On the one hand, it is crucial to approximate the true model as accurately as possible using all the information available. When estimating the hidden drift process the best estimate in a mean-square sense is the conditional mean of the drift given the available information, the so-called \emph{filter}. Observations usually include the stock returns but can also involve external sources of information like news, company reports or ratings. In fact, Merton~\cite{merton_1980} points out that due to the difficulty of estimating expected returns, sources of information other than time series data of market returns are needed to improve estimates. Filtering techniques thus are a way to reduce uncertainty about model parameters.
On the other hand, model uncertainty can be approached by setting up \emph{worst-case optimization} problems. Instead of working with just one particular model, one specifies a range of possible models and tries to optimize the objective, given that for any chosen strategy the worst of all possible models will occur. This leads to robust strategies, i.e.\ strategies that are less vulnerable to the specific choice of the model.
In this paper we combine a worst-case approach with filtering techniques for a utility maximization problem in a financial market with stochastic drift. This is a follow-up paper on Sass and Westphal~\cite{sass_westphal_2020} where a worst-case utility maximization problem for a financial market with constant drift is investigated. In~\cite{sass_westphal_2020} we work with a Black--Scholes market and address an optimization problem of the form
\begin{equation}\label{eq:basic_robust_problem_constant_drift}
\adjustlimits\sup_{\pi\in\mathcal{A}_h(x_0)}\inf_{\mu\in K}\E_\mu\bigl[U(X^\pi_T)\bigr],
\end{equation}
where $U\colon\mathbb{R}_+\to\mathbb{R}$ is a utility function, $X^\pi_T$ denotes the terminal wealth achieved when using strategy $\pi$, and $\mathcal{A}_h(x_0)$ is a class of constrained admissible strategies with initial capital $x_0$. The expectation $\E_\mu[\cdot]$ is with respect to a measure under which the drift of the asset returns is constantly equal to $\mu\in\mathbb{R}^d$, with $d$ denoting the number of risky assets in the market. By $K\subseteq\mathbb{R}^d$ we denote a fixed ellipsoid and speak of the \emph{uncertainty set}. The main result in~\cite{sass_westphal_2020} is a representation of the optimal strategy for~\eqref{eq:basic_robust_problem_constant_drift} in the case of power or logarithmic utility and a corresponding minimax theorem.
In the present paper we generalize the results from Sass and Westphal~\cite{sass_westphal_2020} to a financial market with a stochastic drift process and time-dependent uncertainty sets $K$. This is motivated by the idea that information about the hidden drift process, as e.g.\ obtained from filtering techniques, might change over time. A surplus of information should then be reflected in a smaller uncertainty set.
More precisely, we assume that under the reference measure returns follow the dynamics
\[ \mathrm{d} R_t = \nu_t\,\mathrm{d} t+\sigma\,\mathrm{d} W_t, \]
where the reference drift $(\nu_t)_{t\in[0,T]}$ is adapted to the investor filtration $(\mathcal{G}_t)_{t\in[0,T]}$ representing the investor's information. This is justified by a separation principle where one performs a filtering step before solving the optimization problem, i.e.\ $(\nu_t)_{t\in[0,T]}$ represents the investor's filter for the drift process. We introduce a time-dependent uncertainty set $(K_t)_{t\in[0,T]}$ that is a set-valued stochastic process adapted to $(\mathcal{G}_t)_{t\in[0,T]}$, meaning that the investor knows the realization of $K_t$ at time $t$. In our case, $K_t$ is an ellipsoid in $\mathbb{R}^d$.
It is not obvious how to set up a worst-case optimization problem in this time-dependent setting. The problem lies in the fact that the realization of the uncertainty sets $(K_t)_{t\in[0,T]}$ is not known initially but gets revealed over time. A worst-case drift process $(\mu_t)_{t\in[0,T]}$ is characterized by being the worst one with the property that $\mu_t\in K_t$ for all $t\in[0,T]$. However, optimization with respect to this worst-case drift process is not feasible for an investor since it is not known initially. Instead, it makes sense to consider the following local approach. For any fixed $t\in[0,T]$, the current uncertainty set $K_t$ is known. Given this $K_t$, investors take model uncertainty into account by assuming that in the future the worst possible drift process having values in $K_t$ will be realized, i.e.\ the worst drift process from the class
\[ \mathcal{K}^{(t)}=\bigl\{\mu^{(t)}=(\mu^{(t)}_s)_{s\in[t,T]} \,\big|\, \mu^{(t)}_s\in K_t \text{ and } \mu^{(t)}_s \text{ is }\mathcal{G}_t\text{-measurable for each }s\in[t,T]\bigr\}. \]
Investors then solve at each time $t\in[0,T]$, given $X_t^\pi =x>0$, the local optimization problem
\begin{equation}\label{eq:basic_robust_problem_local}
\adjustlimits \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \E_{\mu^{(t)}}\Bigl[U\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr],
\end{equation}
leading to an optimal strategy $(\pi^{(t),*}_s)_{s\in[t,T]}$. Here $X^{t,x,\pi^{(t)}}_T$ is the terminal wealth for starting at time $t$ with wealth $x$ and following strategy $\pi^{(t)}$. In our continuous-time setting this decision will be revised as soon as $K_t$ changes, possibly continuously in time. The realized optimal strategy of the investor is then given by $\pi^*_t=\pi^{(t),*}_t$ for all $t\in[0,T]$.
The focus of this paper lies in carrying over the results for the robust utility maximization problem with constant drift from Sass and Westphal~\cite{sass_westphal_2020} to the more general model described above. We determine the optimal strategy for~\eqref{eq:basic_robust_problem_local} and prove a local minimax theorem in analogy to Sass and Westphal~\cite[Thm.~3.12]{sass_westphal_2020}.
The difficulty here lies in proving that a worst-case drift exists such that the optimization of the robust objective function is equivalent to optimization with respect to the worst-case drift. The key result of our work is the duality we derive in Theorem~\ref{thm:minimax_theorem} which ensures existence of such a worst-case drift process. Moreover, we are able to specify the form of this drift process explicitly which is what makes it possible to also compute the optimal strategy. This is not an obvious result since duality as in that theorem can only be guaranteed under strong conditions. Results from the literature, e.g.\ from Quenez~\cite{quenez_2004}, do not carry over directly to our setting since the constraint that we put on the admissible strategies leads to a more complicated structure of attainable terminal wealths.
We then show how the time-dependent uncertainty set $(K_t)_{t\in[0,T]}$ can be defined based on the filter $\muhat{}{t}=\E[\mu_t\,|\,\mathcal{G}_t]$ for various investor filtrations $(\mathcal{G}_t)_{t\in[0,T]}$. The construction is motivated by confidence regions.
Finally, we compare the optimal strategies for different investor filtrations $(\mathcal{G}_t)_{t\in[0,T]}$ and investigate which effect a surplus of information has on their performance. By means of a numerical simulation we demonstrate that investors do need to account for model uncertainty by choosing a robustified strategy $\pi^*$. When investors rely on the respective filter only, adding more information leads to a smaller worst-case expected utility since the naive strategy that relies only on the filter is very vulnerable to model misspecifications. Investors need to robustify their strategy by taking model uncertainty into account to be able to profit from additional information. This effect can also be understood as an overconfidence of experts as studied empirically by Heath and Tversky~\cite{heath_tversky_1991}.
\bigskip
Model uncertainty, also called \emph{Knightian uncertainty} in reference to the seminal book by Knight~\cite{knight_1921}, has been addressed in numerous papers. Gilboa and Schmeidler~\cite{gilboa_schmeidler_1989} and Schmeidler~\cite{schmeidler_1989} formulate rigorous axioms on preference relations that account for risk aversion and uncertainty aversion. A robust utility functional in their sense is a mapping
\[ X\mapsto \inf_{Q\in\mathcal{Q}}\E_Q\bigl[U(X)\bigr], \]
where $U$ is a utility function and $\mathcal{Q}$ a convex set of probability measures.
Chen and Epstein~\cite{chen_epstein_2002} give a continuous-time extension of this multiple-priors utility.
Optimal investment decisions under such preferences are investigated in Quenez~\cite{quenez_2004} and Schied~\cite{schied_2005}, building up on Kramkov and Schachermayer~\cite{kramkov_schachermayer_1999, kramkov_schachermayer_2003}. An extension of those results by means of a duality approach is given in Schied~\cite{schied_2007}.
Pflug et al.~\cite{pflug_pichler_wozabal_2012} study risk minimization under model uncertainty.
Papers addressing drift uncertainty in a financial market are Garlappi et al.~\cite{garlappi_uppal_wang_2007} and Biagini and P\i nar~\cite{biagini_pinar_2017}, among others. The latter also focuses on ellipsoidal uncertainty sets. Uncertainty about both drift and volatility is investigated in a recent paper by Pham et al.~\cite{pham_wei_zhou_2018}.
Filtering techniques play a crucial role in utility maximization problems under partial information. There are essentially two models for the drift process that lead to finite-dimensional filters. In the first one the drift is modelled as an Ornstein--Uhlenbeck process, in the second one as a continuous-time Markov chain. The filters are the well-known Kalman and Wonham filter, respectively, see e.g.\ Elliott et al.~\cite{elliott_aggoun_moore_1995} and Liptser and Shiryaev~\cite{liptser_shiryaev_1974}.
\bigskip
The paper is organized as follows. Since this is a follow-up paper on Sass and Westphal~\cite{sass_westphal_2020} that generalizes results for a financial market with constant drift to one with stochastic drift we recap the main results of \cite{sass_westphal_2020} in Section~\ref{sec:recap_of_results_for_constant_drift}. For the convenience of the reader the later sections then refer to Section~2. In Section~\ref{sec:generalized_financial_market_model} we set up the generalized financial market model with stochastic drift process and state our local worst-case optimization problem. Section~\ref{sec:solution_to_the_robust_utility_maximization_problem} solves this problem in several steps. We provide representations of the optimal strategy that will be realized by an investor whose information about the drift process changes continuously in time and of the worst-case drift process. Further, we prove a minimax theorem for the local optimization problems. In Section~\ref{sec:construction_of_uncertainty_sets_via_filters} we explain how filtering techniques can be used to set up time-dependent uncertainty sets, motivated by confidence regions. We also compare the performance of the optimal strategies for different investor filtrations by means of a numerical simulation.
\section{Recap of Results for Constant Drift}\label{sec:recap_of_results_for_constant_drift}
This paper builds up on Sass and Westphal~\cite{sass_westphal_2020} and generalizes results for a financial market with constant drift to a model with a stochastic drift process. For the convenience of the reader we recap the main results of Sass and Westphal~\cite{sass_westphal_2020} in this section. This is in order to show in the following sections what carries over and where we have to provide new arguments. By referring to the cited results in this section, this can be conveniently done in a self-contained way.
\subsection{Financial market model}
The paper~\cite{sass_westphal_2020} deals with a continuous-time financial market with one risk-free and various risky assets. Let $T>0$ denote some finite investment horizon and let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space where the filtration $\mathbb{F}=(\mathcal{F}_t)_{t\in[0,T]}$ satisfies the usual conditions. All processes are assumed to be $\mathbb{F}$-adapted.
The risk-free asset $S^0$ is of the form $S^0_t=\mathrm{e}^{rt}$, $t\in[0,T]$, where $r\in\mathbb{R}$ is the deterministic risk-free interest rate.
Aside from the risk-free asset, investors can also invest in $d\geq 2$ risky assets. Their prices are given by constant initial prices $S_0^i >0$ and evolve according to $\mathrm{d} S_t^i = S_t^i \,\mathrm{d} R_t^i$, $i=1, \ldots, d$, where the return process $R=(R^1,\dots,R^d)^\top$ is defined by
\[ \mathrm{d} R_t = \nu\,\mathrm{d} t + \sigma\,\mathrm{d} W_t, \quad R_0=0, \]
where $W=(W_t)_{t\in[0,T]}$ is an $m$-dimensional Brownian motion under $\mathbb{P}$ with $m\geq d$. Further, $\nu\in\mathbb{R}^d$ and $\sigma\in\mathbb{R}^{d\times m}$, where it is assumed that $\sigma$ has full rank equal to $d$.
Model uncertainty is introduced by assuming that the true drift of the stocks is only known to be an element of some set $K\subseteq\mathbb{R}^d$ with $\nu\in K$ and that investors want to maximize their worst-case expected utility when the drift takes values within $K$. The value $\nu$ can be thought of as an estimate for the drift that was for instance obtained from historical stock prices. Changing the drift from $\nu$ to some $\mu\in K$ can be expressed by a change of measure. For this purpose, let the process $(Z^\mu_t)_{t\in[0,T]}$ be defined by
\[ Z^\mu_t = \exp\Bigl(\theta(\mu)^\top W_t -\frac{1}{2}\lVert\theta(\mu)\rVert^2 t\Bigr), \]
where $\theta(\mu)=\sigma^\top(\sigma\sigma^\top)^{-1}(\mu-\nu)$. One can then define a new measure $\mathbb{P}^\mu$ by setting $\frac{\mathrm{d} \mathbb{P}^\mu}{\mathrm{d} \mathbb{P}} = Z^\mu_T$. Note that since $\theta(\mu)$ is a constant, the process $(Z^\mu_t)_{t\in[0,T]}$ is a strictly positive martingale. Therefore, $\mathbb{P}^\mu$ is a probability measure that is equivalent to $\mathbb{P}$ and it follows from Girsanov's Theorem that the process $(W^\mu_t)_{t\in[0,T]}$, defined by $W^\mu_t = W_t-\theta(\mu)t$, is a Brownian motion under $\mathbb{P}^\mu$. The return dynamics can therefore be rewritten as
\begin{equation*}
\mathrm{d} R_t = \nu\,\mathrm{d} t + \sigma\,\mathrm{d} W_t = \nu\,\mathrm{d} t + \sigma\bigl(\mathrm{d} W^\mu_t+\theta(\mu)\,\mathrm{d} t\bigr) = \mu\,\mathrm{d} t + \sigma\,\mathrm{d} W^\mu_t,
\end{equation*}
hence a change of measure from $\mathbb{P}$ to $\mathbb{P}^\mu$ corresponds to changing the drift in the return dynamics from $\nu$ to $\mu$. In the following, let $\E_\mu[\cdot]$ denote the expectation under measure $\mathbb{P}^\mu$ and $\E[\cdot]=\E_\nu[\cdot]$ the expectation under the reference measure $\mathbb{P}=\mathbb{P}^\nu$.
An investor's trading decisions are described by a self-financing trading strategy $(\pi_t)_{t\in[0,T]}$ with values in $\mathbb{R}^d$. The entry $\pi^i_t$, $i=1, \dots, d$, is the proportion of wealth invested in asset $i$ at time $t$. The corresponding wealth process $(X^\pi_t)_{t\in[0,T]}$ given initial wealth $x_0>0$ can then be described by the stochastic differential equation
\[ \mathrm{d} X^\pi_t = X^\pi_t\Bigl( r\,\mathrm{d} t + \pi_t^\top(\mu-r\mathbf{1}_d)\,\mathrm{d} t + \pi_t^\top \sigma\,\mathrm{d} W^\mu_t \Bigr), \quad X^\pi_0=x_0, \]
for any $\mu\in K$, where $\mathbf{1}_d$ denotes the $d$-dimensional vector with all entries equal to~$1$.
Trading strategies are required to be $\mathbb{F}^R$-adapted, where $\mathbb{F}^R=(\mathcal{F}^R_t)_{t\in[0,T]}$ for $\mathcal{F}^R_t=\sigma((R_s)_{s\in[0,t]})$. The basic admissibility set is defined as
\[ \mathcal{A}(x_0) = \biggl\{(\pi_t)_{t\in[0,T]} \;\bigg|\; \pi \text{ is } \mathbb{F}^R\text{-adapted}, \; X^\pi_0=x_0, \; \E_\mu\biggl[\int_0^T \lVert\sigma^\top\pi_t\rVert^2\,\mathrm{d} t\biggr]<\infty \text{ for all } \mu\in K\biggr\}. \]
The paper Sass and Westphal~\cite{sass_westphal_2020} considers investors with power or logarithmic utility, using the notation $U_\gamma\colon\mathbb{R}_+\to\mathbb{R}$ for $\gamma\in(-\infty,1)$, where $U_\gamma(x)=\frac{x^\gamma}{\gamma}$ for $\gamma\neq 0$ denotes power utility and $U_0(x)=\log(x)$ is the logarithmic utility function.
Investors with a robust approach to the portfolio optimization problem would try to maximize
\[ \inf_{\mu\in K} \E_\mu\bigl[U_\gamma(X^\pi_T)\bigr] \]
among the admissible strategies. It is quite straightforward to show that as soon as $r\mathbf{1}_d\in K$, the strategy $(\pi_t)_{t\in[0,T]}$ with $\pi_t=0$ for all $t\in[0,T]$ is optimal in the class of admissible strategies $\mathcal{A}(x_0)$. This observation is proven in Sass and Westphal~\cite[Prop.~2.1]{sass_westphal_2020} and implies that as the level of uncertainty exceeds a certain threshold, it will be optimal for investors to not invest anything in the stocks and everything in the risk-free asset. For finding less conservative strategies that still take into account model uncertainty a constraint on the admissible strategies is introduced that prevents a pure bond investment. Consider for some $h>0$ the admissibility set
\[ \mathcal{A}_h(x_0)=\bigl\{ \pi\in\mathcal{A}(x_0) \,\big|\, \langle\pi_t,\mathbf{1}_d\rangle = h \text{ for all } t\in[0,T] \bigr\}. \]
Taking $h=1$ would imply that investors are not allowed to invest anything in the risk-free asset. They must then distribute all of their wealth among the risky assets.
Sass and Westphal~\cite{sass_westphal_2020} study the case where the uncertainty set is an ellipsoid in $\mathbb{R}^d$ centered around the reference parameter $\nu$, i.e.\
\begin{equation}\label{eq:uncertainty_ellipsoid}
K=\bigl\{ \mu\in\mathbb{R}^d \,\big|\, (\mu-\nu)^\top \Gamma^{-1}(\mu-\nu) \leq \kappa^2 \bigr\}.
\end{equation}
Here, $\kappa>0$, $\nu\in\mathbb{R}^d$, and $\Gamma\in\mathbb{R}^{d\times d}$ is symmetric and positive definite. For $\Gamma=I_d$ one simply gets a ball in the Euclidean norm with radius $\kappa$ and center $\nu$. Another special case discussed in the literature is $\Gamma=\sigma\sigma^\top$, see e.g.\ Biagini and P\i nar~\cite{biagini_pinar_2017}.
The value of $\kappa$ determines the size of the ellipsoid. Higher values of $\kappa$ correspond to more uncertainty about the true drift. The robust utility maximization problem over the constrained strategies $\pi\in\mathcal{A}_h(x_0)$ can then be written in the form
\begin{equation}\label{eq:robust_problem}
\adjustlimits \sup_{\pi\in\mathcal{A}_h(x_0)} \inf_{\mu\in K} \E_\mu\bigl[U_\gamma(X^\pi_T)\bigr].
\end{equation}
\subsection{Solution of the non-robust problem}
To solve the optimization problem in~\eqref{eq:robust_problem} Sass and Westphal~\cite{sass_westphal_2020} first address the non-robust constrained utility maximization problem under a fixed parameter $\mu\in\mathbb{R}^d$. For better readability the following notation is introduced.
\begin{definition}\label{def:matrix_A_vector_c}
Denote with $D$ the matrix
\[ D =
\begin{pmatrix}
1 & & 0 & -1 \\
&\ddots & &\vdots \\
0 & & 1 & -1
\end{pmatrix}\in\mathbb{R}^{(d-1)\times d} \]
and define the matrix $A\in\mathbb{R}^{d\times d}$ and the vector $c\in\mathbb{R}^d$ by
\begin{align*}
A &= D^\top(D\sigma\sigma^\top D^\top)^{-1}D, \\
c &= e_d-D^\top(D\sigma\sigma^\top D^\top)^{-1}D\sigma\sigma^\top e_d = (I_d-A\sigma\sigma^\top)e_d.
\end{align*}
\end{definition}
The following result gives the optimal strategy for the non-robust problem and can be found in Sass and Westphal~\cite[Prop.~3.4]{sass_westphal_2020}.
\begin{proposition}\label{prop:optimal_strategy_non-robust}
Let $\mu\in\mathbb{R}^d$. Then the optimal strategy for the optimization problem
\[ \sup_{\pi\in\mathcal{A}_h(x_0)} \E_\mu\bigl[U_\gamma(X^\pi_T)\bigr] \]
is the strategy $(\pi_t)_{t\in[0,T]}$ with
\[ \pi_t = \frac{1}{1-\gamma}A\mu +hc \]
for all $t\in[0,T]$, with $A$ and $c$ as in Definition~\ref{def:matrix_A_vector_c}.
\end{proposition}
In the proof the $d$-dimensional constrained problem is reduced to a $(d-1)$-dimensional unconstrained problem. Using the form of the optimal strategy in the $(d-1)$-dimensional market which is known from Merton~\cite{merton_1969} yields the following representation for the optimal expected utility from terminal wealth. This result is given in Sass and Westphal~\cite[Cor.~3.5]{sass_westphal_2020}.
\begin{corollary}\label{cor:optimal_utility_non-robust}
Let $\mu\in\mathbb{R}^d$. Then the optimal expected utility from terminal wealth is
\begin{equation*}
\begin{aligned}
\sup_{\pi\in\mathcal{A}_h(x_0)} &\E_\mu\bigl[U_\gamma(X^\pi_T)\bigr] \\
&=
\begin{dcases}
\frac{x_0^\gamma}{\gamma}\exp\Bigl(\gamma T\Bigl( \widetilde{r}+\frac{1}{2(1-\gamma)}\bigl(\widetilde{\mu}-\widetilde{r}\mathbf{1}_{d-1}\bigr)^\top(\widetilde{\sigma}\widetilde{\sigma}^\top )^{-1}\bigl(\widetilde{\mu}-\widetilde{r}\mathbf{1}_{d-1}\bigr)\Bigr)\Bigr), &\gamma\neq 0,\\
\log(x_0) + \Bigl( \widetilde{r}+\frac{1}{2}\bigl(\widetilde{\mu}-\widetilde{r}\mathbf{1}_{d-1}\bigr)^\top(\widetilde{\sigma}\widetilde{\sigma}^\top )^{-1}\bigl(\widetilde{\mu}-\widetilde{r}\mathbf{1}_{d-1}\bigr) \Bigr)T, &\gamma=0,
\end{dcases}
\end{aligned}
\end{equation*}
where
\begin{equation}\label{eq:recall_substitution_r_mu_sigma}
\begin{aligned}
\widetilde{\sigma}&=D\sigma, \\
\widetilde{r}&=(1-h)r+he_d^\top\mu-\frac{1}{2}(1-\gamma)\lVert h\sigma^\top e_d \rVert^2, \\
\widetilde{\mu}&=D\mu - h(1-\gamma)D\sigma\sigma^\top e_d+\widetilde{r}\mathbf{1}_{d-1}.
\end{aligned}
\end{equation}
\end{corollary}
\subsection{The worst-case parameter}
In a next step one may ask what the worst possible parameter $\mu$ would be for the investor, given that she reacts optimally, i.e.\ by applying the strategy from Proposition~\ref{prop:optimal_strategy_non-robust}. This corresponds to solving the dual problem
\[ \adjustlimits \inf_{\mu\in K} \sup_{\pi\in\mathcal{A}_h(x_0)} \E_\mu\bigl[U_\gamma(X^\pi_T)\bigr]. \]
Note that at this point it is not clear whether equality holds between the original problem and the corresponding dual problem. The following result for the solution of the dual problem is given in Sass and Westphal~\cite[Thm.~3.8]{sass_westphal_2020}. Let us decompose $\Gamma=\tau\tau^\top$ for a nonsingular matrix $\tau\in\mathbb{R}^{d\times d}$.
\begin{theorem}\label{thm:solution_of_the_inf_sup_problem}
Let $0=\lambda_1<\lambda_2\leq\cdots\leq\lambda_d$ denote the eigenvalues of $\tau^\top A\tau$, and let
\[ v_1=\frac{1}{\lVert \tau^{-1}\mathbf{1}_d \rVert}\tau^{-1}\mathbf{1}_d, v_2,\dots,v_d\in\mathbb{R}^d \]
denote the respective orthogonal eigenvectors with $\lVert v_i\rVert=1$ for all $i=1,\dots, d$.
Then
\[ \adjustlimits \inf_{\mu\in K} \sup_{\pi\in\mathcal{A}_h(x_0)} \E_\mu\bigl[U_\gamma(X^\pi_T)\bigr] = \E_{\mu^*}\bigl[U_\gamma(X^{\pi^*}_T)\bigr], \]
where
\[ \mu^*=\nu-\tau\sum_{i=1}^d \biggl( \frac{\lambda_i}{1-\gamma}+\frac{h}{\psi(\kappa)\lVert \tau^{-1}\mathbf{1}_d \rVert} \biggr)^{-1}\biggl\langle h\tau^\top c+\frac{\lambda_i}{1-\gamma}\tau^{-1}\nu, v_i\biggr\rangle v_i \]
for $\psi(\kappa)\in(0,\kappa]$ that is uniquely determined by $\lVert\tau^{-1}(\mu^*-\nu)\rVert=\kappa$, and where $(\pi^*_t)_{t\in[0,T]}$ is defined by
\[ \pi^*_t = \frac{1}{1-\gamma}A\mu^* +hc \]
for all $t\in[0,T]$.
\end{theorem}
The preceding theorem solves the problem
\begin{equation}\label{eq:the_inf_sup_problem}
\adjustlimits \inf_{\mu\in K} \sup_{\pi\in\mathcal{A}_h(x_0)} \E_\mu\bigl[U_\gamma(X^\pi_T)\bigr].
\end{equation}
This is the corresponding dual problem to the original optimization problem
\begin{equation}\label{eq:the_sup_inf_problem}
\adjustlimits \sup_{\pi\in\mathcal{A}_h(x_0)} \inf_{\mu\in K} \E_\mu\bigl[U_\gamma(X^\pi_T)\bigr],
\end{equation}
but in general the values of these two problems do not coincide. There are, of course, special cases in which the supremum and the infimum do interchange. Those results are called \emph{minimax theorems} in the literature. In a portfolio optimization setting that is similar to ours a minimax theorem has been shown in Quenez~\cite{quenez_2004}, building up on the theory by Kramkov and Schachermayer~\cite{kramkov_schachermayer_1999}. Due to the constraint $\langle\pi_t,\mathbf{1}_d\rangle=h$ for all $t\in[0,T]$, the result from Quenez~\cite{quenez_2004} does not apply directly to our setting. It is possible, however, to use the knowledge about the optimal strategy for~\eqref{eq:the_inf_sup_problem} to show that it indeed also solves~\eqref{eq:the_sup_inf_problem} and that in this case, the supremum and the infimum can be interchanged.
\subsection{A minimax theorem}
The following representation of $\pi^*$, given in Sass and Westphal~\cite[Lem.~3.10]{sass_westphal_2020}, is useful for proving a minimax theorem.
\begin{lemma}\label{lem:representation_of_pi_star}
The strategy $\pi^*$ from Theorem~\ref{thm:solution_of_the_inf_sup_problem} satisfies
\[ \pi^*_t = -\frac{h}{\psi(\kappa)\lVert \tau^{-1}\mathbf{1}_d \rVert}\Gamma^{-1}(\mu^*-\nu) \]
for all $t\in[0,T]$.
\end{lemma}
The preceding lemma characterizes the strategy $\pi^*$ that is optimal for the parameter $\mu^*$. Vice versa, $\mu^*$ is also the worst possible drift parameter, given that an investor applies strategy $\pi^*$. This is shown in Sass and Westphal~\cite[Prop.~3.11]{sass_westphal_2020}.
\begin{proposition}\label{prop:mu_star_is_worst_for_pi_star}
The parameter $\mu$ that attains the minimum in
\[ \inf_{\mu\in K} \E_\mu\bigl[U_\gamma(X^{\pi^*}_T)\bigr] \]
is $\mu^*$, i.e.\ $\mu^*$ is the worst possible parameter, given that an investor chooses strategy $\pi^*$.
\end{proposition}
It then follows that the point $(\pi^*,\mu^*)$ is a \emph{saddle point} of the problem, i.e.\ it holds
\[ \E_{\mu^*}\bigl[U_\gamma(X^{\pi}_T)\bigr] \leq \E_{\mu^*}\bigl[U_\gamma(X^{\pi^*}_T)\bigr] \leq \E_{\mu}\bigl[U_\gamma(X^{\pi^*}_T)\bigr] \]
for all $\mu\in K$ and $\pi\in\mathcal{A}_h(x_0)$. This property is essential for proving the following minimax theorem, given in Sass and Westphal~\cite[Thm.~3.12]{sass_westphal_2020}.
\begin{theorem}\label{thm:duality_result}
Let $K=\{ \mu\in\mathbb{R}^d \,|\, (\mu-\nu)^\top \Gamma^{-1}(\mu-\nu) \leq \kappa^2 \}$. Then
\[ \adjustlimits \sup_{\pi\in\mathcal{A}_h(x_0)} \inf_{\mu\in K} \E_\mu\bigl[U_\gamma(X^\pi_T)\bigr] = \E_{\mu^*}\bigl[U_\gamma(X^{\pi^*}_T)\bigr] = \adjustlimits \inf_{\mu\in K} \sup_{\pi\in\mathcal{A}_h(x_0)} \E_\mu\bigl[U_\gamma(X^\pi_T)\bigr], \]
where $\mu^*$ and $\pi^*$ are defined as in Theorem~\ref{thm:solution_of_the_inf_sup_problem}.
\end{theorem}
\section{Generalized Financial Market Model}\label{sec:generalized_financial_market_model}
In the following we generalize the approach from Sass and Westphal~\cite{sass_westphal_2020} to a financial market model where the drift is a stochastic process instead of a constant. To account for a change in information about the drift we also introduce time-dependence in the uncertainty set.
The basic idea is that the available information in the market, for instance the observed asset returns or external sources of information, are used to estimate the true drift based on filtering techniques and to set up a corresponding uncertainty set $K_t$ at any time $t\in[0,T]$. Given $K_t$, investors then take model uncertainty into account by assuming that in the future the worst possible drift process $(\mu^{(t)}_s)_{s\in[t,T]}$ with values in $K_t$ will be realized. In our continuous-time setting the decision about the uncertainty set will be revised as soon as the information about the true drift changes, so in the extreme case continuously in time.
\subsection{Reference model}
Before stating our generalized financial market, we make an observation that justifies the setup of the model.
\begin{remark}
Suppose that the ``true'' dynamics of the $d$-dimensional return process $R$ are given by
\[ \mathrm{d} R_t=\mu_t\,\mathrm{d} t + \sigma\,\mathrm{d} W_t, \quad R_0=0, \]
for some stochastic drift process $(\mu_t)_{t\in[0,T]}$, an $m$-dimensional Brownian motion $(W_t)_{t\in[0,T]}$, $m\geq d$, and some $\sigma\in\mathbb{R}^{d\times m}$ with full rank. Assume further that the information of an investor is given by the investor filtration $\mathbb{G}=(\mathcal{G}_t)_{t\in[0,T]}$. The investor's best estimator for $\mu$ is then the conditional mean $m_t:=\E[\mu_t\,|\,\mathcal{G}_t]$ and one can rewrite the dynamics of the return process as
\[ \mathrm{d} R_t=m_t\,\mathrm{d} t + \sigma\,\mathrm{d} V_t, \]
where the so-called innovations process $(V_t)_{t\in[0,T]}$ is a $\mathbb{G}$-adapted Brownian motion.
For instance, in the setting where an investor observes only the return process $R$, the process $(m_t)_{t\in[0,T]}$ would be the Kalman filter.
In the following, we set up our continuous-time financial market model working directly with the innovations process and therefore assuming a $\mathbb{G}$-adapted drift process.
The separation principle that we use here by filtering first and then performing the optimization is a common approach for dealing with partial information.
\end{remark}
We fix an investment horizon $T>0$ and some filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ where the filtration $\mathbb{F}=(\mathcal{F}_t)_{t\in[0,T]}$ satisfies the usual conditions. All processes are assumed to be $\mathbb{F}$-adapted. We assume that an investor's information is described by the investor filtration $\mathbb{G}=(\mathcal{G}_t)_{t\in[0,T]}$ with $\mathcal{G}_t\subseteq\mathcal{F}_t$ for all $t\in[0,T]$.
We consider, as before, a financial market with one risk-free and $d\geq 2$ risky assets. The risk-free asset $S^0$ evolves as
\[ \mathrm{d} S^0_t = S^0_t r\,\mathrm{d} t, \quad S^0_0 = 1, \]
where $r>0$ is the deterministic risk-free interest rate. The prices of the risky assets evolve according to $\mathrm{d} S_t^i = S_t^i \,\mathrm{d} R_t^i$ for $S_0^i>0$, $i=1, \ldots, d$, where the returns $R^1,\dots,R^d$ of the risky assets follow the dynamics
\begin{equation}\label{eq:dynamics_of_S_time_dependent_drift}
\mathrm{d} R_t = \nu_t\,\mathrm{d} t + \sigma\,\mathrm{d} W_t, \quad R_0=0,
\end{equation}
where $R=(R^1,\dots,R^d)^\top$. Here, $(W_t)_{t\in[0,T]}$ is an $m$-dimensional Brownian motion under $\mathbb{P}$, $m\geq d$. Note that the volatility matrix $\sigma\in\mathbb{R}^{d\times m}$ in~\eqref{eq:dynamics_of_S_time_dependent_drift} is constant. Further, we assume that $\sigma$ has full rank equal to $d$. In contrast to the volatility, the drift might change in the course of time. We assume that $(\nu_t)_{t\in[0,T]}$ is an $\mathbb{R}^d$-valued $\mathbb{G}$-adapted stochastic process and think of $(\nu_t)_{t\in[0,T]}$ as an estimation for the true drift process given all available information. We speak of $(\nu_t)_{t\in[0,T]}$ as the \emph{reference drift}.
\begin{remark}
We could generalize this model by replacing the constant volatility $\sigma$ by a stochastic process $(\sigma_t)_{t\in[0,T]}$ that is \emph{observable} by the investor, i.e.\ $\mathbb{G}$-adapted. Our techniques and the results in the following section carry over to this setting. It would just be necessary to put assumptions on the volatility process to ensure that the change of measure in Section~\ref{subs:uncertainty_sets_and_change_of_measure} below is well-defined.
In a model where the stochastic volatility $(\sigma_t)_{t\in[0,T]}$ is \emph{not} observable by the investor, our techniques do not carry over in general. The reason is that we need models where we can write down the (non-robust) optimal strategy of an investor explicitly. Portfolio optimization under stochastic volatility is much more complicated since one has to deal with truly incomplete markets. Pham and Quenez~\cite{pham_quenez_2001} state assumptions on the model needed to obtain solutions. Similar restrictions might work for our case. These also have to take into account observability issues which essentially require that the volatility process can be observed one-to-one from the quadratic variation of the stock returns.
However, for portfolio optimization the drift is the crucial factor and, at the same time, the one that is difficult to estimate from historical asset price data. Drift processes tend to fluctuate randomly over time and even if they were constant, long time series would be needed to estimate this parameter with a satisfactory degree of precision. On the contrary, volatility can be estimated reasonably well from return observations. For these reasons, our model with constant volatility serves as a suitable first approximation for more general models that allows an exact study of the influence of model uncertainty on the optimal strategy.
\end{remark}
\subsection{Uncertainty sets and change of measure}\label{subs:uncertainty_sets_and_change_of_measure}
As before, we are concerned with investors who are uncertain about the true drift. They are aware that $(\nu_t)_{t\in[0,T]}$ in~\eqref{eq:dynamics_of_S_time_dependent_drift} might not be the true drift process. In utility maximization problems they want to maximize their worst-case expected utility, given that the true drift process is in a way ``close'' to $\nu$.
To model the uncertainty about the drift we specify the ellipsoidal sets
\[ K_t = \bigl\{ \mu\in\mathbb{R}^d \,\big|\, (\mu-\nu_t)^\top\Gamma_t^{-1}(\mu-\nu_t)\leq \kappa_t^2 \bigr\}, \quad t\in[0,T], \]
where $(\Gamma_t)_{t\in[0,T]}$ is a $\mathbb{G}$-adapted stochastic process of symmetric and positive-definite matrices $\Gamma_t\in\mathbb{R}^{d\times d}$ and $(\kappa_t)_{t\in[0,T]}$ is $\mathbb{G}$-adapted with $\kappa_t>0$ for each $t\in[0,T]$.
The set $K_t$ is determined at time $t\in[0,T]$ by taking the available information about the true drift process into account, for example based on filtering techniques. The process $(K_t)_{t\in[0,T]}$ is a $\mathbb{G}$-adapted set-valued process, therefore the investor knows the realization of $K_t$ at time $t\in[0,T]$.
Given this $K_t$, investors then take model uncertainty into account by assuming that in the future the worst possible drift process having values in $K_t$ will be realized. We denote this worst-case future drift by $(\mu^{(t),*}_s)_{s\in[t,T]}$. This allows for some deterministic dynamics given $K_t$, i.e.\ the $\mu^{(t),*}_s$ for any $s\in[t,T]$ are $\mathcal{G}_t$-measurable. The worst-case optimization problem then leads to an optimal strategy $(\pi^{(t),*}_s)_{s\in[t,T]}$, determined at time $t$. In our continuous-time setting this decision will be revised as soon as $K_t$ changes, possibly continuously in time. The realized worst-case drift process $(\mu^*_t)_{t\in[0,T]}$ and optimal strategy $(\pi^*_t)_{t\in[0,T]}$ are then given by
\[ \mu^*_t=\mu^{(t),*}_t, \quad \pi^*_t=\pi^{(t),*}_t \]
for any $t\in[0,T]$. If $\mu^*$ and $\pi^*$ are uniquely determined, then they are by construction $\mathbb{G}$-adapted.
This is not so much a game setting but rather a way how the investor determines the worst case. It is a mixture of using estimation methods and taking model uncertainty into account.
The optimization problem can be derived only locally for each $t\in[0,T]$. In detail, the setup looks as follows. At time $t\in[0,T]$ investors assume that the future drift process will be the worst one within the class
\[ \mathcal{K}^{(t)}=\bigl\{\mu^{(t)}=(\mu^{(t)}_s)_{s\in[t,T]} \,\big|\, \mu^{(t)}_s\in K_t \text{ and } \mu^{(t)}_s \text{ is }\mathcal{G}_t\text{-measurable for each }s\in[t,T]\bigr\}. \]
For each $\mu=\mu^{(t)}\in\mathcal{K}^{(t)}$ we can construct a new measure by defining the $\mathbb{R}^m$-valued process $(\theta_s(\mu))_{s\in[0,T]}$ with
\[ \theta_s(\mu)=
\begin{cases}
0, &s<t,\\
\sigma^\top(\sigma\sigma^\top)^{-1}(\mu_s-\nu_s), &s\geq t,
\end{cases} \]
and
\[ Z^\mu_s = \exp\biggl(\int_0^s\theta_u(\mu)^\top\,\mathrm{d} W_u -\frac{1}{2}\int_0^s\lVert\theta_u(\mu)\rVert^2\,\mathrm{d} u\biggr) \]
for $s\in[0,T]$. We then define the new probability measure $\mathbb{P}^\mu$ by
\[ \frac{\mathrm{d} \mathbb{P}^{\mu}}{\mathrm{d} \mathbb{P}} = Z^\mu_T \]
and note that, under $\mathbb{P}^\mu$, the process $(W^\mu_s)_{s\in[0,T]}$ with
\[ W^\mu_s = W_s-\int_0^s \theta_u(\mu)\,\mathrm{d} u \]
for $s\in[0,T]$ is a Brownian motion by Girsanov's Theorem. Note that due to boundedness of $K_t$ the process $\theta(\mu)$ is bounded and therefore $(Z^\mu_s)_{s\in[0,T]}$ is a true martingale.
The change of measure causes a change in the drift on the interval $[t,T]$ only. For our optimization problems this is the only relevant time interval since we condition on $\mathcal{G}_t$.
For $s\in[t,T]$ we can rewrite the dynamics of the asset returns as
\[ \mathrm{d} R_s = \nu_s\,\mathrm{d} s + \sigma\,\mathrm{d} W_s = \mu_s\,\mathrm{d} s + \sigma\,\mathrm{d} W^\mu_s, \]
which means that under $\mathbb{P}^\mu$ the future drift of the stocks is given by $(\mu_s)_{s\in[t,T]}$. We write $\E_\mu[\cdot]=\E_{\mu^{(t)}}[\cdot]$ for expectation under the measure $\mathbb{P}^\mu$.
\subsection{Local optimization problem}
An investor's behavior in the time interval $[t,T]$ is described by a self-financing trading strategy $\pi^{(t)}=(\pi^{(t)}_s)_{s\in[t,T]}$. The class of admissible trading strategies, given that the investor has wealth $x>0$ at time $t$, is
\begin{equation*}
\begin{aligned}
\mathcal{A}(t,x) = \biggl\{\pi^{(t)}=(\pi^{(t)}_s)_{s\in[t,T]} \;\bigg|&\; \pi^{(t)} \text{ is } \mathbb{G}\text{-adapted}, \; X^\pi_t=x,\\
&\E_{\mu^{(t)}}\biggl[\int_t^T\! \lVert\sigma^\top\pi^{(t)}_s\rVert^2\,\mathrm{d} s\biggr]<\infty \text{ for all } \mu^{(t)}\in \mathcal{K}^{(t)}\biggr\}.
\end{aligned}
\end{equation*}
We will restrict these strategies by imposing, as in Sass and Westphal~\cite{sass_westphal_2020}, a constraint that prevents a pure bond investment. For any $h>0$ we define the set
\[ \mathcal{A}_h(t,x)=\bigl\{ \pi^{(t)}\in\mathcal{A}(t,x) \,\big|\, \langle\pi^{(t)}_s,\mathbf{1}_d\rangle=h \text{ for all }s\in[t,T]\bigr\}. \]
For an investor choosing strategy $\pi=\pi^{(t)}\in\mathcal{A}(t,X^\pi_t)$ the terminal wealth can be written as
\[ X^\pi_T = X^\pi_t\exp\biggl(\int_t^T \Bigl(r+\pi_s^\top(\mu_s-r\mathbf{1}_d)-\frac{1}{2}\lVert\sigma^\top\pi_s\rVert^2\Bigr)\mathrm{d} s + \int_t^T \pi_s^\top\sigma\,\mathrm{d} W^\mu_s\biggr). \]
We are now able to state our utility maximization problem. In the following we write $X^{t,x,\pi^{(t)}}_s$ for the wealth at $s\in[t,T]$ when starting at $t$ with $x$ and using strategy $\pi^{(t)}$. At time $t$ the local optimization problem for $x>0$ then reads
\begin{equation}\label{eq:value_function_robust_power_constrained_time-dependent}
\adjustlimits \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr].
\end{equation}
Here, $U_\gamma$ with $\gamma\in(-\infty,1)$ again denotes the power utility function $U_\gamma(x)=\frac{x^\gamma}{\gamma}$ if $\gamma\neq 0$, and logarithmic utility $U_0(x)=\log(x)$ if $\gamma=0$.
\begin{remark}
In the case where $K_t= \{ \mu\in\mathbb{R}^d \,|\, (\mu-\nu)^\top\Gamma^{-1}(\mu-\nu)\leq \kappa^2 \}$ for all $t\in[0,T]$, i.e.\ where our reference drift is simply a constant $\nu$, and also the matrix $\Gamma_t=\Gamma$ as well as the radius $\kappa_t=\kappa$ are constant in time, we obtain the setting from Section~\ref{sec:recap_of_results_for_constant_drift} as a special case.
\end{remark}
\section{Solution to the Robust Utility Maximization Problem}\label{sec:solution_to_the_robust_utility_maximization_problem}
In this section we solve~\eqref{eq:value_function_robust_power_constrained_time-dependent} by computing the optimal strategy $\pi^{(t),*}$ and the worst-case drift $\mu^{(t),*}$ and prove a minimax theorem in analogy to Theorem~\ref{thm:duality_result}. We proceed as in the setting with constant drift in Section~\ref{sec:recap_of_results_for_constant_drift}. Looking at the local optimization problem for a fixed $t\in[0,T]$ enables us to reduce the drift uncertainty to $K_t$ and make use of our results for constant drift. At the end of this section we explain which strategy will be realized by an investor whose information about the drift process changes continuously in time and how this strategy is naturally obtained from the solution to the local optimization problems.
\subsection{Solution to the non-robust problem}
As a first step towards solving~\eqref{eq:value_function_robust_power_constrained_time-dependent} we compute the optimal strategy for an investor given a particular future drift $\mu^{(t)}\in\mathcal{K}^{(t)}$.
Due to the constraint on the admissible strategies, which prevents a pure bond investment, we reduce this problem to a less-dimensional unconstrained problem for which standard results from Merton~\cite{merton_1969} apply.
This is only the first step in solving the robust problem, but knowing the optimal strategy given a fixed drift will later enable us to compute the worst-case drift process and prove a minimax theorem.
\begin{proposition}\label{prop:optimal_strategy_non-robust_time-dependent}
Let $t\in[0,T]$, $x>0$ and $\mu^{(t)}\in\mathcal{K}^{(t)}$. Then the optimal strategy for the optimization problem
\[ \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr] \]
is the strategy $(\pi^{(t)}_s)_{s\in[t,T]}$ with
\[ \pi^{(t)}_s = \frac{1}{1-\gamma}A\mu^{(t)}_s+hc \]
for all $s\in[t,T]$, where $A\in\mathbb{R}^{d\times d}$ and $c\in\mathbb{R}^d$ are as introduced in Definition~\ref{def:matrix_A_vector_c}.
\end{proposition}
\begin{proof}
The proof works along the lines of the proof of Proposition~\ref{prop:optimal_strategy_non-robust}. We take an arbitrary strategy $\pi=\pi^{(t)}\in\mathcal{A}_h(t,x)$ and recall that we can write the terminal wealth given $X_t^\pi =x$ under strategy $\pi$ as
\[ X^{t,x,\pi}_T = x\,\exp\biggl(\int_t^T \Bigl(r+\pi_s^\top(\mu^{(t)}_s-r\mathbf{1}_d)-\frac{1}{2}\lVert\sigma^\top\pi_s\rVert^2\Bigr)\mathrm{d} s + \int_t^T \pi_s^\top\sigma\,\mathrm{d} W^\mu_s\biggr). \]
We now proceed exactly as in the proof of Proposition~\ref{prop:optimal_strategy_non-robust}, replacing the constant $\mu$ by the $\mathcal{G}_t$-measurable $(\mu^{(t)}_s)_{s\in[t,T]}$, and perform the same transformation to a $(d-1)$-dimensional unconstrained financial market.
We can deduce that $\E_{\mu^{(t)}}[U_\gamma(X^{t,x,\pi}_T)]$ equals the expected utility of terminal wealth, conditional on $X_t^\pi=x$, in an unconstrained financial market with $d-1$ risky assets, where the future drift process is $(\widetilde{\mu}_s)_{s\in[t,T]}$, the risk-free interest rate is $(\widetilde{r}_s)_{s\in[t,T]}$ and the volatility matrix is $\widetilde{\sigma}\in\mathbb{R}^{(d-1)\times m}$. These transformed market parameters have the form
\begin{align*}
\widetilde{\sigma}&=D\sigma, \\
\widetilde{r}_s&=(1-h)r+he_d^\top\mu^{(t)}_s-\frac{1}{2}(1-\gamma)\lVert h\sigma^\top e_d \rVert^2, \\
\widetilde{\mu}_s&=D\mu^{(t)}_s - h(1-\gamma)D\sigma\sigma^\top e_d+\widetilde{r}_s\mathbf{1}_{d-1}.
\end{align*}
Note that since the $(\mu^{(t)}_s)_{s\in[t,T]}$ are $\mathcal{G}_t$-measurable, so are $(\widetilde{r}_s)_{s\in[t,T]}$ and $(\widetilde{\mu}_s)_{s\in[t,T]}$, in particular the market parameters in the transformed market can be observed by the investor.
In this $(d-1)$-dimensional unconstrained financial market we can deduce from Merton~\cite{merton_1969} that the optimal strategy is of the form
\begin{equation}\label{eq:optimal_pi_tilde_time-dependent}
\widetilde{\pi}_s = \frac{1}{1-\gamma}(\widetilde{\sigma}\widetilde{\sigma}^\top)^{-1}(\widetilde{\mu}_s-\widetilde{r}_s\mathbf{1}_{d-1})
= \frac{1}{1-\gamma}(D\sigma\sigma^\top D^\top)^{-1}\bigl(D\mu^{(t)}_s - h(1-\gamma)D\sigma\sigma^\top e_d\bigr)
\end{equation}
for every $s\in[t,T]$.
For the logarithmic utility case, this is immediate, for power utility, this needs to be shown.
Merton~\cite{merton_1969} yields the form of the optimal strategy in a Black--Scholes market with constant parameters. This result can be extended to a market where the risk-free interest rate as well as drift and volatility of the stocks are not necessarily constant but still observable by the investor, see Westphal~\cite[App.~B]{westphal_2019} for a complete proof. A similar result has been proven in Karatzas et al.~\cite{karatzas_lehoczky_shreve_xu_1991} for complete markets with deterministic market coefficients and for incomplete markets with totally unhedgeable market coefficients.
Now we can return to our original market and obtain that the optimal strategy fulfills
\begin{equation*}
\begin{aligned}
\pi^{(t)}_s &= D^\top\widetilde{\pi}_s+he_d\\
&= D^\top\frac{1}{1-\gamma}(D\sigma\sigma^\top D^\top)^{-1}\bigl(D\mu^{(t)}_s - h(1-\gamma)D\sigma\sigma^\top e_d\bigr)+he_d \\
&= \frac{1}{1-\gamma}D^\top(D\sigma\sigma^\top D^\top)^{-1}D\mu^{(t)}_s +h\bigl(I_d-D^\top(D\sigma\sigma^\top D^\top)^{-1}D\sigma\sigma^\top\bigr)e_d \\
&= \frac{1}{1-\gamma}A\mu^{(t)}_s+hc
\end{aligned}
\end{equation*}
for all $s\in[t,T]$, where we have used the notation for $A$ and $c$ from Definition~\ref{def:matrix_A_vector_c}. Note that $(\pi^{(t)}_s)_{s\in[t,T]}$ is indeed admissible due to boundedness of $K_t$.
\end{proof}
The preceding proposition states the form of the investor's optimal strategy under the assumption that a specific future drift process $(\mu^{(t)}_s)_{s\in[t,T]}$ is given. The explicit form can be used to compute also the expected utility obtained when applying the optimal strategy.
\begin{corollary}
Let $t\in[0,T]$ and $\mu^{(t)}\in\mathcal{K}^{(t)}$. Then the optimal expected utility from terminal wealth is
\begin{equation*}
\begin{aligned}
&\sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr]\\
&\quad=
\begin{dcases}
\frac{x^\gamma}{\gamma}\exp\biggl(\gamma\! \int_t^T\!\Bigl( \widetilde{r}_s+\frac{1}{2(1-\gamma)}\bigl(\widetilde{\mu}_s-\widetilde{r}_s\mathbf{1}_{d-1}\bigr)^\top(\widetilde{\sigma}\widetilde{\sigma}^\top )^{-1}\bigl(\widetilde{\mu}_s-\widetilde{r}_s\mathbf{1}_{d-1}\bigr)\Bigr)\mathrm{d} s\!\biggr), &\gamma\neq 0,\\
\log(x) + \int_t^T\Bigl( \widetilde{r}_s+\frac{1}{2}\bigl(\widetilde{\mu}_s-\widetilde{r}_s\mathbf{1}_{d-1}\bigr)^\top(\widetilde{\sigma}\widetilde{\sigma}^\top )^{-1}\bigl(\widetilde{\mu}_s-\widetilde{r}_s\mathbf{1}_{d-1}\bigr) \Bigr)\mathrm{d} s, &\gamma=0,
\end{dcases}
\end{aligned}
\end{equation*}
where
\begin{equation*}
\begin{aligned}
\widetilde{\sigma}&=D\sigma, \\
\widetilde{r}_s&=(1-h)r+he_d^\top\mu^{(t)}_s-\frac{1}{2}(1-\gamma)\lVert h\sigma^\top e_d \rVert^2, \\
\widetilde{\mu}_s&=D\mu^{(t)}_s - h(1-\gamma)D\sigma\sigma^\top e_d+\widetilde{r}_s\mathbf{1}_{d-1}.
\end{aligned}
\end{equation*}
\end{corollary}
\begin{proof}
The representation in the corollary follows, just like in the proof of Corollary~\ref{cor:optimal_utility_non-robust}, by the fact that we have reduced our constrained utility maximization problem to a $(d-1)$-dimensional unconstrained problem where the parameters of our transformed financial market are exactly those that are listed in the corollary.
We have seen that the optimal strategy in this $(d-1)$-dimensional market fulfills
\[ \widetilde{\pi}_s = \frac{1}{1-\gamma}(\widetilde{\sigma}\widetilde{\sigma}^\top)^{-1}(\widetilde{\mu}_s-\widetilde{r}_s\mathbf{1}_{d-1}) \]
for all $s\in[t,T]$. Plugging this optimal strategy in yields the expression from the corollary.
\end{proof}
\subsection{The worst-case drift process}
In the following, we compute the worst-case future drift process that is determined at time $t\in[0,T]$, i.e.\ the drift process $\mu^{(t)}\in\mathcal{K}^{(t)}$ for which
\[ \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr] \]
is minimized. Due to the previous corollary we see that this is equivalent to the minimization of the integral
\begin{equation}\label{eq:what_has_to_be_minimized_time-dependent}
\int_t^T\Bigl( \widetilde{r}_s+\frac{1}{2}\bigl(\widetilde{\mu}_s-\widetilde{r}_s\mathbf{1}_{d-1}\bigr)^\top(\widetilde{\sigma}\widetilde{\sigma}^\top )^{-1}\bigl(\widetilde{\mu}_s-\widetilde{r}_s\mathbf{1}_{d-1}\bigr) \Bigr)\mathrm{d} s.
\end{equation}
When plugging the representations for $\widetilde{\mu}$, $\widetilde{r}$ and $\widetilde{\sigma}$ back in, we obtain an expression that depends on $(\mu^{(t)}_s)_{s\in[t,T]}$ again. By the same calculations as in the setting with constant drift we deduce that minimizing~\eqref{eq:what_has_to_be_minimized_time-dependent} is equivalent to minimizing
\[ \int_t^T\Bigl(\frac{1}{2(1-\gamma)}(\mu^{(t)}_s)^\top A\mu^{(t)}_s+hc^\top\mu^{(t)}_s \Bigr)\mathrm{d} s. \]
But the minimization of this integral is equivalent to a pointwise minimization of
\[ K_t\ni\mu \mapsto \frac{1}{2(1-\gamma)}\mu^\top A\mu+hc^\top\mu. \]
Now it is straightforward to see that we can use the results from Section~\ref{sec:recap_of_results_for_constant_drift} to obtain the worst-case drift process $(\mu^{(t),*}_s)_{s\in[t,T]}$. Here, $\mu^{(t),*}_s$ is for any $s\in[t,T]$ obtained as the minimizer of the above function on $K_t$. Recall that the uncertainty set is an ellipsoid of the form
\[ K_t = \{ \mu\in\mathbb{R}^d \,|\, (\mu-\nu_t)^\top\Gamma_t^{-1}(\mu-\nu_t)\leq \kappa_t^2 \}. \]
We have assumed that $\Gamma_t$ is a symmetric positive-definite matrix in $\mathbb{R}^{d\times d}$. In the following we use the representation $\Gamma_t=\tau_t\tau_t^\top$ where $\tau_t\in\mathbb{R}^{d\times d}$ is a nonsingular matrix.
\begin{corollary}\label{cor:solution_of_the_inf_sup_problem_time-dependent}
We fix some $t\in[0,T]$ and let $0=\lambda_{t,1}<\lambda_{t,2}\leq\cdots\leq\lambda_{t,d}$ denote the eigenvalues of $\tau_t^\top A\tau_t$, and
\[ v_{t,1}=\frac{1}{\lVert \tau_t^{-1}\mathbf{1}_d \rVert}\tau_t^{-1}\mathbf{1}_d, v_{t,2},\dots,v_{t,d}\in\mathbb{R}^d \]
the respective orthogonal eigenvectors with $\lVert v_{t,i}\rVert=1$ for all $i=1,\dots, d$.
Then
\[ \adjustlimits \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr] = \E_{\mu^{(t),*}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t),*}}_T\bigr)\Bigr], \]
where
\[ \mu^{(t),*}_s=\nu_t-\tau_t\sum_{i=1}^d \biggl(\frac{\lambda_{t,i}}{1-\gamma}+\frac{h}{\psi_t(\kappa_t)\lVert \tau_t^{-1}\mathbf{1}_d \rVert}\biggr)^{-1}\Bigl\langle h\tau_t^\top c+\frac{\lambda_{t,i}}{1-\gamma}\tau_t^{-1}\nu_t, v_{t,i}\Bigr\rangle\, v_{t,i} \]
for all $s\in[t,T]$, and where $\psi_t(\kappa_t)\in(0,\kappa_t]$ is uniquely determined by $\lVert\tau_t^{-1}(\mu^{(t),*}_s-\nu_t)\rVert=\kappa_t$. The strategy $(\pi^{(t),*}_s)_{s\in[t,T]}$ has the form
\[ \pi^{(t),*}_s = \frac{1}{1-\gamma}A\mu^{(t),*}_s +hc \]
for all $s\in[t,T]$.
\end{corollary}
\begin{proof}
We have seen that the worst-case drift process $(\mu^{(t),*}_s)_{s\in[t,T]}$ is the one where $\mu^{(t),*}_s$ is for any $s\in[t,T]$ equal to the minimizer of the function
\[ \mu \mapsto \frac{1}{2(1-\gamma)}\mu^\top A\mu+hc^\top\mu \]
over all $\mu\in K_t$. So we can do the minimization as in Section~\ref{sec:recap_of_results_for_constant_drift}. We know that the matrix $\tau_t^\top A\tau_t\in\mathbb{R}^{d\times d}$ is symmetric and positive definite with
\[ \mathrm{ker}(\tau_t^\top A\tau_t)=\mathrm{span}(\{\tau_t^{-1}\mathbf{1}_d\}). \]
Now the representation of $\mu^{(t),*}_s$ follows as in Theorem~\ref{thm:solution_of_the_inf_sup_problem}. The form of the optimal strategy $\pi^{(t),*}$ then follows from Proposition~\ref{prop:optimal_strategy_non-robust_time-dependent}.
\end{proof}
The preceding corollary shows that the problem
\[ \adjustlimits \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr] \]
is solved by drift process $(\mu^{(t),*}_s)_{s\in[t,T]}$ and strategy $(\pi^{(t),*}_s)_{s\in[t,T]}$. Note that both the worst-case drift process and the optimal strategy are constant on $[t,T]$ and $\mathcal{G}_t$-measurable. This is due to the setup of the model in which investors assume that the future drift process will take values in the ellipsoid $K_t$ only.
The problem above is the dual to our original problem
\[ \adjustlimits \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr]. \]
To ensure that $\mu^{(t),*}$ and $\pi^{(t),*}$ are also a solution to this problem we have to show that $\mu^{(t),*}$ is the worst drift process in the set $\mathcal{K}^{(t)}$, given that an investor chooses trading strategy $\pi^{(t),*}$. In that case, the infimum and the supremum interchange and we can deduce that $\pi^{(t),*}$ and $\mu^{(t),*}$ also establish a solution to our original robust optimization problem.
\subsection{A minimax theorem}
We proceed as in Section~\ref{sec:recap_of_results_for_constant_drift} and note that the strategy $\pi^{(t),*}$ from the previous corollary satisfies
\[ \pi^{(t),*}_s = -\frac{h}{\psi_t(\kappa_t)\lVert\tau_t^{-1}\mathbf{1}_d\rVert}\Gamma_t^{-1}\bigl(\mu^{(t),*}_s-\nu_t\bigr) \]
for all $s\in[t,T]$. This can be proven by analogy with Lemma~\ref{lem:representation_of_pi_star}. This observation helps to prove the following proposition.
\begin{proposition}
The drift process $(\mu^{(t)}_s)_{s\in[t,T]}$ that attains for any $x>0$ the minimum in
\[ \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t),*}}_T\bigr)\Bigr] \]
is $(\mu^{(t),*}_s)_{s\in[t,T]}$, i.e.\ $\mu^{(t),*}$ is the worst possible drift process, given that an investor chooses the strategy $\pi^{(t),*}$.
\end{proposition}
\begin{proof}
We take an arbitrary $\mu=\mu^{(t)}\in\mathcal{K}^{(t)}$. Note that in case $\gamma\neq 0$ we can write
\begin{equation*}
\begin{aligned}
&\E_{\mu}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t),*}}_T\bigr)\Bigr]\\
&= \frac{x^\gamma}{\gamma}\mathrm{e}^{\gamma r(T-t)}\E_{\mu}\biggl[\exp\biggl(\!\gamma\!\int_t^T\!\!\! \Bigl((\pi^{(t),*}_s)\!^\top\!(\mu_s-r\mathbf{1}_d) -\frac{1}{2}\lVert\sigma^\top\!\pi^{(t),*}_s\rVert^2\Bigr)\mathrm{d} s + \!\gamma\!\int_t^T \!\!\!(\pi^{(t),*}_s)\!^\top\! \sigma\,\mathrm{d} W^\mu_s \!\biggr)\biggr] \\
&= \frac{x^\gamma}{\gamma}\mathrm{e}^{\gamma r(T-t)}\exp\biggl(\gamma\int_t^T \Bigl((\pi^{(t),*}_s)^\top(\mu_s-r\mathbf{1}_d) -\frac{1-\gamma}{2}\lVert\sigma^\top\pi^{(t),*}_s\rVert^2\Bigr)\mathrm{d} s\biggl).
\end{aligned}
\end{equation*}
In case $\gamma=0$ we have
\[ \E_{\mu}\Bigl[\log\bigl(X^{t,x,\pi^{(t),*}}_T\bigr)\Bigr]
=\log(x)+ r(T-t)+\int_t^T \Bigl((\pi^{(t),*}_s)^\top(\mu_s-r\mathbf{1}_d) -\frac{1}{2}\lVert\sigma^\top\pi^{(t),*}_s\rVert^2\Bigr)\mathrm{d} s. \]
In both cases, the drift process $(\mu_s)_{s\in[t,T]}\in \mathcal{K}^{(t)}$ that minimizes this expression is the one that minimizes
\[ \int_t^T (\pi^{(t),*}_s)^\top\mu_s\,\mathrm{d} s. \]
Since $(\pi^{(t),*}_s)_{s\in[t,T]}$ is constant, we find the minimizer as the minimizer of $(\pi^{(t),*}_s)^\top\mu_s$. Recall that
\[ \pi^{(t),*}_s = -\frac{h}{\psi_t(\kappa_t)\lVert\tau_t^{-1}\mathbf{1}_d\rVert}\Gamma_t^{-1}\bigl(\mu^{(t),*}_s-\nu_t\bigr). \]
It follows that
\[ (\pi^{(t),*}_s)^\top\Gamma_t\pi^{(t),*}_s = \frac{h^2}{\psi_t(\kappa_t)^2\lVert\tau_t^{-1}\mathbf{1}_d\rVert^2}\bigl(\mu^{(t),*}_s-\nu_t\bigr)^\top\Gamma_t^{-1}\bigl(\mu^{(t),*}_s-\nu_t\bigr) = \frac{h^2\kappa_t^2}{\psi_t(\kappa_t)^2\lVert\tau_t^{-1}\mathbf{1}_d\rVert^2}. \]
Knowing that $\psi_t(\kappa_t)>0$ we can deduce
\[ \sqrt{(\pi^{(t),*}_s)^\top\Gamma_t\pi^{(t),*}_s}=\frac{h\kappa_t}{\psi_t(\kappa_t)\lVert\tau_t^{-1}\mathbf{1}_d\rVert}. \]
The drift process $\mu^{(t),*}_s$ at time $s$ can thus be rewritten in the form
\[ \mu^{(t),*}_s=\nu_t-\frac{\psi_t(\kappa_t)\lVert\tau_t^{-1}\mathbf{1}_d\rVert}{h}\Gamma_t\pi^{(t),*}_s = \nu_t-\frac{\kappa_t}{\sqrt{(\pi^{(t),*}_s)^\top\Gamma_t\pi^{(t),*}_s}}\Gamma_t\pi^{(t),*}_s. \]
This is exactly the vector that minimizes $(\pi^{(t),*}_s)^\top\mu$ over all $\mu\in K_t$, see the proof of \cite[Prop.~3.11]{sass_westphal_2020}.
Hence, $\mu^{(t),*}$ is the drift process that minimizes the expected utility of terminal wealth for an investor who chooses strategy $\pi^{(t),*}$.
\end{proof}
The previous proposition establishes an equilibrium result. By definition, the strategy $\pi^{(t),*}$ is optimal for the drift $\mu^{(t),*}$. Due to the proposition, it also holds that $\mu^{(t),*}$ is the worst drift given that an investor chooses strategy $\pi^{(t),*}$. Hence, we see that $(\pi^{(t),*},\mu^{(t),*})$ is a saddle point of the optimization problem
\[ \adjustlimits \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr]. \]
In particular, the supremum and infimum can be interchanged. We obtain the following minimax theorem.
\begin{theorem}\label{thm:minimax_theorem}
Let $t\in[0,T]$. Then for $x>0$
\begin{equation*}
\begin{aligned}
\adjustlimits \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr]
&= \E_{\mu^{(t),*}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t),*}}_T\bigr)\Bigr] \\
&= \adjustlimits \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \E_{\mu^{(t),*}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t),*}}_T\bigr)\Bigr],
\end{aligned}
\end{equation*}
where $\mu^{(t),*}$ and $\pi^{(t),*}$ are defined as in Corollary~\ref{cor:solution_of_the_inf_sup_problem_time-dependent}.
\end{theorem}
\begin{proof}
The proof is analogous to the proof of Theorem~\ref{thm:duality_result}.
\end{proof}
The previous theorem solves our original local optimization problem~\eqref{eq:value_function_robust_power_constrained_time-dependent} for a fixed time $t\in[0,T]$. It shows that the best strategy for an investor in this robust optimization problem is the strategy $(\pi^{(t),*}_s)_{s\in[t,T]}$ with
\[ \pi^{(t),*}_s=\frac{1}{1-\gamma}A\mu^{(t),*}_s+hc \]
for all $s\in[t,T]$, where $(\mu^{(t),*}_s)_{s\in[t,T]}$ is defined as in Corollary~\ref{cor:solution_of_the_inf_sup_problem_time-dependent}. The process $(\mu^{(t),*}_s)_{s\in[t,T]}$ can be interpreted as the worst possible realization of the future drift process from the investor's point of view at time $t$. The worst-case drift and optimal strategy in this setting are constant on $[t,T]$. Since these do not depend on $X_t^\pi=x$, this would also be true for the unconditional case.
This is due to the assumption of the investor that the future drift will take values in the set $K_t$ only, where $K_t$ is determined at time $t$ using all available information, i.e.\ $K_t$ is $\mathcal{G}_t$-measurable.
In our continuous-time setting it is likely that the information about the unobservable true drift process changes continuously, therefore also the uncertainty set $K_t$ will be updated continuously in time. At each time $t\in[0,T]$, the investor will revise both the uncertainty set and the optimization problem
\[ \adjustlimits \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr]. \]
The strategy that is realized by the investor can then be found as $(\pi^*_t)_{t\in[0,T]}$ with $\pi^*_t=\pi^{(t),*}_t$ for any $t\in[0,T]$. It has the form
\[ \pi^*_t=\frac{1}{1-\gamma}A\mu^*_t +hc \]
where $(\mu^*_t)_{t\in[0,T]}$ is constructed via $\mu^*_t=\mu^{(t),*}_t$ for all $t\in[0,T]$. Note that the processes $(\mu^*_t)_{t\in[0,T]}$ and $(\pi^*_t)_{t\in[0,T]}$ are uniquely determined, $\mathbb{G}$-adapted and in general non-constant.
In the special case where $K_t = K_0$ for all $t\in[0,T]$, i.e.\ where our reference drift is simply a constant $\nu$, and also the matrix $\Gamma_t=\Gamma$ as well as the radius $\kappa_t=\kappa$ are constant in time, also $(\mu^*_t)_{t\in[0,T]}$ and $(\pi^*_t)_{t\in[0,T]}$ are constant in time. The constant values are the ones that we also get in the setting with constant drift and uncertainty set in Theorem~\ref{thm:solution_of_the_inf_sup_problem}.
\section{Construction of Uncertainty Sets via Filters}\label{sec:construction_of_uncertainty_sets_via_filters}
In the preceding sections we have seen how the duality approach from Sass and Westphal~\cite{sass_westphal_2020} carries over to a financial market where the drift is not necessarily constant. The generalized model allows for local uncertainty sets of the form
\[ K_t = \bigl\{ \mu\in\mathbb{R}^d \,\big|\, (\mu-\nu_t)^\top\Gamma_t^{-1}(\mu-\nu_t)\leq \kappa_t^2 \bigr\}, \quad t\in[0,T]. \]
We have fixed an investor filtration $\mathbb{G}=(\mathcal{G}_t)_{t\in[0,T]}$ describing the investor's information in the course of time. Our model then assumes that the processes $\nu=(\nu_t)_{t\in[0,T]}$, $\Gamma=(\Gamma_t)_{t\in[0,T]}$ and $\kappa=(\kappa_t)_{t\in[0,T]}$ are $\mathbb{G}$-adapted. Recall that $\nu$ takes values in $\mathbb{R}^d$, $\Gamma$ in the set of symmetric and positive-definite matrices in $\mathbb{R}^{d\times d}$ and $\kappa$ on the positive real line.
We motivated the reference drift $\nu$ as an estimation for the true drift, based on the information available to the investor. Here we want to make this more specific by considering the filter.
\subsection{Confidence regions as uncertainty sets}
The filter is the conditional distribution of $\mu$ given the available information $\mathbb{G}$. We take $\nu$ to be the conditional expectation of the drift given $\mathbb{G}$, i.e.\ $\nu_t=\muhat{}{t}:=\E[\mu_t\,|\,\mathcal{G}_t]$ for every $t\in[0,T]$. The conditional covariance matrix
\[ \gam{}{t} := \E\bigl[(\mu_t-\muhat{}{t})(\mu_t-\muhat{}{t})^\top\,\big|\,\mathcal{G}_t\bigr] \]
measures how close the estimator $\muhat{}{t}$ is to the true drift. Note that by construction both $(\muhat{}{t})_{t\in[0,T]}$ and $(\gam{}{t})_{t\in[0,T]}$ are $\mathbb{G}$-adapted processes. The key idea for constructing uncertainty sets based on the filter is to create confidence regions centered around $\muhat{}{t}$, shaped by $\gam{}{t}$ for every $t\in[0,T]$.
Let us assume that the drift process and the investor filtration are such that the filter is normally distributed, more precisely
\[ \mu_t\,|\,\mathcal{G}_t \sim \mathcal{N}(\muhat{}{t},\gam{}{t}). \]
By applying a simple transformation we deduce that
\[ (\mu_t-\muhat{}{t})^\top(\gam{}{t})^{-1}(\mu_t-\muhat{}{t}) \]
given $\mathcal{G}_t$ is $\chi^2$-distributed with $d$ degrees of freedom. We fix some $\eta\in(0,1)$ and observe that a $(1-\eta)$-confidence region can be obtained from
\[ 1-\eta = \mathbb{P}\Bigl((\mu_t-\muhat{}{t})^\top(\gam{}{t})^{-1}(\mu_t-\muhat{}{t}) \leq \chi^2_{d,1-\eta} \,\Big|\,\mathcal{G}_t\Bigr). \]
Here, $\chi^2_{d,1-\eta}$ denotes the $(1-\eta)$-quantile of the $\chi^2$-distribution with $d$ degrees of freedom. This motivates the choice of
\[ K_t = \bigl\{ \mu\in\mathbb{R}^d \,\big|\, (\mu-\muhat{}{t})^\top(\gam{}{t})^{-1}(\mu-\muhat{}{t}) \leq \chi^2_{d,1-\eta} \bigr\}, \quad t\in[0,T], \]
i.e.\ taking $\nu_t=\muhat{}{t}$, $\Gamma_t=\gam{}{t}$ and $\kappa_t=\sqrt{\chi^2_{d,1-\eta}}$ for every $t\in[0,T]$.
If indeed $\mu_t$ given $\mathcal{G}_t$ is normally distributed, we additionally know that at any fixed time $t\in[0,T]$ the probability that $\mu_t\in K_t$, conditional on $\mathcal{G}_t$, is equal to $1-\eta$. Note that $K_t$ is still a reasonable uncertainty set for $\mu_t$ in the case where the assumption about the normal distribution of the filter is not fulfilled.
\subsection{Comparison of different investor filtrations}
The preceding section explains how time-dependent uncertainty sets can be created based on filters. We now apply this to a model with an unobservable Ornstein--Uhlenbeck drift process and unbiased, normally distributed expert opinions arriving at discrete points in time. The setting is based on Gabih et al.~\cite{gabih_kondakji_sass_wunderlich_2014} as well as Sass et al.~\cite{sass_westphal_wunderlich_2017, sass_westphal_wunderlich_2021}.
Returns in this setting are modelled as
\[ \mathrm{d} R_t = \mu_t\,\mathrm{d} t+\sigma_R\,\mathrm{d} W^R_t, \]
where $W^R=(W^R_t)_{t\in[0,T]}$ is an $m$-dimensional Brownian motion with $m\geq d$ and where we assume that $\sigma_R\in\mathbb{R}^{d\times m}$ has full rank. The drift process $\mu$ is defined by the Ornstein--Uhlenbeck dynamics
\[ \mathrm{d} \mu_t = \alpha (\delta - \mu_t)\,\mathrm{d} t + \beta\,\mathrm{d} B_t, \]
where $\alpha$ and $\beta\in\mathbb{R}^{d\times d}$, $\delta\in\mathbb{R}^d$ and $B=(B_t)_{t\in [0,T]}$ is a $d$-dimensional Brownian motion that is independent of $W^R$. The matrices $\alpha$ and $\beta\beta^\top$ are assumed to be symmetric and positive definite. We further make the assumption that $\mu_0\sim\mathcal{N}(m_0,\Sigma_0)$ for some $m_0\in\mathbb{R}^d$ and some symmetric and positive-semidefinite matrix $\Sigma_0\in\mathbb{R}^{d\times d}$, and that $\mu_0$ is independent of the Brownian motions $W^R$ and $B$, i.e.\ $\mu$ is independent of $W^R$.
Information about the drift process can be drawn from return observations. An additional source of information in this model are expert opinions that arrive at discrete points in time and give an unbiased estimate of the state of the drift at that time point.
We assume that the expert opinions arrive at the information dates $(T_k)_{k\in I}$ and that an expert opinion at time $T_k$ is of the form
\[ Z_k = \mu_{T_k}+(\Gamma_k)^{1/2}\varepsilon_k, \]
where the matrices $\Gamma_k\in\mathbb{R}^{d\times d}$ are symmetric and positive definite and the $\varepsilon_k$ are multivariate $\mathcal{N}(0,I_d)$-distributed and independent of the Brownian motions in the market and of~$\mu_0$. The sequence of information dates $(T_k)_{k\in I}$ is also independent of the $(\varepsilon_k)_{k\in I}$ and the Brownian motions as well as of $\mu_0$. In particular, given $\mu_{T_k}$ the expert opinion is multivariate $\mathcal{N}(\mu_{T_k},\Gamma_k)$-distributed.
The model then gives rise to various investor filtrations $\mathbb{G}=(\mathcal{G}_t)_{t\in[0,T]}$. We consider the cases
\begin{alignat*}{3}
&\mathbb{G}=\mathbb{F}^N && =(\mathcal{F}^N_t)_{t\in[0,T]} && \text{ where } \mathcal{F}^N_t=\sigma(\mathcal{N}_{\mathbb{P}}), \\
&\mathbb{G}=\mathbb{F}^R && =(\mathcal{F}^R_t)_{t\in[0,T]} && \text{ where } \mathcal{F}^R_t=\sigma((R_s)_{s\in[0,t]})\vee\sigma(\mathcal{N}_{\mathbb{P}}), \\
&\mathbb{G}=\mathbb{F}^E && =(\mathcal{F}^E_t)_{t\in[0,T]} && \text{ where } \mathcal{F}^E_t=\sigma((T_k,Z_k)_{T_k\leq t})\vee\sigma(\mathcal{N}_{\mathbb{P}}), \\
&\mathbb{G}=\mathbb{F}^C && =(\mathcal{F}^C_t)_{t\in[0,T]} && \text{ where } \mathcal{F}^C_t=\sigma((R_s)_{s\in[0,t]})\vee\sigma((T_k,Z_k)_{T_k\leq t})\vee\sigma(\mathcal{N}_{\mathbb{P}})
\end{alignat*}
for the investor filtrations, where we write $\mathcal{N}_{\mathbb{P}}$ for the set of null sets under $\mathbb{P}$, i.e.\ we work with the filtrations that are augmented by null sets. We speak of the investor with filtration $\mathbb{F}^H$, $H\in\{N,R,E,C\}$, as the $H$-investor. Note that the $N$-investor observes neither returns nor expert opinions and only has knowledge about the market parameters. The $R$-investor observes only the return process, the $E$-investor only the discrete-time expert opinions, and the $C$-investor the combination of both.
\begin{example}
Based on one realization of the model's stochastic processes, fixing one information setting $H\in\{N,R,E,C\}$, we obtain one realization of the filter, leading to a time-dependent uncertainty set $K^H$. For illustration purposes we plot in Figure~\ref{fig:uncertainty_sets_various_investor_filtrations} against time a realization of the different filters with resulting uncertainty sets in a market with $d=1$ stock.
The various subplots are all based on the same realization of the drift process $\mu$, returns $R$ and expert opinions $Z_k$. As a first case we consider in Figure~\ref{subf:H=E_n=0} the degenerate information setting $H=N$, corresponding to an investor who observes neither the return process nor the expert opinions. The only knowledge the investor has are the model parameters. The conditional mean is in this case constantly equal to the long-term mean $\delta$ of the drift process. The resulting uncertainty set converges very fast to a fixed interval centered around $\delta$.
For $H=R$, the uncertainty set moves up and down along with the conditional mean as can be seen in Figure~\ref{subf:H=R}. In Figures~\ref{subf:H=E_n=10} and~\ref{subf:H=C_n=10} we have equidistant information dates with expert opinions. The corresponding uncertainty set jumps at information dates along with the conditional mean, due to the updates caused by an incoming expert opinion. In the case $H=E$ shown in Figure~\ref{subf:H=E_n=10}, no further information between the arrival times of the expert opinions is used. Therefore, the filter evolves deterministically in direction of the long-term mean between the jump times. In the case $H=C$ in Figure~\ref{subf:H=C_n=10}, in addition to the expert opinions the return observations are used. It also becomes apparent from the plots that the conditional variance decreases at information dates, leading to a shrinking uncertainty set.
Neither of the information filtrations leads to a perfect uncertainty set in the sense that the true drift stays in that uncertainty set at any point in time. By the setup of the uncertainty set there is always a positive probability that the true drift process moves out of the uncertainty set at some point in time.
\begin{figure}[ht]
\begin{subfigure}{.44\textwidth}
\includegraphics{confidence_region_N}
\caption{$\mathbb{G}=\mathbb{F}^N$}\label{subf:H=E_n=0}
\end{subfigure}%
\begin{subfigure}{.44\textwidth}
\includegraphics{confidence_region_R}
\caption{$\mathbb{G}=\mathbb{F}^R$}\label{subf:H=R}
\end{subfigure}%
\begin{subfigure}{.12\textwidth}
\centering
\includegraphics{legend_confidence_regions}
\end{subfigure}
\newline
\begin{subfigure}{.44\textwidth}
\includegraphics{confidence_region_E}
\caption{$\mathbb{G}=\mathbb{F}^E$}\label{subf:H=E_n=10}
\end{subfigure}%
\begin{subfigure}{.44\textwidth}
\includegraphics{confidence_region_C}
\caption{$\mathbb{G}=\mathbb{F}^C$}\label{subf:H=C_n=10}
\end{subfigure}
\caption{Uncertainty sets based on filters for various investor filtrations $\mathbb{F}^H$. Each subplot is based on the same realization of the drift and return process and expert opinions. Based on this realization, the filter of the $H$-investor can be computed. The uncertainty set $K^H$ is then determined according to the filter realization.}
\label{fig:uncertainty_sets_various_investor_filtrations}
\end{figure}
\end{example}
We now give a numerical example to illustrate the effect that the worst-case optimization among uncertainty sets created from filters has for the various investor filtrations considered before. We create for a fixed realization of the drift process $\mu$, of the return process $R$ and the expert opinions $Z_k$ a time-dependent uncertainty set for each of the corresponding filters. The aim is to compare the robust strategies that take into account model uncertainty with the ``naive'' strategies that rely on the respective drift estimates, only.
\paragraph{Model parameters.}
We want to apply our worst-case utility maximization problem, in particular also imposing the constraint $\langle\pi_t,\mathbf{1}_d\rangle=h$ on the investor's strategies. For that purpose we take a market with $d=2$ stocks here. We fix an investment horizon of $T=1$ and take $h=1$. Moreover, we assume that investors start with an initial wealth of $x_0=1$, use power utility functions $U_\gamma$ with $\gamma=0.5$ and a confidence level $\eta=0.1$ to create their uncertainty sets. Further parameters of the market are given in Table~\ref{tab:market_parameters_for_utility_study}.
\begin{table}[ht]
\centering
\begin{tabular}{llll}
\hline
\addlinespace[1ex]
mean reversion speed of drift process &$\alpha$ &$=$ &$\begin{pmatrix}
3 & 0\\
0 & 2
\end{pmatrix}$ \\
\addlinespace[1ex]
volatility of drift process &$\beta$ &$=$ &$\begin{pmatrix}
0.50 & 0.25\\
0.25 & 0.50
\end{pmatrix}$ \\
\addlinespace[1ex]
mean reversion level of drift process &$\delta$ &$=$ &$\begin{pmatrix}
0.02\\
0.03
\end{pmatrix}$ \\
\addlinespace[1ex]
initial mean of drift process &$m_0$ &$=$ &$\begin{pmatrix}
0.02\\
0.03
\end{pmatrix}$ \\
\addlinespace[1ex]
initial variance of drift process &$\Sigma_0$ &$=$ &$\begin{pmatrix}
0.01 & 0\\
0 & 0.01
\end{pmatrix}$ \\
\addlinespace[1ex]
volatility of returns &$\sigma_R$ &$=$ &$\begin{pmatrix}
0.10 & 0.05\\
0.05 & 0.01
\end{pmatrix}$ \\
\addlinespace[1ex]
volatility of continuous expert &$\sigma_J$ &$=$ &$\begin{pmatrix}
0.10 & 0.05\\
0.05 & 0.01
\end{pmatrix}$ \\
\addlinespace[1ex]\hline
\end{tabular}
\caption{Market parameters for numerical example.}\label{tab:market_parameters_for_utility_study}
\end{table}
\paragraph{Simulation study.}
For the given model parameters we simulate a drift process, the return process $R$ and $n=10$ discrete-time expert opinions arriving at deterministic and equidistant information dates on $[0,T]$. We then obtain a realization of the filters $(\muhat{H}{}, \gam{H}{})$ for any of the information settings $H$ from above. As before, this leads to one time-dependent uncertainty set for each of the investors.
We can then determine the worst-case drift process $(\mu^*_t)_{t\in[0,T]}$ and the optimal strategy $(\pi^*_t)_{t\in[0,T]}$ that is realized by the investor who solves at each time point the local optimization problem
\[ \adjustlimits \sup_{\pi^{(t)}\in\mathcal{A}_h(t,x)} \inf_{\mu^{(t)}\in \mathcal{K}^{(t)}} \E_{\mu^{(t)}}\Bigl[U_\gamma\bigl(X^{t,x,\pi^{(t)}}_T\bigr)\Bigr], \]
where the information at $t$ is now given by $\mathcal{G}_t = \mathcal{F}^H_t$.
Recall that $(\mu^*_t)_{t\in[0,T]}$ and $(\pi^*_t)_{t\in[0,T]}$ are calculated from the solutions of the local optimization problems via
\[ \pi^*_t=\pi^{(t),*}_t, \quad \mu^*_t=\mu^{(t),*}_t \]
for all $t\in[0,T]$.
The value of each investor's worst-case optimization is then equal to (writing now $X_T^\pi$ for $X_T^{0,x_0,\pi}$)
\begin{equation}\label{eq:worst-case_expected_utility}
\E_{\mu^*}\bigl[U_\gamma(X^{\pi^*}_T)\bigr].
\end{equation}
The quantity in~\eqref{eq:worst-case_expected_utility} is the worst-case expected utility from the $H$-investor's point of view when using the robust strategy $\pi^{*}$. For comparison, we also compute
\[ \E_{\mu^*}\bigl[U_\gamma(X^{\hat{\pi}}_T)\bigr], \quad \E_{\nu}\bigl[U_\gamma(X^{\pi^*}_T)\bigr] \quad\text{and}\quad \E_{\nu}\bigl[U_\gamma(X^{\hat{\pi}}_T)\bigr], \]
where $\nu=\muhat{H}{}$ is the conditional mean of the $H$-investor's filter and $\hat{\pi}$ is the corresponding optimal strategy given that the drift equals $\muhat{H}{}$, i.e.\
\[ \hat{\pi}_t=\frac{1}{1-\gamma}A\muhat{H}{t}+hc. \]
We repeat this simulation $10\,000$ times where in each iteration a new drift process, a new return process and new expert opinions are simulated based on the parameters given above. Table~\ref{tab:comparison_of_utility_for_different_investors} gives the sample mean of the various expected utilities over all simulations and in brackets the corresponding sample standard deviation.
\begin{table}[ht]
\centering
\begin{tabular}{cc|r|r|r|r}
$H$ &$n$ &\multicolumn{1}{c}{$\E_{\mu^*}\bigl[U_\gamma(X^{\pi^*}_T)\bigr]$} &\multicolumn{1}{|c}{$\E_{\mu^*}\bigl[U_\gamma(X^{\hat{\pi}}_T)\bigr]$} &\multicolumn{1}{|c}{$\E_{\nu}\bigl[U_\gamma(X^{\pi^*}_T)\bigr]$} &\multicolumn{1}{|c}{$\E_{\nu}\bigl[U_\gamma(X^{\hat{\pi}}_T)\bigr]$}\rule[-0.9ex]{0pt}{0pt}\\\hline
$N$ & &1.6179 \textit{(0.0000)} &1.5996 \textit{(0.0000)} &2.0196 \textit{\phantom{00}(0.0000)} &2.0426 \textit{\phantom{00\,00}(0.0000)}\rule{0pt}{2.6ex}\\
$R$ & &1.7086 \textit{(0.1057)} &0.7754 \textit{(0.3737)} &2.2362 \textit{\phantom{00}(2.4692)} &25.9029 \textit{\phantom{00\,}(732.4104)}\\
$E$ &$10$ &1.7055 \textit{(0.1117)} &0.8170 \textit{(0.3870)} &2.2393 \textit{\phantom{00}(3.4208)} &21.1610 \textit{\phantom{00\,}(530.6829)}\\
$C$ &$10$ &1.7854 \textit{(0.4027)} &0.6891 \textit{(0.3752)} &4.5313 \textit{(134.5858)} &264.0838 \textit{(19\,288.2826)}\\
\end{tabular}
\caption{Comparison of utility for different investors.}\label{tab:comparison_of_utility_for_different_investors}
\end{table}
\paragraph{Observations.}
When comparing the worst-case expected utility $\E_{\mu^*}[U_\gamma(X^{\pi^*}_T)]$ among the investors we see that the information setting $H=N$, which corresponds to only knowing the model parameters, gives the lowest value. The observation of returns or of $n=10$ expert opinions increases this value. The combination of return observation and discrete-time expert opinions yields a considerably larger worst-case expected utility.
In the next column, $\E_{\mu^*}[U_\gamma(X^{\hat{\pi}}_T)]$ measures the expected utility when using strategy $\hat{\pi}$, given that the true drift is actually the worst-case drift $\mu^*$. The values are in any case smaller than the corresponding expected utility when using the robust strategy $\pi^*$. What is striking is that the information setting $H=N$, i.e.\ only knowledge of the model parameters, gives the best expected utility here. Adding more information, from return observations or expert opinions, and using the optimal strategy based on the filter leads to a smaller worst-case expected utility. This shows that for the worst-case optimization problem it is dangerous for investors to rely on their estimates of the drift, i.e.\ the conditional mean of the filter, only. They need to robustify their strategy by taking into account model uncertainty to be able to profit from any additional information. This effect can be linked to overconfidence of experts as studied empirically by Heath and Tversky~\cite{heath_tversky_1991}, seeing that more knowledge about the drift process leads to a worse expected utility in the non-robust case due to taking more risky strategies.
The last two columns show the expected utility when using strategy $\pi^*$, respectively $\hat{\pi}$, given that the true drift was actually the conditional mean $\nu=\muhat{H}{}$. Of course, when compared to the expected utility given the worst-case drift $\mu^*$, the expected utility given $\nu$ is much higher. Not surprisingly, the performance of $\hat{\pi}$ given drift $\nu$ is on average extremely good. However, we also notice the very large sample standard deviation. In comparison to that, we see that the robust strategies $\pi^*$ perform reasonably well given drift $\nu$, even though they are tailored for the worst-case drift in the respective uncertainty set. At the same time, the sample standard deviation is much smaller than for strategy $\hat{\pi}$.
\paragraph{Conclusions.}
In conclusion, we see that a surplus of information, either from return observations or expert opinions, results in better strategies in general. However, investors do need to account for model uncertainty by choosing a robustified strategy $\pi^*$ instead of relying on the respective filter only. The naive strategy $\hat{\pi}$ performs extremely well if the true drift coincides with the conditional mean $\muhat{H}{}$, but it is much more vulnerable to model misspecifications than the robust strategy~$\pi^*$.
|
{
"timestamp": "2021-06-01T02:21:46",
"yymm": "2009",
"arxiv_id": "2009.14559",
"language": "en",
"url": "https://arxiv.org/abs/2009.14559"
}
|
\section{Introduction}
The local behavior of the perturbed dark energy has become an object of study in a series of works of the last decade. The impact of perturbation and clusterization of dark energy on the matter dynamics was studied on clusters of galaxies and galaxy scales (for example, in \cite{Creminelli2010,Pace10,Tsizh2015,novosyadlyj_2016}) and on the astrophysical scale (\cite{Babichev2013,novosyadlyj_2014,smerechynskyi19,tsizh14,Kamiab11,Alavirad13,Pani11,Yazadjiev15,Sbisa20,Kpadonou15}). In general, studies have shown, that the local impact of minimally coupled perturbed dark energy is mostly negligible, except some specific value regions of its parameters space, which can be used for ruling out these values.
In particular, the dark energy models as well as the modified gravity theories are tested as an altering factor of the compact objects' properties. Usually, in these works the authors investigate how the gravitational potential is altered due to hidden component or alternative gravitation theory and how it changes the characteristics of a compact object. The comparison of the theoretical predictions with the corresponding observational data can give some constraint on the value of parameters of the theory. There are also a number of papers where the probing objects are, in particular, neutron stars (NS). For example, the gravitational aether theory is tested with NSs in \cite{Kamiab11}. In \cite{Alavirad13} the effect of a logarithmic f(R) theory on relativistic stars is studied. Similarly, the alternative theories of gravity are tested by NSs in \cite{Pani11} (Einstein-Dilaton-Gauss-Bonnet gravity), \cite{Yazadjiev15,Sbisa20} (R-squared gravity) and \cite{Kpadonou15} (f(T) gravity).
The stationary accretion of dark energy onto the Schwarzschild black hole was studied in \cite{Babichev2013}. We studied the static solutions of dark energy dynamical equations in the vicinity of compact objects and found that only relativistic objects with the lowest ratio ``radius to gravitational radius'' disturb the density of dynamical dark energy noticeably \cite{novosyadlyj_2014}. In the paper \cite{smerechynskyi19} we have investigated how the dynamical dark energy inside white dwarfs can change their mass-radius relation and have found that the squared effective speed of sound $c_s^2$ of dark energy must be larger than $\sim 5 \cdot 10^{-4}$ in order to satisfy the observed mass-radius relation for white dwarfs. In the paper \cite{tsizh14}, we have used the uncertainty of determination of gravitating mass in the Solar system as
an upper limit on the amount of clustered dark energy
and obtained the similar constraints for the value of $c_s^2 \geq 2 \cdot 10^{-4}$.
There are plenty of works which constrain the dark matter
parameters and theories of gravity based on the reduction of NS maximal mass caused by the accumulation of the dark matter in its interior (see \cite{Lavallaz10, Guver13, Ellis18,Ivanytskyi20}). In particular, authors of \cite{Deliyergiyev19} analyzed compact objects that contain dark matter admixed with ordinary matter made of neutron star and white dwarf materials and dependence of maximum radius of such objects on dark matter properties. In this work we use a similar approach towards another part of the dark sector -- minimally coupled dark energy.
The paper has the following structure. In Section~\ref{sect_2} we describe the equations of state for NS matter and hence present the model which governs the matter distribution inside NS. Section~\ref{sect_3} is devoted to the description of the dark energy model and the radial distribution of the dark energy inside NS and its dependence on the dark energy parameters. In Section~\ref{sect_4} we deduce the constraint on the effective speed of sound for the dark energy using the NSs and in Section~\ref{sect_5} we give our conclusions.
\section{Equation of state for the neutron star matter }
\label{sect_2}
The NSs exist due to the pressure of the degenerate gas of Fermi-particles, neutrons, much like the electrons in the case of white dwarfs, though, nuclear forces play an important role in the former ones.
The state of the matter inside white dwarfs is well known, but we are still uncertain about the composition and equation of state (EoS) for the matter at the higher densities that correspond to the NS interiors \cite{haensel_book_2007}.
Based on a given EoS one can yield the maximum mass of NS configuration, the so-called Tolman-Oppenheimer-Volkoff mass limit \cite{Tolman, Oppenheimer}.
The last one is similar to the Chandrasekhar mass limit \cite{Chandrasekhar_1931a, Chandrasekhar_1931b, Chandrasekhar_1935}, but strongly depends on the incorporated physics, resulting in different stiffness of EoS.
The maximum mass limit for NS is approximately $1.5 M_{\odot}$ for soft equations of state and reaches $3 M_{\odot}$ in the case of stiff ones \cite{bombaci_1996, alsing_2018}.
The maximal known mass of observed NSs is a crucial value
for testing
different EoSs
(see, for example, \cite{alsing_2018, zhou_2019,annala_2018,zhang_2018,zhang_2019}).
For the description of the NS interior we have exploited three unified EoSs developed by the Brussels-Montreal group \cite{goriely_2010, pearson_2011, pearson_2012,potekhin_2013}.
Despite the matter being under different physical conditions and states, such an EoS is valid throughout all parts of a neutron star -- from the outer envelope to its crust and core.
We will use the same denotations BSk19, BSk20 and BSk21 for the considered EoSs as in \cite{potekhin_2013}.
They differ by its stiffness, BSk21 is the stiffest one and BSk19 is the softest one.
The analytical representation of the equations of state with variables $\xi=\lg(\rho_m/\textrm{g}\cdot\textrm{cm}^{-3})$ and
$\zeta = \lg(p_m/\textrm{dyn}\cdot\textrm{cm}^{-2})$ ($\lg$ denotes $\log_{10}$) is defined by the following parametrization:
\begin{eqnarray}
\begin{aligned}
\zeta =
\frac{a_1+a_2\xi+a_3\xi^3}{1+a_4\,\xi}\,
\frac{1}{\exp\left[a_5(\xi-a_6)\right]+1}
+\frac{a_7+a_8\xi}{\exp\left[a_9(a_6-\xi)\right]+1}
+ \frac{a_{10}+a_{11}\xi}{\exp\left[a_{12}(a_{13}-\xi)\right]+1}
\\
+ \frac{a_{14}+a_{15}\xi}{\exp\left[a_{16}(a_{17}-\xi)\right]+1}
+ \frac{a_{18}}{1+ [a_{19}\,(\xi-a_{20})]^2}
+ \frac{a_{21}}{1+ [a_{22}\,(\xi-a_{23})]^2}
,
\end{aligned}
\label{fit.P}
\end{eqnarray}
where $p_m$ and $\rho_m$ are the local pressure and density of the NS (or baryonic) matter, respectively.
It is the same for three considered cases, only the values of coefficients $a_1$-$a_{23}$ are different (see \cite{potekhin_2013}).
Such representation of EoS simplifies its usage for the consideration of the inner structure of NSs. The fitting procedure introduces the errors of the macroscopic NS characteristics, but they are far below the observational uncertainties \cite{potekhin_2013}.
\begin{figure}
\centering\includegraphics[width=.6\textwidth]{M_R_NS.pdf}
\caption{Mass-radius relation for neutron stars with three considered equations of state.}
\label{m-r}
\end{figure}
Using these equations of state, one can solve the hydrostatic equilibrium equation to obtain the corresponding mass-radius relations, shown in Fig.~\ref{m-r}.
As we can see, the stiffer the EoS is, the higher maximal mass of NS corresponds to it.
These values (in solar mass units) as well as the corresponding central densities $\rho_m(0)\equiv\rho_c$ of NSs are given in Table~\ref{m-r_tab}.
\begin{table}[h!]
\centering
\caption{The maximum masses and corresponding central densities of neutron stars for three equation of states BSk19, BSk20, BSk21 \cite{potekhin_2013}.}
\begin{tabular}{ccc}
\hline
\hline
EoS&$M_{max}/M_{\odot}$& $\rho_c$, $10^{15}$ g/cm$^3$\\
\hline
BSk19& $1.86$& $3.48$ \\
\hline
BSk20& $2.16$& $2.69$ \\
\hline
BSk21& $2.27$& $2.27$ \\
\hline
\hline
\end{tabular}
\label{m-r_tab}
\end{table}
The maximal known masses for observed NSs are: $2.14^{+0.10}_{-0.09} M_{\odot}$ for PSR J0740+6620 (here and below the confidence interval is 68.3\%), obtained from the measurement of the relativistic Shapiro delay \cite{cromartie_2019}; and $2.27_{-0.15}^{+0.17} M_{\odot}$ for PSR J2215+5135, yielded by the simultaneous fitting of radial velocity curves and three-band light curves \cite{linares_2018, linares_2019}.
It is worth mentioning that there are objects which are reported to have masses even larger. For instance, pulsar PSR B1957+20 with mass $\sim 2.4 M_{\odot}$ \cite{kerkwijk2010}, but systematic uncertainties of the mass determination are large. It is argued, that the accuracy of this measurement is partly limited by optical flares and variable emission lines of companion's stellar wind \cite{romani_2015,linares_2019}. There are evidences indicating significantly lower mass of that pulsar ($\sim1.8 M_{\odot}$ \cite{horvath_2017,luo_2020}).
Thus, to date, the maximum mass of observed NSs, in general, agrees with estimates on the upper bound of NS masses made within different approaches \cite{alsing_2018, shibata_2019,lawrence_2015,fryer_2015,margalit_2017,rezzolla_2018,ruiz_2018,abbott_2017}.
One can infer from the comparison of theoretical maximum masses from Table~\ref{m-r_tab} and observed ones, that the maximum mass values predicted by the equations of state BSk20 and BSk21 are inside the range inferred from observations and BSk19 does not allow the existence of such massive NSs. Probably, there is a discrepancy for BSk20 EoS and PSR J2215+5135 mass, if no evidence of significant uncertainty of its mass determination is found. In the following sections we will use all three mentioned equations of state, bearing in mind that BSk19 can not explain the existence of the most massive among known NSs and we consider it only for the sake of consistency and analysis.
\section{Dark energy inside a neutron star}
\label{sect_3}
\subsection{Dark energy model}
It was shown that the minimally-coupled scalar field model of dark energy with barotropic EoS
\begin{equation}
\label{eq1a}
p_{de}=w(\rho_{de})c^2\rho_{de},
\end{equation}
where $p_{de}$ and $\rho_{de}$ are pressure and density of dark energy, respectively, can agglomerate inside and in vicinity of a compact object when the EoS parameter $w$ and the squared effective speed of sound $c_s^2$ ($c_s$ in the units of speed of light) are related as \cite{Babichev2013, novosyadlyj_2014}
\begin{equation}
w=c_s^2-(c_s^2-w_{\infty})\frac{\rho_{\infty}}{\rho_{de}}.\label{w-rho}
\end{equation}
Here $\rho_{\infty}$ is the background density of dark energy (at $r\rightarrow \infty$), which in our case is equal to $10^{-23}$g/cm$^3$ \cite{novosyadlyj_2013}. The value of background dark energy density was chosen to be higher than the cosmological value in $\Lambda$CDM model. The reason for this is that dark energy is assumed to undergo clusterization process along with dark matter during initial perturbation growth. We considered only the quintessence type of dark energy with $w_{\infty}>-1$. The model, described by equation (\ref{w-rho}), implies the constant effective speed of sound of dark energy.
The hydrodynamical representation of the scalar field dark energy as a perfect or imperfect fluid with barotropic EoS is usually used in cosmology. The Lagrangian of the field $\mathcal{L}(X,U)$ with kinetic term $X$ and potential $U$, is connected to phenomenological hydrodynamical quantities as follows \cite{armendariz-picon_1999}
$$ c^2 \rho_{de} = 2X\mathcal{L}_{,X} - \mathcal{L},\quad p_{de} = \mathcal{L}, \quad
w = \frac{p_{de}}{c^2\rho_{de}} = \frac{\mathcal{L}}{2X\mathcal{L}_{,X}}, \quad c_s^2 = \frac{\delta p_{de}}{c^2\delta\rho_{de}} = \frac{\mathcal{L}_{,X}}{2X\mathcal{L}_{,XX} - \mathcal{L}_{,X}}. $$
The scalar field dark energy with conditions $c_s^2=const>0$ and $w<0$ in stationary Minkowski or Schwarzschild world is governed by the Klein-Gordon or hydrodynamical continuity equations. In \cite{novosyadlyj_2014} we have shown how the eq. (3) is deduced in the framework of these conditions and in \cite{Sergijenko14} how the scalar field variables are related with hydrodynamical ones for this dark energy model.
\subsection{Distribution of the dark energy inside a neutron star}
In order to estimate the influence of the dynamical dark energy on the equilibrium condition in a NS we suppose that it is the non-rotational non-magnetic star which is in static equilibrium: the pressure gradient of baryon matter balances the gravitational attraction of the total mass in a given sphere in the star, as well as the pressure gradient of the dark energy balances the gravitational attraction of the same mass. The Einstein and conservation law equations for minimally coupled baryonic and dark energy are used.
Therefore, we considered a spherically symmetric object for which the space-time metric can be written in the form
\begin{equation}
\label{eq2}
ds^2=e^{\nu(r)}c^2d\tau^2-e^{\lambda(r)}dr^2-r^2\left(d\theta^2+\sin^2{\theta}d\varphi^2\right).
\end{equation}
If we limit ourselves to the case of static configuration of dark energy inside a NS, the components of metric will not depend on time and can be obtained from the Einstein equations with the boundary condition $\lambda(r=0) = 0$ \cite{novosyadlyj_2014}
\begin{equation}
\label{metric_func}
\begin{aligned}
e^{-\lambda(r)}&=1-\frac{8\pi G}{c^2r}\int\limits_0^r \left[\rho_m(r')+\rho_{de}(r')\right]r'^2dr',\\
\nu(r)+\lambda(r)&=-\frac{8\pi G}{c^2}\int\limits_r^{\infty} \left[\rho_m(r')+\rho_{de}(r')+\frac{p_m(r')+p_{de}(r')}{c^2}\right] e^{\lambda(r')}r'dr'.
\end{aligned}
\end{equation}
Here $\rho_m$, $p_m$ are the local density and pressure of baryonic matter and $\rho_{de}$, $p_{de}$ denote the corresponding characteristics of dark energy. Other boundary conditions are the following: $\nu(\infty) = -\lambda(\infty)$, $\rho_m(R_+)=0$, $\rho_{de}(\infty)=\rho_{\infty}$.
With the metric functions given in (\ref{metric_func}) we numerically solved the equilibrium equations for both baryonic matter and dark energy
\begin{eqnarray}
\label{equilibr_eq}
\begin{aligned}
\frac{dp_m}{dr}+\frac12(\rho_m c^2+p_m)\frac{d\nu}{dr}=0,\\
\frac{dp_{de}}{dr}+\frac12(\rho_{de} c^2+p_{de})\frac{d\nu}{dr}=0,
\end{aligned}
\end{eqnarray}
applying the iterative procedure. On the initial step we evaluated the gravitational potential without the dark energy influence, and, thereafter, found the distribution of the dark energy in such potential. Then we solved the system of equations for baryonic matter and dark energy densities and their joint gravitational potential, and compared the results with ones obtained in the previous step. Then we re-evaluated the distribution of dark energy and baryonic matter in the new potential. Such procedure was repeated until the solutions converged or the iteration limit exceeded (for more details see~\cite{smerechynskyi19}).
\begin{figure}[h!]
\begin{minipage}{.32\textwidth}
\includegraphics[width=1\textwidth]{rho_de_r_rho0_2_cs2_1e-2_bsk20_x2_v1.pdf}
\centering{(a)}
\end{minipage}
\begin{minipage}{.32\textwidth}
\includegraphics[width=1\textwidth]{rho_de_r_rho0_2_cs2_1e-1_bsk20_x2_v1.pdf}
\centering{(b)}
\end{minipage}
\begin{minipage}{.32\textwidth}
\includegraphics[width=1\textwidth]{rho_de_r_rho0_5_cs2_1e-1_bsk20_x2_v1.pdf}
\centering{(c)}
\end{minipage}
\caption{The relative deviation $\delta_{de} (r) = (\rho_{de}(r)-\rho_{\infty})/\rho_{\infty}$ of dark energy density as a function of radial coordinate $r$ inside a neutron star with radius $R$ for the case of BSk20 EoS: (a) $\rho_c = 2\rho_n$, $c_s^2=0.01$; (b) $\rho_c = 2\rho_n$, $c_s^2=0.1$; (c) $\rho_c = 5\rho_n$, $c_s^2=0.1$. Line types correspond to different values of the parameter $w_{\infty}$: from $-0.80$ for top curve to $-0.99$ for bottom curve. The upper x-axis corresponds to matter density at given radial coordinates.}
\label{rho_r_20}
\end{figure}
The solutions of the system of equations (\ref{metric_func}--\ref{equilibr_eq}) for the dark energy component are shown in Fig.~\ref{rho_r_20} in the form of radial dependence of the relative deviation of the dark energy density $\delta_{de}(r)$ inside a star. The results correspond to BSk20 EoS for two different values of effective speed of sound $c_s^2 = 0.01$ and $0.1$, and central density of the NS matter $\rho_c=2\rho_n$ and $5\rho_n$, where $\rho_n=2.8\cdot 10^{14}$~g/cm$^3$ is the so called normal nuclear density \cite{haensel_book_2007}.
The line types correspond to the different values of the parameter $w_{\infty}$ given in the figure.
The upper x-axis corresponds to the matter density at the given radial coordinates.
One can see, that the relative deviation of the dark energy density in NS is very sensitive to the value of $c_s^2$ and is increasing as the latter one is decreasing. It follows from the comparison of Figs.~\ref{rho_r_20}a and \ref{rho_r_20}b, corresponding to the same value of $\rho_c$. $\delta_{de}(r)$ increases also with increasing central density of baryonic matter $\rho_c$ at constant effective speed of sound of dark energy $c_s^2$ (Figs.~\ref{rho_r_20}b and \ref{rho_r_20}c).
Also, one can infer that lowering $w_{\infty}$ causes a smaller deviation of the dark energy density from the background one. This makes no surprise: it is well known that the dark energy with $w_{\infty}=-1$ is not perturbed at all, so one would expect the deviation to be smaller as $w_{\infty}$ approaches -1 and vanishes at $w_{\infty}=-1$.
The solutions for the case of the stiffest of considered EoSs, namely BSk21, are illustrated in Fig.~\ref{rho_r_21}. Similar dependencies of the relative deviation of the dark energy density on parameters $w_{\infty}$, $c_s^2$ and $\rho_c$ can be inferred from Figs.~\ref{rho_r_21}a--\ref{rho_r_21}c. However, it should be noted that the amount of dark energy inside a NS is larger for the case of stiffer EoS assuming the same values of other parameters.
\begin{figure}[h!]
\begin{minipage}{.32\textwidth}
\includegraphics[width=1\textwidth]{rho_de_r_rho0_2_cs2_1e-2_bsk21_x2_v1.pdf}
\centering{(a)}
\end{minipage}
\begin{minipage}{.32\textwidth}
\includegraphics[width=1\textwidth]{rho_de_r_rho0_2_cs2_1e-1_bsk21_x2_v1.pdf}
\centering{(b)}
\end{minipage}
\begin{minipage}{.32\textwidth}
\includegraphics[width=1\textwidth]{rho_de_r_rho0_5_cs2_1e-1_bsk21_x2_v1.pdf}
\centering{(c)}
\end{minipage}
\caption{The same as in Fig.~\ref{rho_r_20}, but for the case of BSk21 equation of state.}
\label{rho_r_21}
\end{figure}
\section{Influence of dark energy on neutron star and parameter constraints}
\label{sect_4}
The density distributions of dark energy obtained in the previous section give us the possibility to calculate its total Lagrangian mass inside a star. Fig.~\ref{m_de-rho} illustrates the dark energy mass $M_{de}$ (in solar mass units) as a function of central density $\rho_c$ of baryonic matter for the values of $c_s^2$ ranging from $0.008$ to $0.013$ (depicted with different colors). The dependence on the parameter $w_{\infty}$ is shown by shadowed regions between solid lines corresponding to $w_{\infty}=-0.99$ and dash-dotted lines ($w_{\infty}=-0.80$). The dark energy mass rises steeply with central matter density with an exception of the region near $\rho_c/\rho_n\approx1$ and then reaches saturation level which is in the range of one percent of total mass for all considered values of $c_s^2$ and $w_{\infty}$ (see zoomed-in part of the figure in the upper left corner). The amount of dark energy is higher for smaller values of $c_s^2$ and larger values of $w_{\infty}$. Moreover, the results for $M_{de}$ are more sensitive to the change of $c_s^2$.
\begin{figure}[h!]
\centering\includegraphics[width=.6\textwidth]{Mde_rho_NS_BSk21_modified.pdf}
\caption{The dark energy mass (in solar mass units and logarithmic scale) as a function of the neutron star central density for the case of BSk21 equation of state. The colors correspond to different values of $c_s^2$: from $0.008$ (top) to $0.013$ (bottom). The results for $w_{\infty}=-0.99$ are depicted with solid lines and ones for $w_{\infty}=-0.80$ are depicted with dash-dotted lines, shadowed area between these lines corresponds to the results for in-between values of $w_{\infty}$. Zoomed-in region of $M_{de}/M_{sun}\sim 10^{-2}$ is shown in the upper left corner of the figure in the linear scale.}
\label{m_de-rho}
\end{figure}
Aiming to constrain the parameters of dark energy we have studied its influence on the NS mass. The dark energy does not reveal itself until a certain value of central matter density is reached because of very strong dependence on the ratio of gravitational radius to stellar surface one\footnote{The ratio $M_{de}/M\ll1$ for objects with $R\ll r_g$ for $c_s^2>10^{-4}$ \cite{novosyadlyj_2014}, where $R$ is radius of object, $r_g$ is its gravitational radius. So, the dynamical dark energy practically does not influence the gravitational field in the normal stars.}. This value depends on the chosen EoS for the NS matter and the parameters of dark energy. Thus, in order to analyze the impact of each parameter, we have consecutively fixed all of them except one. As was mentioned above, in our calculations we adopted $\rho_{\infty} = 10^{-23}$g/cm$^3$, and this parameter was not changed at all.
The total mass of a NS configuration (including the dark energy inside) as a function of its central matter density is shown in Fig.~\ref{m-rho}a for three considered equations of state (labeled respectively) and the same values of squared effective speed of sound $c_s^2$ as in Fig.~\ref{m_de-rho}, here the EoS parameter remained fixed ($w_{\infty}=-0.8$). Similarly to white dwarfs, the quintessence type of dark energy reduces the NS mass, acting matter-like, and contributing to the joint gravitational potential.
The corresponding mass-radius relations for two considered EoSs for NS matter are shown in Fig.~\ref{m-rho}b.
\begin{figure}[h!]
\begin{minipage}{.51\textwidth}
\centering\includegraphics[width=\textwidth]{M_rho_NS_w_08.pdf}
\centering{(a)}
\end{minipage}
\begin{minipage}{.48\textwidth}
\centering\includegraphics[width=\textwidth]{M_R_NS_w_08.pdf}
\centering{(b)}
\end{minipage}
\caption{(a) The total mass of neutron star configuration (including the dark energy inside) as a function of its central matter density for three considered equations of state for NS matter and different values of $c_s^2$ and $w_{\infty}=-0.8$ for dark energy (see text for details); (b) Mass-radius relation for two equations of state of NS matter in models with dark energy with different values of $c_s^2$ (depicted with different colors) and without it (black solid lines).}
\label{m-rho}
\end{figure}
As we can see from both figures, the influence of dark energy becomes crucial at some value of $\rho_c$ which depends on $c_s^2$.
At the central matter densities higher than this ``turn-off'' point, the dark energy content causes abrupt deviation from the model without the latter one (black solid lines).
The amount of dark energy accumulated inside a neutron star with central matter density near the turn-off point can be seen in Fig.~\ref{m_de-m_m_r} for the model with $\rho_c = 8.5\rho_n$, $c_s^2=0.012$ and BSk21 EoS as an example. On the left panel (Fig.~\ref{m_de-m_m_r}a) the ratio between dark energy mass $m_{de}(r)$ and mass of baryonic matter $m_m(r)$ is shown as functions of dimensionless radial coordinate for given values of the parameter $w_{\infty}$ (depicted with different line styles and colors). Because of the concentration of dark energy towards the center (see Fig.~\ref{rho_r_21}), it dominates in the central part of the object (except the model with $w_{\infty}=-0.99$ for which $m_{de}<m_m$ even in stellar center). But with growing radial coordinate $r$, the ratio decreases and reaches the values less than 0.01 on the surface for all considered $w_{\infty}$. On the right panel (Fig.~\ref{m_de-m_m_r}b) we can see the radial dependencies of dark energy mass (dash-dotted lines), mass of NS matter (solid lines) and total mass, which is the sum of the previous two (dotted lines), in units of solar mass for two values of $w_{\infty}$ (depicted with different colors).
The baryonic masses are only less than 1 percent lower at $r=R$ than total ones (see zoomed-in region in the upper right corner of the figure) and both baryonic and total masses are slightly lower for the model with $w_{\infty}=-0.8$, while the amount of dark energy is higher in this case. Therefore, more than 99 percent of the total mass of stable NS consists of baryonic mass.
Returning to Fig.~\ref{m-rho}, at the higher central matter densities the amount of dark energy is so high, that the mass of the matter drops and in this region of $\rho_c$ there are no stable equilibrium configurations. The pressure of baryonic matter can no more resist gravitation force from potential strengthened by dark energy. Therefore, for a given value of $c_s^2$ we obtain a corresponding existence region for central matter densities of NSs, roughly constrained by this turn-off point.
\begin{figure}[h!]
\begin{minipage}{.49\textwidth}
\centering\includegraphics[width=\textwidth]{M_de_M_m_rho0_85_cs2_12e-3.pdf}
\centering{(a)}
\end{minipage}
\begin{minipage}{.49\textwidth}
\centering\includegraphics[width=\textwidth]{M_total_M_m_rho0_85_cs2_12e-3_modified.pdf}
\centering{(b)}
\end{minipage}
\caption{The dark energy to baryonic mass ratio $m_{de}(r)/m_m(r)$ as functions of dimensionless radial coordinate for given values of $w_{\infty}$ (depicted with different colors and line types);
(b) The radial dependencies of dark energy mass (dash-dotted lines), mass of baryonic matter (solid lines) and total mass, which is the sum of previous two (dotted lines), in units of solar mass for two values of $w_{\infty}$ (depicted with different colors). Zoomed-in surface region $(0.95 - 1)$ $r/R$ is shown in the upper right corner. The results correspond to the choice of BSk21 EoS. Other parameters have values $\rho_c = 8.5\rho_n$, $c_s^2=0.012$.}
\label{m_de-m_m_r}
\end{figure}
This fact can be used to find a lower bound on the parameter $c_s^2$, but for that purpose one needs the observational data on NS masses. With grey color in Fig.~\ref{m-rho}a we depicted the upper bound range for maximum NS mass given in papers~\cite{margalit_2017,rezzolla_2018,ruiz_2018}, the results of which are based on the observations of the binary NS merger GW170817 \cite{abbott_2017}. The masses of the most massive NSs, PSR J0740+6620 and PSR J2215+5135, are indicated with labeled straight lines. In addition, 90\% credible region for maximum mass, obtained in paper~\cite{alsing_2018} with help of Bayesian model selection analysis for NS mass distribution, is shown as the green filled area.
One can immediately conclude from Fig.~\ref{m-rho}a, that the accuracy of NS mass determination or/and the increase of the number of NSs with known masses are crucial for setting a tight constraint on the effective sound speed of dark energy. Considering the upper bounds for maximum NS mass (grey region) we found that $c_s^2$ should be larger than $0.009$ in the case of BSk21 EoS, and $0.013$ for BSk20.
Assuming the mass estimation of PSR J0740+6620 is reliable, we can constrain $c_s^2\gtrsim0.012$ for BSk21. It should be mentioned, that these constraints depend on the considered EoS and the accuracy of the mass determination. In general, for a fixed NS model, the higher the maximum mass of NSs is found, the higher is the lower limit for $c_s^2$ value. On the other hand, such high maximum masses make possible and at some point even require stiffer equations of state (to allow their existence), and in that case the opposite is true: the model with stiffer EoS of NS lowers the bound on $c_s^2$.
\begin{figure}[h!]
\begin{minipage}[t]{.49\textwidth}
\centering\includegraphics[width=1\textwidth]{M_rho_NS_cs2_9e-3.pdf}
\centering{(a)}
\end{minipage}
\begin{minipage}[t]{.49\textwidth}
\centering\includegraphics[width=1\textwidth]{M_rho_NS_cs2_12e-3.pdf}
\centering{(b)}
\end{minipage}
\caption{Similar as in Figure~\ref{m-rho}a, but for different values of $w_{\infty}$ and fixed $c_s^2$ (see text for details):
(a) $c_s^2=0.009$; (b) $c_s^2=0.012$.}
\label{m-rho-cs2}
\end{figure}
As a next step, we investigated how the location of a turn-off point depends on parameter $w_{\infty}$ at fixed value of $c_s^2$. The results for the total mass of NS as a function of $\rho_c$ for $c_s^2=0.009$ are illustrated in Fig.~\ref{m-rho-cs2}a and for $c_s^2=0.012$ -- in Fig.~\ref{m-rho-cs2}b. As in Fig.~\ref{m-rho}a, the results of the model without dark energy are shown with black solid lines. We considered the values of $w_{\infty}$ ranging from $-0.80$ to $-0.99$. For these values of $w_{\infty}$, the central matter density $\rho_c$ corresponding to the turn-off point varies approximately on 3.9\% and 4.0\% for BSk21 and BSk20, respectively, in the case $c_s^2=0.009$, while 3.5\% and 5.0\% are corresponding variations for the case $c_s^2=0.012$. The values
of the turn-off point for different values of dark energy parameters $c_s^2$ and $w_{\infty}$ and three considered EoSs
are given in Table~\ref{rho_cs_w_tab}. Thus, one can infer from Fig.~\ref{m-rho-cs2} and Table~\ref{rho_cs_w_tab} that the results are
less sensitive to the choice of $w_{\infty}$ than $c_s^2$
for all considered equations of state.
\begin{table}[h!]
\centering
\caption{The values of turn-off point -- the central matter density of NS $\rho_c$ (in units of normal nuclear density $\rho_n$) at which the dark energy with different values of $c_s^2$ and $w_{\infty}$ reveals itself on the mass -- central density dependence (see Figs.~\ref{m-rho}a and \ref{m-rho-cs2}). The results are given for three considered equations of state BSk19, BSk20 and BSk21.}
\begin{tabular}{c|cccccc}
\hline
\hline
\multicolumn{7}{c}{BSk21}\\
\hline
\backslashbox{$w_{\infty}$}{$c_s^2$} & $0.008$ & $0.009$ & $0.010$ & $0.011$ & $0.012$ & $0.013$\\
\hline
$-0.80$ & $4.5$ & $5.1$ & $5.8$ & $6.6$ & $7.6$ & $8.6$\\
$-0.90$& $4.5$ & $5.2$ & $5.9$ & $6.7$ & $7.6$ & $8.7$\\
$-0.99$& $4.6$ & $5.3$ & $6.0$ & $6.9$ & $7.9$ & $9.0$\\
\hline
\hline
\multicolumn{7}{c}{BSk20}\\
\hline
\backslashbox{$w_{\infty}$}{$c_s^2$} & $0.008$ & $0.009$ & $0.010$ & $0.011$ & $0.012$ & $0.013$\\
\hline
$-0.80$ & $5.1$ & $5.7$ & $6.4$ & $7.1$ & $7.9$ & $8.9$\\
$-0.90$& $5.1$ & $5.7$ & $6.4$ & $7.2$ & $8.0$ & $9.0$\\
$-0.99$& $5.2$ & $5.9$ & $6.6$ & $7.4$ & $8.3$ & $9.3$\\
\hline
\hline
\multicolumn{7}{c}{BSk19}\\
\hline
\backslashbox{$w_{\infty}$}{$c_s^2$} & $0.008$ & $0.009$ & $0.010$ & $0.011$ & $0.012$ & $0.013$\\
\hline
$-0.80$ & $6.8$ & $7.7$ & $8.7$ & $9.8$ & $11.1$ & $12.6$\\
$-0.90$& $6.8$ & $7.8$ & $8.8$ & $10.0$ & $11.3$ & $12.7$\\
$-0.99$& $7.0$ & $8.0$ & $9.1$ & $10.3$ & $11.6$ & $13.2$\\
\hline
\hline
\end{tabular}
\label{rho_cs_w_tab}
\end{table}
\section{Conclusions}
\label{sect_5}
In this paper we have analyzed the impact of the dynamical scalar field quintessence dark energy on the NS. The density distribution was found from the numerical solutions of the conservation equation for NS matter and dark energy in joint potential, which corresponds to the static equilibrium of NS. We studied how this distribution depends on the parameters of dark energy (EoS $w_\infty$ and squared effective speed of sound $c_s^2$) and also, on the central density of baryonic matter $\rho_c$. We have found, that the relative deviation of the dark energy density $\delta_{de}(r)$ inside a neutron star increases as $w_\infty$ and $\rho_c$ grow and as $c_s$ decreases.
We have also established, that there is a turn-off point in the dependence of NS mass on its central matter density determined by the amount of dark energy inside a star.
When exceeding this turn-off point, the dark energy with certain set of parameters ($c_s^2$ and $w_{\infty}$) makes impossible a stable static solution of equilibrium equation to exist, meaning that considered combination of parameters ($M$ and $\rho_c$ or $M-R$ relation) is impossible.
Using this, we have established, that current limitation on the maximal mass of NSs allows one to constrain the minimal value of speed of sound $c_s$ relying on certain NS model. We have used the Brussels-Montreal EoS for NS matter with the set of parameters BSk20-21 and the estimations of maximal NS mass from binary NS merger GW170817 to obtain the lowest possible minimum value for $c_s^2\gtrsim 10^{-2}$. We have also found that the dependence of the turn-off point on the background EoS parameter of dark energy $w_\infty$ is weak, so we didn't use it to establish constraints on $w_\infty$.
This constraint is stronger than one obtained from consideration of "white dwarfs + dark energy" system ($c_s^2\gtrsim 10^{-4}$), which means that NSs are more suitable objects for studying this dark component of the Universe. On the other hand, large uncertainties in the determination of masses of NSs, as well as knowledge of the matter state inside them, postpone obtaining interesting results for further perspective.
\section*{Acknowledgements}
This work was supported by the projects of Ministry of Education and Science of Ukraine $\Phi\Phi$-63Hp (No. 0117U007190) and “Formation and characteristics of elements of the structure of the multicomponent Universe, gamma radiation of supernova remnants and observations of variable stars” (No. 0119U002210).
|
{
"timestamp": "2020-10-01T02:16:27",
"yymm": "2009",
"arxiv_id": "2009.14612",
"language": "en",
"url": "https://arxiv.org/abs/2009.14612"
}
|
"\\section{Introduction}\n\\label{sec:intro}\nMulti-task learning (MTL) is a branch of supervised le(...TRUNCATED)
| {"timestamp":"2020-10-01T02:16:58","yymm":"2009","arxiv_id":"2009.14635","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\\label{sec:01}\n\nIn this paper, we will introduce a DNN based algorithm fo(...TRUNCATED)
| {"timestamp":"2020-10-27T01:16:16","yymm":"2009","arxiv_id":"2009.14597","language":"en","url":"http(...TRUNCATED)
|
"\\part{Time- and Frequency-Domain Signals in Accelerators}\n\\section{Introduction and Recapitulati(...TRUNCATED)
| {"timestamp":"2020-10-01T02:13:34","yymm":"2009","arxiv_id":"2009.14544","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\\subsection{The status quo}\nThe most important Lie group in macroscopic (...TRUNCATED)
| {"timestamp":"2020-12-22T02:23:23","yymm":"2009","arxiv_id":"2009.14613","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\\label{sec:introduction} \n\nMuch of the commercial success of the group-(...TRUNCATED)
| {"timestamp":"2020-10-01T02:16:57","yymm":"2009","arxiv_id":"2009.14634","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\nThe Friedmann-Lema\\^itre-Robertson-Walker (FLRW) metric, based on\nthe co(...TRUNCATED)
| {"timestamp":"2020-10-01T02:15:16","yymm":"2009","arxiv_id":"2009.14579","language":"en","url":"http(...TRUNCATED)
|
"\n\\section{Introduction}\n\nThe methodological melting pot that is the interdisciplinary field of (...TRUNCATED)
| {"timestamp":"2020-12-02T02:18:21","yymm":"2009","arxiv_id":"2009.14545","language":"en","url":"http(...TRUNCATED)
|
End of preview.
No dataset card yet
- Downloads last month
- 6