Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Lemma 6.2 Uniformly for \( n \leq L\left( p\right) \) ,\n\n\[ \frac{d}{dp}{h}_{p}\left( n\right) \asymp {n}^{2}{\widehat{\pi }}_{p}\left( n\right) \] | Proof. Clearly\n\n\[ \frac{d}{dp}{h}_{p}\left( n\right) = \mathop{\sum }\limits_{x}{P}_{p}\left( {x\text{ is pivotal }}\right) . \]\n\nUsing the previous estimates, we see that the contribution of the \( O\left( {n}^{2}\right) \) points \( x \) that are at distance more than \( n/4 \) of the boundary of the parallelogr... | Yes |
Corollary 6.3 There exist absolute constants \( c,{c}^{\prime } \) and \( {p}_{1} > 1/2 \) such that for all \( {p}_{0} \in \left( {1/2,{p}_{1}}\right) \) , \[ c \leq L{\left( {p}_{0}\right) }^{2}{\int }_{1/2}^{{p}_{0}}{\widehat{\pi }}_{p}\left( {L\left( {p}_{0}\right) }\right) {dp} \leq {c}^{\prime }. \] | Proof. We integrate the identity of the previous lemma from \( p = 1/2 \) to \( p = {p}_{0} \) for \( n = L\left( {p}_{0}\right) \leq L\left( p\right) \). Note that the definition of \( L\left( p\right) \) shows that \( {h}_{{p}_{0}}\left( {L\left( {p}_{0}\right) }\right) - {h}_{1/2}\left( {L\left( {p}_{0}\right) }\rig... | Yes |
Lemma 6.3 Uniformly for \( {p}^{\prime } \in \left( {1/2,{p}_{0}}\right) \) , \[ {\widehat{\pi }}_{{p}^{\prime }}\left( {L\left( {p}_{0}\right) }\right) \asymp {\widehat{\pi }}_{1/2}\left( {L\left( {p}_{0}\right) }\right) . \] | If we furthermore combine this with the corollary derived in the previous paragraph, we get that \[ 1 \asymp {\int }_{1/2}^{{p}_{0}}{dpL}{\left( {p}_{0}\right) }^{2}{\widehat{\pi }}_{1/2}\left( {L\left( {p}_{0}\right) }\right) = \left( {{p}_{0} - 1/2}\right) \times L{\left( {p}_{0}\right) }^{2} \times {\widehat{\pi }}_... | Yes |
Lemma 6.4 When \( p \rightarrow 1/2 + \), one has\n\n\[ \n{P}_{p}\left( {0 \leftrightarrow \partial {\Lambda }_{L\left( p\right) }}\right) = L{\left( p\right) }^{-5/{48} + o\left( 1\right) }.\n\] | End of the proof of the theorem. | No |
Proposition 1.5. Every transition matrix on a finite state space has a random mapping representation. | Proof. Let \( P \) be the transition matrix of a Markov chain with state space \( \mathcal{X} = \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) . Take \( \Lambda = \left\lbrack {0,1}\right\rbrack \) ; our auxiliary random variables \( Z,{Z}_{1},{Z}_{2},\ldots \) will be uniformly chosen in this interval. Set \( {F}_{j, k}... | Yes |
Proposition 1.7. If \( P \) is aperiodic and irreducible, then there is an integer \( {r}_{0} \) such that \( {P}^{r}\left( {x, y}\right) > 0 \) for all \( x, y \in \mathcal{X} \) and \( r \geq {r}_{0} \) . | Proof. We use the following number-theoretic fact: any set of non-negative integers which is closed under addition and which has greatest common divisor 1 must contain all but finitely many of the non-negative integers. (See Lemma 1.30 in the Notes of this chapter for a proof.) For \( x \in \mathcal{X} \), recall that ... | Yes |
Consider the graph \( G \) shown in Figure 1.4. The transition matrix of simple random walk on \( G \) is | \[ P = \left( \begin{matrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{4} & \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{matrix}\right) \] | Yes |
Proposition 1.19. If \( P \) is an irreducible transition matrix and \( \pi \) is the unique probability distribution solving \( \pi = {\pi P} \), then for all states \( z \) , | Proof. Let \( {\widetilde{\pi }}_{z}\left( y\right) \) equal \( \widetilde{\pi }\left( y\right) \) as defined in (1.19), and write \( {\pi }_{z}\left( y\right) = {\widetilde{\pi }}_{z}\left( y\right) /{\mathbf{E}}_{z}{\tau }_{z}^{ + } \) . Proposition 1.14 implies that \( {\pi }_{z} \) is a stationary distribution, so ... | No |
Proposition 1.20. Let \( P \) be the transition matrix of a Markov chain with state space \( \mathcal{X} \) . Any distribution \( \pi \) satisfying the detailed balance equations (1.29) is stationary for \( P \) . | Proof. Sum both sides of (1.29) over all \( y \) :
\[
\mathop{\sum }\limits_{{y \in \mathcal{X}}}\pi \left( y\right) P\left( {y, x}\right) = \mathop{\sum }\limits_{{y \in \mathcal{X}}}\pi \left( x\right) P\left( {x, y}\right) = \pi \left( x\right) ,
\]
since \( P \) is stochastic. | Yes |
Proposition 1.23. Let \( \left( {X}_{t}\right) \) be an irreducible Markov chain with transition matrix \( P \) and stationary distribution \( \pi \) . Write \( \left( {\widehat{X}}_{t}\right) \) for the time-reversed chain with transition matrix \( \widehat{P} \) . Then \( \pi \) is stationary for \( \widehat{P} \), a... | Proof. To check that \( \pi \) is stationary for \( \widehat{P} \), we simply compute\n\n\[ \n\mathop{\sum }\limits_{{y \in \mathcal{X}}}\pi \left( y\right) \widehat{P}\left( {y, x}\right) = \mathop{\sum }\limits_{{y \in \mathcal{X}}}\pi \left( y\right) \frac{\pi \left( x\right) P\left( {x, y}\right) }{\pi \left( y\rig... | Yes |
Proposition 1.28. If \( \pi \) is stationary for the finite transition matrix \( P \), then \( \pi \left( {y}_{0}\right) = 0 \) for all inessential states \( {y}_{0} \) . | Proof. Let \( \mathcal{C} \) be an essential communicating class. Then\n\n\[ \n{\pi P}\left( \mathcal{C}\right) = \mathop{\sum }\limits_{{z \in \mathcal{C}}}\left( {\pi P}\right) \left( z\right) = \mathop{\sum }\limits_{{z \in \mathcal{C}}}\left\lbrack {\mathop{\sum }\limits_{{y \in \mathcal{C}}}\pi \left( y\right) P\l... | Yes |
Proposition 1.29. The transition matrix \( P \) has a unique stationary distribution if and only if there is a unique essential communicating class. | Proof. Suppose that there is a unique essential communicating class \( \mathcal{C} \) . Recall that \( {P}_{\mathcal{C}} \) is the restriction of the matrix \( P \) to the states in \( \mathcal{C} \), and that \( {P}_{\mid \mathcal{C}} \) is a transition matrix, irreducible on \( \mathcal{C} \) with a unique stationary... | Yes |
Assume that a gambler making fair unit bets on coin flips will abandon the game when her fortune falls to 0 or rises to \( n \) . Let \( {X}_{t} \) be gambler’s fortune at time \( t \) and let \( \tau \) be the time required to be absorbed at one of 0 or \( n \) . Assume that \( {X}_{0} = k \), where \( 0 \leq k \leq n... | Proof. Let \( {p}_{k} \) be the probability that the gambler reaches a fortune of \( n \) before ruin, given that she starts with \( k \) dollars. We solve simultaneously for \( {p}_{0},{p}_{1},\ldots ,{p}_{n} \) . Clearly \( {p}_{0} = 0 \) and \( {p}_{n} = 1 \), while\n\n\[ \n{p}_{k} = \frac{1}{2}{p}_{k - 1} + \frac{1... | No |
Consider a collector attempting to collect a complete set of coupons. Assume that each new coupon is chosen uniformly and independently from the set of \( n \) possible types, and let \( \tau \) be the (random) number of coupons collected when the set first contains every type. Then\n\n\[\n\\mathbf{E}\\left( \\tau \\ri... | The expectation \( \\mathbf{E}\\left( \\tau \\right) \) can be computed by writing \( \\tau \) as a sum of geometric random variables. Let \( {\\tau }_{k} \) be the total number of coupons accumulated when the collection first contains \( k \) distinct coupons. Then\n\n\[\\tau = {\\tau }_{n} = {\\tau }_{1} + \\left( {{... | Yes |
Proposition 2.4. Let \( \tau \) be a coupon collector random variable, as in Proposition 2.3. For any \( c > 0 \) , \[ \mathbf{P}\{ \tau > \lceil n\log n + {cn}\rceil \} \leq {e}^{-c}. \] | Proof. Let \( {A}_{i} \) be the event that the \( i \) -th type does not appear among the first \( \lceil n\log n + {cn}\rceil \) coupons drawn. Observe first that \[ \mathbf{P}\{ \tau > \lceil n\log n + {cn}\rceil \} = \mathbf{P}\left( {\mathop{\bigcup }\limits_{{i = 1}}^{n}{A}_{i}}\right) \leq \mathop{\sum }\limits_{... | Yes |
Proposition 2.8. Every birth-and-death chain is reversible. | Proof. A function \( w \) on \( \mathcal{X} \) satisfies the detailed balance equations (1.29) if and only if\n\n\[ \n{p}_{k - 1}{w}_{k - 1} = {q}_{k}{w}_{k} \n\]\n\nfor \( 1 \leq k \leq n \) . For our birth-and-death chain, a solution is given by \( {w}_{0} = 1 \) and\n\n\[ \n{w}_{k} = \mathop{\prod }\limits_{{i = 1}}... | Yes |
Proposition 2.12. Let \( P \) be the transition matrix of a random walk on a finite group \( G \) and let \( U \) be the uniform probability distribution on \( G \) . Then \( U \) is a stationary distribution for \( P \) . | Proof. Let \( \mu \) be the increment distribution of the random walk. For any \( g \in G, \)\n\n\[ \mathop{\sum }\limits_{{h \in G}}U\left( h\right) P\left( {h, g}\right) = \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{k \in G}}P\left( {{k}^{-1}g, g}\right) = \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{k \in... | Yes |
Proposition 2.13. Let \( \mu \) be a probability distribution on a finite group \( G \) . The random walk on \( G \) with increment distribution \( \mu \) is irreducible if and only if \( S = \{ g \in G : \mu \left( g\right) > 0\} \) generates \( G \) . | Proof. Let \( a \) be an arbitrary element of \( G \) . If the random walk is irreducible, then there exists an \( r > 0 \) such that \( {P}^{r}\left( {\mathrm{{id}}, a}\right) > 0 \) . In order for this to occur, there must be a sequence \( {s}_{1},\ldots ,{s}_{r} \in G \) such that \( a = {s}_{r}{s}_{r - 1}\ldots {s}... | Yes |
Proposition 2.14. The random walk on a finite group \( G \) with increment distribution \( \mu \) is reversible if \( \mu \) is symmetric. | Proof. Let \( U \) be the uniform probability distribution on \( G \) . For any \( g, h \in G \) , we have that\n\n\[ \nU\left( g\right) P\left( {g, h}\right) = \frac{\mu \left( {h{g}^{-1}}\right) }{\left| G\right| }\;\text{ and }\;U\left( h\right) P\left( {h, g}\right) = \frac{\mu \left( {g{h}^{-1}}\right) }{\left| G\... | Yes |
Proposition 2.16. Let \( P \) be the transition matrix of a transitive Markov chain on a finite state space \( \mathcal{X} \) . Then the uniform probability distribution on \( \mathcal{X} \) is stationary for \( P \) . | Proof. Fix \( x, y \in \mathcal{X} \) and let \( \varphi : \mathcal{X} \rightarrow \mathcal{X} \) be a transition-probability-preserving bijection for which \( \varphi \left( x\right) = y \) . Let \( U \) be the uniform probability on \( \mathcal{X} \) . Then\n\n\[ \mathop{\sum }\limits_{{z \in \mathcal{X}}}U\left( z\r... | Yes |
Proposition 4.2. Let \( \mu \) and \( \nu \) be two probability distributions on \( \mathcal{X} \) . Then\n\n\[ \parallel \mu - \nu {\parallel }_{\mathrm{{TV}}} = \frac{1}{2}\mathop{\sum }\limits_{{x \in \mathcal{X}}}\left| {\mu \left( x\right) - \nu \left( x\right) }\right| . \] | Proof. Let \( B = \{ x : \mu \left( x\right) \geq \nu \left( x\right) \} \) and let \( A \subset \mathcal{X} \) be any event. Then\n\n\[ \mu \left( A\right) - \nu \left( A\right) \leq \mu \left( {A \cap B}\right) - \nu \left( {A \cap B}\right) \leq \mu \left( B\right) - \nu \left( B\right) . \]\n\nThe first inequality ... | Yes |
Proposition 4.5. Let \( \mu \) and \( \nu \) be two probability distributions on \( \mathcal{X} \) . Then the total variation distance between them satisfies\n\n\[ \parallel \mu - \nu {\parallel }_{\mathrm{{TV}}} = \frac{1}{2}\sup \left\{ {\mathop{\sum }\limits_{{x \in \mathcal{X}}}f\left( x\right) \mu \left( x\right) ... | Proof. If \( \mathop{\max }\limits_{{x \in \mathcal{X}}}\left| {f\left( x\right) }\right| \leq 1 \), then\n\n\[ \frac{1}{2}\left| {\mathop{\sum }\limits_{{x \in \mathcal{X}}}f\left( x\right) \mu \left( x\right) - \mathop{\sum }\limits_{{x \in \mathcal{X}}}f\left( x\right) \nu \left( x\right) }\right| \leq \frac{1}{2}\m... | Yes |
Proposition 4.15. For a reversible Markov chain,\n\n\[ \n{d}^{\left( \infty \right) }\left( {2t}\right) = {\left\lbrack {d}^{\left( 2\right) }\left( t\right) \right\rbrack }^{2} = \mathop{\max }\limits_{{x \in \mathcal{X}}}{q}_{2t}\left( {x, x}\right) - 1.\n\] | Proof. First observe that\n\n\[ \n{P}^{2t}\left( {x, y}\right) = \mathop{\sum }\limits_{{z \in \mathcal{X}}}{P}^{t}\left( {x, z}\right) {P}^{t}\left( {z, y}\right) .\n\]\n\nDividing both sides by \( \pi \left( y\right) \) and using reversibility yields\n\n\[ \n{q}_{2t}\left( {x, y}\right) = \mathop{\sum }\limits_{{z \i... | Yes |
Let \( Q \) be an irreducible transition matrix and consider the lazy chain with transition matrix \( P = \left( {Q + I}\right) /2 \) . The distributions at time \( t \) and \( t + 1 \) satisfy\n\n\[ \n{\begin{Vmatrix}{P}^{t}\left( x, \cdot \right) - {P}^{t + 1}\left( x, \cdot \right) \end{Vmatrix}}_{\mathrm{{TV}}} \le... | Proof. Let \( \left( {{N}_{t},{M}_{t}}\right) \) be a coupling of the Binomial \( \left( {t,\frac{1}{2}}\right) \) distribution with the Binomial \( \left( {t + 1,\frac{1}{2}}\right) \) distribution, and let \( \left( {Z}_{t}\right) \) be a Markov chain with transition matrix \( Q \) started from \( x \) and independen... | Yes |
Proposition 6.1. Let \( \left( {X}_{t}\right) \) be the random walk on \( {\mathcal{S}}_{n} \) corresponding to the top-to-random shuffle on \( n \) cards. Given at time \( t \) that there are \( k \) cards under the original bottom card, each of the \( k \) ! possible orderings of these cards are equally likely. There... | Proof. When \( t = 0 \), there are no cards under the original bottom card, and the claim is trivially valid. Now suppose that the claim holds at time \( t \) . There are two possibilities at time \( t + 1 \) : either a card is placed under the original bottom card, or not. In the second case, the cards under the origi... | Yes |
Proposition 6.11. If \( \tau \) is a strong stationary time for starting state \( x \), then\n\n\[ \n{\begin{Vmatrix}{P}^{t}\left( x, \cdot \right) - \pi \end{Vmatrix}}_{\mathrm{{TV}}} \leq {\mathbf{P}}_{x}\{ \tau > t\} .\n\] | We break the proof into two lemmas. It will be convenient to introduce a parameter \( {s}_{x}\left( t\right) \), called separation distance and defined by\n\n\[ \n{s}_{x}\left( t\right) \mathrel{\text{:=}} \mathop{\max }\limits_{{y \in \mathcal{X}}}\left\lbrack {1 - \frac{{P}^{t}\left( {x, y}\right) }{\pi \left( y\righ... | No |
Proposition 6.14. If there exists a halting state for starting state \( x \), then \( \tau \) is an optimal strong stationary time for \( x \), i.e. | Proof. If \( y \) is a halting state for starting state \( x \) and the stopping time \( \tau \) , then inequality (6.9) is an equality for every \( t \) . Therefore, if there exists a halting state for starting state \( x \), then (6.8) is also an equality. | Yes |
Proposition 6.21. For every starting state \( x \), there exists a strong stationary time \( \tau \) such that, for all \( t \geq 0 \) , \[ {s}_{x}\left( t\right) = {\mathbf{P}}_{x}\{ \tau > t\} \] | Proof. Fix \( x \in \mathcal{X} \), and let \( {a}_{t} \mathrel{\text{:=}} \mathop{\min }\limits_{y}\frac{{P}^{t}\left( {x, y}\right) }{\pi \left( y\right) } = 1 - {s}_{x}\left( t\right) \) . Note that \( {a}_{t} \) is nondecreasing. (See Exercise 6.4.) If there exists a strong stationary time \( \tau \) satisfying (6.... | No |
Proposition 7.9. For \( f : \mathcal{X} \rightarrow \mathbb{R} \), define \( {\sigma }_{ \star }^{2} \mathrel{\text{:=}} \max \left\{ {{\operatorname{Var}}_{\mu }\left( f\right) ,{\operatorname{Var}}_{\nu }\left( f\right) }\right\} \) . If\n\n\[ \left| {{E}_{\nu }\left( f\right) - {E}_{\mu }\left( f\right) }\right| \ge... | Proof of Proposition 7.9. Suppose without loss of generality that \( {E}_{\mu }\left( f\right) \leq \) \( {E}_{\nu }\left( f\right) \) . If \( A = \left( {{E}_{\mu }\left( f\right) + r{\sigma }_{ \star }/2,\infty }\right) \), then Chebyshev’s inequality yields that\n\n\[ \mu {f}^{-1}\left( A\right) \leq \frac{4}{{r}^{2... | Yes |
Proposition 8.4. Let \( 0 < \varepsilon < 1 \) . For the random transposition chain on an \( n \) -card deck,\n\n\[ \n{t}_{\operatorname{mix}}\left( \varepsilon \right) \geq \frac{n - 1}{2}\log \left( {\frac{1 - \varepsilon }{6}n}\right) .\n\] | Proof. It is well known (and easily proved using indicators) that the expected number of fixed points in a uniform random permutation in \( {\mathcal{S}}_{n} \) is 1, regardless of the value of \( n \) .\n\nLet \( F\left( \sigma \right) \) denote the number of fixed points of the permutation \( \sigma \) . If \( \sigma... | Yes |
Proposition 8.5. Let \( \tau \) be the time required for the two decks to coincide. Then, no matter the initial configurations of the two decks, \( \mathbf{E}\left( \tau \right) < \frac{{\pi }^{2}}{6}{n}^{2} \) . | Proof. Decompose\n\n\[ \tau = {\tau }_{1} + \cdots + {\tau }_{n} \]\n\nwhere \( {\tau }_{i} \) is the number of transpositions between the first time that \( {a}_{t} \) is greater than or equal to \( i - 1 \) and the first time that \( {a}_{t} \) is greater than or equal to \( i \) . (Since \( {a}_{0} \) can be greater... | Yes |
Proposition 8.6. In the random transposition shuffle, let \( {R}_{t} \) and \( {L}_{t} \) be the cards chosen by the right and left hands, respectively, at time t. Assume that when \( t = 0 \), no cards have been marked. At time \( t \), mark card \( {R}_{t} \) if both of the following are true:\n\n- \( {R}_{t} \) is u... | Proof. It is clear that \( \tau \) is a stopping time. To show that it is a strong stationary time, we prove the following subclaim by induction on \( t \) . Let \( {V}_{t} \subseteq \left\lbrack n\right\rbrack \) be the set of cards marked at or before time \( t \), and let \( {U}_{t} \subseteq \left\lbrack n\right\rb... | Yes |
Proposition 8.11. Let \( \tau \) be the number of inverse riffle shuffles required for all cards to have different bitstring labels. Then \( \tau \) is a strong stationary time. | Proof. Condition on the event that \( \tau = t \) . Since the bitstrings are generated by independent fair coin flips, every possible assignment \( {}^{1} \) of strings of length \( t \) to cards is equally likely. Since the labeling bitstrings are distinct, the permutation is fully determined by the labels. Hence the ... | Yes |
Proposition 8.12. For the riffle shuffle on an \( n \) -card deck, \( {t}_{\operatorname{mix}} \leq 2{\log }_{2}\left( {{4n}/3}\right) \) for sufficiently large \( n \) . | Proof. Consider inverse riffle shuffling an \( n \) -card deck and let \( \tau \) be the stopping time defined in Proposition 8.11. If \( \tau \leq t \), then different labels have been assigned to all \( n \) cards after \( t \) inverse riffle shuffles. Hence\n\n\[ \mathbf{P}\left( {\tau \leq t}\right) = \mathop{\prod... | Yes |
Proposition 8.13. Fix \( 0 < \varepsilon ,\delta < 1 \) . Consider riffle shuffling an \( n \) -card deck. For sufficiently large \( n \) ,\n\n\[{t}_{\operatorname{mix}}\left( \varepsilon \right) \geq \left( {1 - \delta }\right) {\log }_{2}n.\]\n\n(8.7) | Proof. There are at most \( {2}^{n} \) possible states accessible in one step of the time-reversed chain, since we can generate a move using \( n \) independent unbiased bits. Thus \( {\log }_{2}\Delta \leq n \), where \( \Delta \) is the maximum out-degree defined in (7.1). The state space has size \( n \) !, and Stir... | Yes |
Proposition 9.1. Let \( \left( {X}_{t}\right) \) be a Markov chain with irreducible transition matrix \( P \), let \( B \subset \mathcal{X} \), and let \( {h}_{B} : B \rightarrow \mathbb{R} \) be a function defined on \( B \) . The function \( h : \mathcal{X} \rightarrow \mathbb{R} \) defined by \( h\left( x\right) \ma... | Proof. We first show that \( h\left( x\right) = {\mathbf{E}}_{x}{h}_{B}\left( {X}_{{\tau }_{B}}\right) \) is a harmonic extension of \( {h}_{B} \) . Clearly \( h\left( x\right) = {h}_{B}\left( x\right) \) for all \( x \in B \) . Suppose that \( x \in \mathcal{X} \smallsetminus B \) . Then\n\n\[ h\left( x\right) = {\mat... | Yes |
Proposition 9.4 (Node law/cycle law/strength). If \( \theta \) is a flow from a to \( z \) satisfying the cycle law\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{m}r\left( \overrightarrow{{e}_{i}}\right) \theta \left( \overrightarrow{{e}_{i}}\right) = 0 \]\n\n\( \left( {9.10}\right) \)\n\nfor any cycle \( \overrightarrow{{e}_... | Proof. The function \( f = \theta - I \) satisfies the node law at all nodes and the cycle law. Suppose \( f\left( \overrightarrow{{e}_{1}}\right) > 0 \) for some oriented edge \( \overrightarrow{{e}_{1}} \) . By the node law, \( {e}_{1} \) must lead to some oriented edge \( \overrightarrow{{e}_{2}} \) with \( f\left( ... | Yes |
Proposition 9.5. For any \( a, z \in \mathcal{X} \) with \( a \neq z \) , \[ {\mathbf{P}}_{a}\left\{ {{\tau }_{z} < {\tau }_{a}^{ + }}\right\} = \frac{1}{c\left( a\right) \mathcal{R}\left( {a \leftrightarrow z}\right) } = \frac{\mathcal{C}\left( {a \leftrightarrow z}\right) }{c\left( a\right) }.\] | Proof. Applying Proposition 9.1 to \( B = \{ a, z\} \) and \( {h}_{B} = {\mathbf{1}}_{\{ z\} } \) yields that \[ x \mapsto {\mathbf{E}}_{x}{h}_{B}\left( {X}_{{\tau }_{B}}\right) = {\mathbf{P}}_{x}\left\{ {{\tau }_{z} < {\tau }_{a}}\right\} \] is the unique harmonic function on \( \mathcal{X} \smallsetminus \{ a, z\} \)... | Yes |
Proposition 9.16. If \( \\left\\{ {\\Pi }_{k}\\right\\} \) are disjoint edge-cutsets which separate nodes a and \( z \), then\n\n\[ \n\\mathcal{R}\\left( {a \\leftrightarrow z}\\right) \\geq \\mathop{\\sum }\\limits_{k}{\\left( \\mathop{\\sum }\\limits_{{e \\in {\\Pi }_{k}}}c\\left( e\\right) \\right) }^{-1}.\n\]\n\n\(... | Proof. Let \( \\theta \) be a unit flow from \( a \) to \( z \) . For any \( k \), by the Cauchy-Schwarz inequality\n\n\[ \n\\mathop{\\sum }\\limits_{{e \\in {\\Pi }_{k}}}c\\left( e\\right) \\cdot \\mathop{\\sum }\\limits_{{e \\in {\\Pi }_{k}}}r\\left( e\\right) \\theta {\\left( e\\right) }^{2} \\geq {\\left( \\mathop{... | Yes |
Proposition 9.17. Let \( a = \left( {1,1}\right) \) be the lower left-hand corner of \( {B}_{n} \), and let \( z = \left( {n, n}\right) \) be the upper right-hand corner of \( {B}_{n} \) . Suppose each edge of \( {B}_{n} \) has unit conductance. The effective resistance \( \mathcal{R}\left( {a \leftrightarrow z}\right)... | We separate the proof into the lower and upper bounds.\n\nProof of lower bound in (9.25). Let \( {\Pi }_{k} \) be the edge set\n\n\[ {\Pi }_{k} = \left\{ {\{ v, w\} \in E\left( {B}_{n}\right) : \parallel v{\parallel }_{\infty } = k,\parallel w{\parallel }_{\infty } = k + 1}\right\} ,\]\n\nwhere \( {\begin{Vmatrix}\left... | Yes |
Proposition 10.7 (Commute Time Identity). Let \( \\left( {G,\\{ c\\left( e\\right) \\} }\\right) \) be a network, and let \( \\left( {X}_{t}\\right) \) be the random walk on this network. For any nodes \( a \) and \( b \) in \( V \) , | Proof. By (10.13), \[ \\frac{{G}_{{\\tau }_{a, b}}\\left( {a, a}\\right) }{{\\mathbf{E}}_{a}\\left( {\\tau }_{a, b}\\right) } = \\pi \\left( a\\right) = \\frac{c\\left( a\\right) }{{c}_{G}}. \] By definition, after visiting \( b \), the chain does not visit \( a \) until time \( {\\tau }_{a, b} \), so \( {G}_{{\\tau }_... | No |
Proposition 10.10. For a random walk on a transitive connected network \( \langle G,\{ c\left( e\right) \} \rangle \), for any vertices \( a, b \in V \) , \n\n\[ \n{\mathbf{E}}_{a}\left( {\tau }_{b}\right) = {\mathbf{E}}_{b}\left( {\tau }_{a}\right) \n\] | Proof. Suppose \( \xi \) and \( \eta \) are finite strings with letters in \( V \), that is, \( \xi \in {V}^{m} \) and \( \eta \in {V}^{n} \) . We say that \( \xi \preccurlyeq \eta \) if and only if \( \xi \) is a subsequence of \( \eta \) . \n\nLet \( {\tau }_{ab} \) be the time required to first visit \( a \) and the... | Yes |
Proposition 10.20. Let \( G = \left( {V, E}\right) \) be an Eulerian directed graph. Let \( m = \) \( \left| E\right| \), and assume that there exists a directed path of length \( \ell \) from vertex \( x \) to vertex y. Then\n\n\[{\mathbf{E}}_{x}\left( {\tau }_{y}\right) + {\mathbf{E}}_{y}\left( {\tau }_{x}\right) \le... | Proof. It is enough to prove this for the case where there is a directed edge \( \left( {x, y}\right) \), since otherwise \( {\mathbf{E}}_{x}\left( {\tau }_{y}\right) \) is bounded by the sum of the expected hitting times along the path from \( x \) to \( y \), and similarly for \( {\mathbf{E}}_{y}\left( {\tau }_{x}\ri... | Yes |
Proposition 10.21. Consider the simple random walk on the torus \( {\mathbb{Z}}_{n}^{d} \). There exist constants \( 0 < {c}_{d} \leq {C}_{d} < \infty \) such that if \( x \) and \( y \) are at distance \( k \geq 1 \), then\n\n\[ {c}_{d}{n}^{d} \leq {\mathbf{E}}_{x}\left( {\tau }_{y}\right) \leq {C}_{d}{n}^{d}\;\text{ ... | Proof of Proposition 10.21. First, the lower bounds. For \( j \geq 0 \), let \( {\Pi }_{j} \) be the edge-boundary of the cube of side-length \( {2j} \) centered at \( x \), i.e., the set of edges connecting the cube to its complement. For \( 1 \leq j \leq k/d \), the edges in \( {\Pi }_{j} \) form an edge-cutset separ... | No |
Proposition 10.25. Let \( P \) be the transition matrix for a finite reversible chain on state space \( \mathcal{X} \) with stationary distribution \( \pi \) .\n\n(i) For all \( t \geq 0 \) and \( x \in \mathcal{X} \) we have \( {P}^{{2t} + 2}\left( {x, x}\right) \leq {P}^{2t}\left( {x, x}\right) \) . | Proof. (i) Since \( {P}^{{2t} + 2}\left( {x, x}\right) = \mathop{\sum }\limits_{{y, z \in \mathcal{X}}}{P}^{t}\left( {x, y}\right) {P}^{2}\left( {y, z}\right) {P}^{t}\left( {z, x}\right) \), we have\n\n\[ \pi \left( x\right) {P}^{{2t} + 2}\left( {x, x}\right) = \mathop{\sum }\limits_{{y, z \in \mathcal{X}}}{P}^{t}\left... | Yes |
For lazy random walk on a simple graph with \( m \) edges and \( n \) vertices, \[ {t}_{\text{hit }} \leq {4nm} \leq 2{n}^{3}, \] and \[ {t}_{\operatorname{mix}}^{\left( \infty \right) } \leq {16nm} + 1 \leq 8{n}^{3}\text{, so}{t}_{\operatorname{mix}} \leq {8nm} + 1 \leq 4{n}^{3}\text{.} \] | Proof. Since \( {t}_{\text{hit }} \leq \mathop{\max }\limits_{{a, b}}{t}_{a \leftrightarrow b} \), this result follows from Proposition 10.16 together with Theorem 10.22. (The extra factor of 2 comes from the laziness of the walk.) | Yes |
Proposition 10.29. Let \( H \) be the graph obtained by gluing together two copies of \( G \) at the vertex \( {v}_{ \star } \) as defined above. Let \( {\tau }_{\text{couple }}^{G} \) be the time for a coupling of two random walks on \( G \) to meet. Then there is a coupling of two random walks on \( H \) which has a ... | Outline of PROOF. Given a starting point in \( H \), a random walk in \( G \) can be lifted to a random walk in \( H \) in a unique way. (At \( {v}_{ \star } \), the particle moves to each copy with equal probability.) Applying this lifting to the given coupling in \( G \) yields a coupling in \( H \) where at time \( ... | Yes |
Proposition 11.4. Let \( A \subset \mathcal{X} \) . Set \( {t}_{\min }^{A} = \mathop{\min }\limits_{{a, b \in A, a \neq b}}{\mathbf{E}}_{a}\left( {\tau }_{b}\right) \) . Then\n\n\[ \n{t}_{\text{cov }} \geq \mathop{\max }\limits_{{A \subseteq \mathcal{X}}}{t}_{\min }^{A}\left( {1 + \frac{1}{2} + \cdots + \frac{1}{\left|... | Proof. Fix an initial state \( x \in A \) and let \( \sigma \) be a uniform random permutation of the elements of \( A \), chosen independently of the chain trajectory. Let \( {T}_{k} \) be the first time at which all of \( \sigma \left( 1\right) ,\sigma \left( 2\right) ,\ldots ,\sigma \left( k\right) \) have been visi... | Yes |
Proposition 11.9. The cover time satisfies\n\n\[ \n{t}_{\text{cov }} \geq \left( {\log 2}\right) k{2}^{k}\left( {1 + o\left( 1\right) }\right) .\n\] | Proof. Fix \( j = \left\lceil {{\log }_{2}k}\right\rceil \) and let \( A \subseteq \{ 0,1{\} }^{k} \) consist of those bitstrings that end with \( j \) zeroes followed by a 1 . Fix \( a, b \in A \), where \( a \neq b \) . By Lemma 11.8, we have\n\n\[ \n{\mathbf{E}}_{a}{\tau }_{b}^{ + } \geq \left( {1 - \theta }\right) ... | Yes |
Proposition 11.10. The cover time satisfies\n\n\[ \n{t}_{\text{cov }} \leq \left( {\log 2}\right) k{2}^{k}\left( {1 + o\left( 1\right) }\right) .\n\] | Proof. We partition the state space \( \{ 0,1{\} }^{k} \) into two sets. Fix \( j = \left\lceil {{\log }_{2}k}\right\rceil \) and let \( B \) be the set of all strings \( b \in \{ 0,1{\} }^{k} \) with the following property: any bitstring that is both a suffix and a prefix of \( b \) must have length less than \( k - j... | Yes |
Lemma 12.17 (Random Target Lemma for Reversible Chains). For an irreducible reversible Markov chain, \[ {t}_{ \odot } = \mathop{\sum }\limits_{{i = 2}}^{n}\frac{1}{1 - {\lambda }_{i}}. \] | Proof. Since \( {t}_{ \odot } = {t}_{ \odot }^{a} \) for all \( a \in \mathcal{X} \) by Lemma 10.1, \[ {t}_{ \odot } = \mathop{\sum }\limits_{a}\pi \left( a\right) {t}_{ \odot }^{a} = \mathop{\sum }\limits_{{a, x \in \mathcal{X}}}\pi \left( x\right) \pi \left( a\right) {\mathbf{E}}_{a}\left( {\tau }_{x}\right) = \matho... | Yes |
Theorem 13.10 (Sinclair and Jerrum (1989), Lawler and Sokal (1988)). Let \( {\lambda }_{2} \) be the second largest eigenvalue of a reversible transition matrix \( P \), and let \( \gamma = 1 - {\lambda }_{2} \) . Then\n\n\[\n\frac{{\Phi }_{ \star }^{2}}{2} \leq \gamma \leq 2{\Phi }_{ \star }\n\]\n\n(13.6) | Proof of the upper bound in Theorem 13.10. By Lemmas 13.7 and 13.6,\n\n\[\n\gamma = \mathop{\min }\limits_{\substack{{f \neq 0} \\ {{E}_{\pi }\left( f\right) = 0} }}\frac{\mathop{\sum }\limits_{{x, y \in \mathcal{X}}}\pi \left( x\right) P\left( {x, y}\right) {\left\lbrack f\left( x\right) - f\left( y\right) \right\rbra... | Yes |
Corollary 13.21 (Method of Canonical Paths). Let \( P \) be a reversible and irreducible transition matrix with stationary distribution \( \pi \) . Suppose \( {\Gamma }_{xy} \) is a choice of \( E \) -path for each \( x \) and \( y \), and let\n\n\[ \nB = \mathop{\max }\limits_{{e \in E}}\frac{1}{Q\left( e\right) }\mat... | Proof. Let \( \widetilde{P}\left( {x, y}\right) = \pi \left( y\right) \), and observe that the stationary measure for \( \widetilde{P} \) is clearly \( \widetilde{\pi } = \pi \) . For \( f \in {\mathbb{R}}^{\mathcal{X}} \) such that \( 0 = {E}_{\pi }\left( f\right) = \langle f,\mathbf{1}{\rangle }_{\pi } \), \n\n\[ \n\... | Yes |
Proposition 13.35. When \( \left\{ {G}_{n}\right\} \) is a \( \left( {d,\alpha }\right) \) -expander family, the lazy random walks on \( \left\{ {G}_{n}\right\} \) satisfy \( {t}_{\operatorname{mix}}\left( {G}_{n}\right) = O\left( {\log \left| {V\left( {G}_{n}\right) }\right| }\right) \) . | Proof. Theorem 13.10 implies that for all \( {G}_{n} \) the spectral gap for the simple random walk satisfies \( \gamma \geq {\alpha }^{2}/2 \) . Since each \( {G}_{n} \) is regular, the stationary distribution of the lazy random walk is uniform, and Theorem 12.4 tells us that for the lazy walk \( {t}_{\operatorname{mi... | Yes |
Proposition 13.37. For the family \( \\left\\{ {G}_{n}\\right\\} \) of random multigraphs described in (13.38), | Proof. Assume that \( \\delta < {0.03} \) . We first show that with probability tending to 1 as \( n \\rightarrow \\infty \), every subset of \( A \) of size \( k \\leq n/2 \) has more than \( \\left( {1 + \\delta }\\right) k \) neighbors. Note that every edge in \( {G}_{n} \) connects a vertex in \( A = \\left\\{ {{a}... | Yes |
Let \( \left( {M}_{t}\right) \) be a non-negative martingale with respect to \( \left\{ {\mathcal{F}}_{t}\right\} \), and define\n\n\[ \n{T}_{h} \mathrel{\text{:=}} \min \left\{ {t \geq 0 : {M}_{t} = 0\text{ or }{M}_{t} \geq h}\right\} .\n\]\n\nAssume that\n\n(i) \( \operatorname{Var}\left( {{M}_{t + 1} \mid {\mathcal{... | Proof. For \( h \geq {M}_{0} \), we have that \( \{ \tau \geq t\} \subseteq \left\{ {{T}_{h} \geq t}\right\} \cup \left\{ {{M}_{{T}_{h}} \geq h}\right\} \), whence\n\n\[ \n\mathbf{P}\{ \tau \geq t\} \leq \mathbf{P}\left\{ {{T}_{h} \geq t}\right\} + \mathbf{P}\left\{ {{M}_{{T}_{h}} \geq h}\right\}\n\]\n\n(17.34)\n\nWe f... | Yes |
Proposition 17.20. Let \( {\left( {Z}_{t}\right) }_{t \geq 0} \) be a non-negative supermartingale with respect to \( \left\{ {\mathcal{F}}_{t}\right\} \), and let \( \tau \) be a stopping time for \( \left\{ {\mathcal{F}}_{t}\right\} \) . Suppose that (i) \( {Z}_{0} = k \), (ii) there exists \( B \) such that \( \left... | The proof follows the same outline as the proof of Proposition 17.19 and is left to the reader in Exercise 17.4. | No |
Proposition 18.4. For a sequence of irreducible aperiodic reversible Markov chains with relaxation times \( \left\{ {t}_{\text{rel }}^{\left( n\right) }\right\} \) and mixing times \( \left\{ {t}_{\text{mix }}^{\left( n\right) }\right\} \), if there is a pre-cutoff, then \( {t}_{\text{mix }}^{\left( n\right) }/\left( {... | Proof. If \( {t}_{\text{mix }}^{\left( n\right) }/\left( {{t}_{\text{rel }}^{\left( n\right) } - 1}\right) \) does not tend to infinity, then there is an infinite set of integers \( J \) and a constant \( {c}_{1} > 0 \) such that \( \left( {{t}_{\text{rel }}^{\left( n\right) } - 1}\right) /{t}_{\text{mix }}^{\left( n\r... | Yes |
Proposition 21.3. Suppose that \( P \) is the transition matrix of an irreducible Markov chain \( \left( {X}_{t}\right) \) . Define \( G\left( {x, y}\right) \mathrel{\text{:=}} {\mathbf{E}}_{x}\left( {\mathop{\sum }\limits_{{t = 0}}^{\infty }{\mathbf{1}}_{\left\{ {X}_{t} = y\right\} }}\right) = \mathop{\sum }\limits_{{... | Proof. (i) \( \Leftrightarrow \) (iii). Every time the chain visits \( x \), it has the same probability of eventually returning to \( x \), independent of the past. Thus the number of visits to \( x \) is a geometric random variable with success probability \( 1 - {\mathbf{P}}_{x}\left\{ {{\tau }_{x}^{ + } < \infty }\... | Yes |
Proposition 21.6. Let \( \langle G,\{ c\left( e\right) \} \rangle \) be a network. The following are equivalent:\n\n(i) The weighted random walk on the network is transient.\n\n(ii) There is some node a with \( \mathcal{C}\left( {a \leftrightarrow \infty }\right) > 0 \) . (Equivalently, \( \mathcal{R}\left( {a \leftrig... | Proof. That (i) and (ii) are equivalent follows from (21.2), and (21.3) implies the equivalence of (ii) and (iii). | No |
Proposition 21.7 (Nash-Williams). If there exist disjoint edge-cutsets \( \left\{ {\Pi }_{n}\right\} \) that separate a from \( \infty \) and satisfy\n\n\[\n\mathop{\sum }\limits_{n}{\left( \mathop{\sum }\limits_{{e \in {\Pi }_{n}}}c\left( e\right) \right) }^{-1} = \infty\n\]\n\nthen the weighted random walk on \( \lef... | Proof. Recall the definition of \( {z}_{n} \) given in the beginning of this section. The assumption (21.4) implies that \( \mathcal{R}\left( {a \leftrightarrow {z}_{n}}\right) \rightarrow \infty \) . Consequently, by Proposition 9.5, \( {\mathbf{P}}_{a}\left\{ {{\tau }_{{z}_{n}} < {\tau }_{a}^{ + }}\right\} \rightarro... | No |
Proposition 21.15. Let \( P \) be irreducible and suppose the Markov chain with transition matrix \( P \) is recurrent. Let \( \pi \) and \( \mu \) be two measures satisfying \( \pi = {\pi P} \) and \( \mu = {\mu P} \) . Then \( \mu = {c\pi } \) for some constant \( c \) . | Proof. Let \( h = \mu /\pi \) . Then \( h \) is harmonic for \( \widehat{P} \), the time-reversal of \( P \) . Since \( {\widehat{P}}^{t}\left( {x, x}\right) = {P}^{t}\left( {x, x}\right) \) for all \( t \geq 1 \), the \( \widehat{P} \) -chain is also recurrent. The conclusion follows from the fact that all such functi... | Yes |
Proposition 22.7. The following conditions are equivalent:\n\n(i) \( P \) is a monotone chain.\n\n(ii) If \( \mu \preccurlyeq \nu \), then \( {\mu P} \preccurlyeq {\nu P} \) .\n\n(iii) For every pair of comparable states \( x, y \in \mathcal{X} \) with \( x \preccurlyeq y \), there exists a coupling \( \left( {X, Y}\ri... | Proof. (i) \( \Rightarrow \) (ii). Let \( f \) be an increasing function. Then \( {Pf} \) is increasing, so\n\n\[ \left( {\mu P}\right) f = \mu \left( {Pf}\right) \leq \nu \left( {Pf}\right) = \left( {\nu P}\right) f. \]\n\n(ii) \( \Rightarrow \) (iii). If \( x \preccurlyeq y \), then \( {\delta }_{x}P \preccurlyeq {\d... | Yes |
Proposition 23.1. Let \( {G}_{n} \) denote the \( n \) -path with loops at 1 and \( n \) . Let \( {t}_{\text{rel }} \) be the relaxation time for the interchange process on the \( n \) -path, and let \( {t}_{\text{rel }} \) (single) be the relaxation time for the random walk on \( {G}_{n} \) with delay probability \( 1... | Proof. Let \( \varphi \left( j\right) = \cos \left( {\pi \left( {{2j} - 1}\right) /{2n}}\right) \) for \( j = 1,2,\ldots, n \) be the second eigenfunction for the simple random walk on \( {G}_{n} \), with eigenvalue \( {\lambda }_{2} = \cos \left( {\pi /n}\right) \) . (See (12.21).) If \( {\sigma }_{1} \) is the permut... | Yes |
Proposition 24.4. For all chains we have that\n\n\[ \n{t}_{\mathrm{G}} \leq 4{t}_{\text{stop }} + 1 \n\] | Proof of Proposition 24.4. We fix \( x \) . Let \( \tau \) be a stationary time, so that the distribution of \( {X}_{\tau } \) when started from \( x \) is \( \pi \) . Then \( \tau + s \) is also a stationary time for all \( s \geq 1 \) . Hence, if \( {Z}_{t} \) is a geometric random variable independent of \( \tau \) ... | Yes |
Proposition 24.8. For reversible chains,\n\n(i)\n\n\[ \n{t}_{\text{stop }} \leq 8{t}_{\mathrm{G}} \n\]\n\n(ii)\n\n\[ \n{t}_{\text{stop }} \leq 4\left( {{t}_{\text{Ces }} + 1}\right) \n\] | Proof. Consider the chain with transition matrix \( R\left( {x, y}\right) = {\mathbf{P}}_{x}\left\{ {{X}_{G} = y}\right\} \) , where \( G \) is geometric with mean \( t \) . Set \( t = {t}_{\mathrm{G}} \) so that\n\n\[ \n{\begin{Vmatrix}{\mathbf{P}}_{x}\left\{ {X}_{G} = \cdot \right\} - \pi \end{Vmatrix}}_{\mathrm{{TV}... | Yes |
Proposition 24.22. Given an irreducible Markov chain \( \left( {X}_{t}\right) \) with finite state space \( \mathcal{X} \) and stationary distribution \( \pi \), let \( A, C \subseteq \mathcal{X} \) with \( A \cap C = \varnothing \) . Then\n\n\[ \pi \left( A\right) \leq \frac{{d}^{ + }\left( {A, C}\right) }{{d}^{ + }\l... | Proof. Define\n\n\[ \tau = \min \left\{ {t > {\tau }_{C} : {X}_{t} \in A}\right\} .\n\]\n\nConsider a Markov chain on \( A \) defined as follows: for each \( x, y \in A \), let \( Q\left( {x, y}\right) = \) \( {\mathbf{P}}_{x}\left\{ {{X}_{\tau } = y}\right\} \) . Let \( \mu \) denote a stationary distribution of this ... | Yes |
Proposition 24.23. If \( P \) is an irreducible transition matrix, then for any positive eigenvalue \( \lambda > 0 \) , \[ 1 - \lambda \geq \frac{1}{{t}_{\mathrm{G}} + 1} \] In particular, for reversible lazy chains, \[ {t}_{\mathrm{{rel}}} \leq {t}_{\mathrm{G}} + 1 \] | Proof. Let \( K\left( {x, y}\right) = {\mathbf{P}}_{x}\left\{ {{X}_{Z} = y}\right\} \), where \( Z \) is geometric with mean \( t = {t}_{\mathrm{G}} \) . Any eigenvalue \( \lambda \) of \( P \) gives the eigenvalue for the \( K \) -chain \[ \widetilde{\lambda } = \mathop{\sum }\limits_{{k = 1}}^{\infty }{\lambda }^{k}{... | Yes |
Proposition 24.24. Let \( P \) be an irreducible transition matrix on the state space \( \mathcal{X} \) and let \( \widetilde{P}\left( {x, y}\right) = \theta \left( x\right) P\left( {x, y}\right) + \left( {1 - \theta \left( x\right) }\right) {\delta }_{x}\left( y\right) \) . Assume that \( \theta \left( x\right) \geq {... | Proof. Note that one can construct the \( \widetilde{P} \) -chain from the \( P \) -chain \( \left( {X}_{t}\right) \) by repeating the state \( {X}_{t} \) for \( {D}_{t} \) steps, where the conditional distribution of \( {D}_{t} \), given \( {X}_{t} = x \), is geometric \( \left( {\theta \left( x\right) }\right) \) .\n... | Yes |
Corollary 6.8 (Sufficiency in Theorem 6.6): If \( N = D \) and \( \sigma \left( t\right) \) is nonsingular for Lebesgue-almost-every \( t \in \left\lbrack {0, T}\right\rbrack \) almost surely, then the financial market is complete. | Proof. We verify the condition of Proposition 6.2. Let \( B \) be an \( \mathcal{F}\left( T\right) \) -measurable random variable satisfying (6.3), and define the Lévy \( {P}_{0} \) -martingale\n\n\[ \n{M}_{0}\left( t\right) = {E}_{0}\left\lbrack {\left. \frac{B}{{S}_{0}\left( T\right) }\right| \;\mathcal{F}\left( t\ri... | Yes |
Example 3.1 (Forward contract to purchase a stock that pays no dividends): Suppose the contract is to purchase one share of the first stock, i.e., \( B = {S}_{1}\left( T\right) \) . If the first stock pays no dividends and \( \sigma \left( \cdot \right) \) satisfies the Novikov condition (1.5.17) with \( n = 1 \), then... | \[ {V}^{FC}\left( {t;q}\right) = {S}_{1}\left( t\right) - q{S}_{0}\left( t\right) \cdot {E}_{0}\left\lbrack {1/{S}_{0}\left( T\right) \mid \mathcal{F}\left( t\right) }\right\rbrack ,\;0 \leq t \leq T. \] (3.3) If in addition \( {S}_{0}\left( T\right) \) is nonrandom, the hedging portfolio is particularly simple. The ag... | Yes |
Example 3.3 (Forward price of a stock that pays no dividends): If \( B = \) \( {S}_{1}\left( T\right) \), the first stock pays no dividends, and \( \sigma \left( \cdot \right) \) satisfies the Novikov condition (1.5.17) with \( n = 1 \), then (3.3) and (3.5) yield | \[ f\left( t\right) = \frac{{S}_{1}\left( t\right) /{S}_{0}\left( t\right) }{{E}_{0}\left\lbrack {1/{S}_{0}\left( T\right) \mid \mathcal{F}\left( t\right) }\right\rbrack },\;0 \leq t \leq T. \] | Yes |
Example 3.4 (Forward price of a stock with nonrandom dividend rate): If \( B = {S}_{1}\left( T\right) \), the dividend rate process \( {\delta }_{1}\left( \cdot \right) \) is nonrandom and the processes \( {\sigma }_{11}\left( \cdot \right) ,\ldots ,{\sigma }_{1N}\left( \cdot \right) \) are uniformly bounded, then the ... | To see this, observe from (1.5.16) that the process \( \frac{{S}_{1}\left( t\right) }{{S}_{0}\left( t\right) }\exp \left\{ {{\int }_{0}^{t}{\delta }_{1}\left( u\right) {du}}\right\} \) is a \( {P}_{0} \) -martingale, so the numerator of (3.4) is\n\n\[ {E}_{0}\left\lbrack {{S}_{1}\left( T\right) /{S}_{0}\left( T\right) ... | Yes |
Example 3.5 (Forward price of a stock with nonrandom dividend payments, when the money market is nonrandom): If \( B = {S}_{1}\left( T\right) \), if the dividend-payment \( \rho \left( \cdot \right) \triangleq {\delta }_{1}\left( \cdot \right) {S}_{1}\left( \cdot \right) \) and money-market prices \( {S}_{0}\left( \cdo... | From (3.4) we have\n\n\[ f\left( t\right) = {S}_{0}\left( T\right) \left\lbrack {\frac{{S}_{1}\left( t\right) }{{S}_{0}\left( t\right) } - {\int }_{t}^{T}\frac{\rho \left( u\right) }{{S}_{0}\left( u\right) }{du}}\right\rbrack . \]\n\n(3.8) | No |
Corollary 3.9 (Forward-futures spread): Under the conditions of Theorem 3.7, we have\n\n\\[ \nf\\left( t\\right) = \\varphi \\left( t\\right) + \\frac{{\\operatorname{Cov}}_{0}\\left\\lbrack {B,1/{S}_{0}\\left( T\\right) \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack }{{E}_{0}\\left\\lbrack {1/{S}_{0}\\left( T\\r... | Proof. Because\n\n\\[ \n{\\operatorname{Cov}}_{0}\\left\\lbrack {X, Y \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack = {E}_{0}\\left\\lbrack {{XY} \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack - {E}_{0}\\left\\lbrack {X \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack \\cdot {E}_{0}\\left\\lbrack {Y \\... | Yes |
Example 4.1 (European call option): A European call option on the first stock in our market is the ECC given by \( C\left( t\right) = 0,\;0 \leq t < T \) and \( C\left( T\right) = {\left( {S}_{1}\left( T\right) - q\right) }^{ + } \) . The nonrandom constant \( q > 0 \) is called the strike price, and \( T \) is the exp... | With \( \varphi \left( x\right) \triangleq {\left( {x}_{1} - q\right) }^{ + } \), the Gaussian integration in (4.6) can be carried out explicitly, to yield\n\n\[ \n{u}^{ECC}\left( {s,{x}_{1};q}\right) = \left\{ \begin{array}{ll} {x}_{1}{e}^{-{\delta }_{1}s}\Phi \left( {{\rho }_{ + }\left( {s,{x}_{1};q}\right) }\right) ... | Yes |
The European put option confers to its holder the right to sell a stock at a future time at a prespecified price. We model a put on the first stock as the ECC with \( C\left( t\right) = 0 \) for \( 0 \leq t < T \) and \( C\left( T\right) = {\left( q - {S}_{1}\left( T\right) \right) }^{ + } \) . Because \( {\left( q - {... | First note from Remark 1.5.11 that\n\n\[ \n{e}^{-\left( {r - {\delta }_{1}}\right) t}{S}_{1}\left( t\right) = {S}_{1}\left( 0\right) + {\int }_{0}^{t}{e}^{-\left( {r - {\delta }_{1}}\right) u}\left\lbrack {d{S}_{1}\left( u\right) - \left( {r - {\delta }_{1}}\right) {S}_{1}\left( u\right) {du}}\right\rbrack \n\]\n\n\[ \... | Yes |
Consider an ECC of the form \( C\left( t\right) = 0 \) for \( 0 \leq t < T \) and \( C\left( T\right) = G\left( \omega \right) \) , where \( G : C\left( \left\lbrack {0, T}\right\rbrack \right) \rightarrow \mathbb{R} \) is a functional satisfying under \( {P}_{0} \) the conditions (E.4)-(E.6) of Appendix E. Then from t... | \[ {V}^{ECC}\left( t\right) = {e}^{-r\left( {T - t}\right) }{E}_{0}\left\lbrack {G\left( {W}_{0}\right) \mid \mathcal{F}\left( t\right) }\right\rbrack \] \[ = {e}^{-r\left( {T - t}\right) }{E}_{0}G\left( {W}_{0}\right) \] \[ + {e}^{-r\left( {T - t}\right) }{\int }_{0}^{t}{E}_{0}\left\lbrack {\partial G\left( {{W}_{0};(... | Yes |
Theorem 6.7 (McKean (1965)): Under the assumption (6.7), the value process for a perpetual American call option is given by\n\n\[ \n{V}^{AC}\left( {t;\infty }\right) = g\left( {S\left( t\right) }\right) ,\;0 \leq t < \infty ,\n\]\n\nwhere the function \( g \) is\n\n\[ \ng\left( x\right) = \left\{ \begin{array}{ll} \lef... | Proof. Itô's rule for convex functions (e.g., Karatzas and Shreve (1991), Theorem 3.6.22 and Problem 3.6.7(i)) implies\n\n\[ \nd\left( {{e}^{-{rt}}g\left( {S\left( t\right) }\right) }\right) = {e}^{-{rt}}S\left( t\right) {g}^{\prime }\left( {S\left( t\right) }\right) {\sigma d}{W}_{0}\left( t\right) - {e}^{-{rt}}\left(... | Yes |
Example 7.5 (Subsistence consumption): Suppose\n\n\[ \n{U}_{1}\left( c\right) = \left\{ \begin{array}{ll} \log \left( {c - \bar{c}}\right) , & \bar{c} < c < \infty , \\ - \infty , & - \infty < c \leq \bar{c}, \end{array}\right.\n\]\n\nwhere \( \bar{c} \) is a positive constant that consumption must exceed at all times.... | If \( {S}_{0}\left( \cdot \right) \) is deterministic, we can derive the optimal portfolio explicitly. Under this condition,\n\n\[ \n{X}_{1}\left( t\right) = \frac{T - t}{T{H}_{0}\left( t\right) }\left( {x - \bar{c}{h}_{1}}\right) + \bar{c}{S}_{0}\left( t\right) \left( {{\int }_{t}^{T}\frac{du}{{S}_{0}\left( u\right) }... | Yes |
Example 7.9 (Portfolio insurance): Suppose\n\n\\[ \n{U}_{2}\\left( x\\right) = \\left\\{ \\begin{array}{ll} \\log \\left( {x - \\bar{x}}\\right) , & \\bar{x} < x < \\infty , \\\\ - \\infty , & - \\infty < x \\leq \\bar{x}, \\end{array}\\right. \n\\]\n\nwhere \\( \\bar{x} \\) is a positive constant below which terminal ... | We have \\( {\\mathcal{Y}}_{2}\\left( x\\right) = 1/\\left( {x - \\bar{x}{h}_{2}}\\right) \\) for \\( x > \\bar{x}{h}_{2} \\), and the optimal consumption and wealth processes are \\( {c}_{2}\\left( t\\right) \\equiv 0 \\) and\n\n\\[ \n{X}_{2}\\left( t\\right) = \\frac{1}{{H}_{0}\\left( t\\right) }\\left\\{ {x - \\bar{... | Yes |
Theorem 8.11 (Hamilton-Jacobi-Bellman equation): Under Assumptions 8.1 and 8.2, the value function \( V\\left( {t, x}\\right) \) of (8.29),(8.30) is of class \( {C}^{1,2} \) on the set \( D \) of (8.10), continuous on the set \( \\{ \\left( {t, x}\\right) \\in \) \( \\left\\lbrack {0, T}\\right\\rbrack \\times \\left( ... | Proof. Differentiating (8.9) and (8.29) and using the formula (8.35), we obtain for \( \\left( {t, x}\\right) \\in D \) ,\n\n\[ {\\mathcal{X}}_{t}\\left( {t,\\mathcal{Y}\\left( {t, x}\\right) }\\right) + {\\mathcal{X}}_{y}\\left( {t,\\mathcal{Y}\\left( {t, x}\\right) }\\right) {\\mathcal{Y}}_{t}\\left( {t, x}\\right) =... | Yes |
Theorem 8.12 (Convex dual of \( V\left( {t, \cdot }\right) \) ): Let Assumptions 8.1 and 8.2 hold. Then, for each \( t \in \left\lbrack {0, T}\right\rbrack \), the function \( V\left( {t, \cdot }\right) \) satisfies all the conditions of Definition 4.1, and\n\n\[ \mathcal{X}\left( {t,\infty }\right) = \inf \{ x \in \ma... | Proof. All the claims (8.40)-(8.43) made here for fixed \( t \in \lbrack 0, T) \) are contained in Theorem 6.11, taking \( T \) in that theorem to be \( T - t \) here. When \( t = T,\left( {8.40}\right) - \left( {8.43}\right) \) and (8.45) follow directly from the definitions.\n\nEquation (8.42), Lemma 8.4, and Lemma 8... | Yes |
Consider the case that \( r\left( \cdot \right) = \) \( r > 0,\theta \left( \cdot \right) = \theta \neq 0 \), and \( \sigma \left( \cdot \right) = \sigma \) are constants, and \( A\left( \cdot \right) \equiv 0 \) . Set \( \gamma = \frac{1}{2}\parallel \theta {\parallel }^{2} > 0 \) . Assume that\n\n\[ \n{U}_{1}\left( {... | The functions of Theorem 8.12 can be computed explicitly, following Karatzas, Lehoczky, and Shreve (1987), as follows. Denote by \( {\lambda }_{ + } \) and \( {\lambda }_{ - } \) the respective positive and negative roots of the quadratic equation \( \gamma {\lambda }^{2} - \) \( \left( {r - \alpha - \gamma }\right) \l... | Yes |
Theorem 9.20 (Hamilton-Jacobi-Bellman equation): Let Assumptions 9.9 and 9.13 hold. Then the value function \( {V}_{\infty } \) is twice continuously differentiable on \( \left( {{\mathcal{X}}_{\infty }\left( \infty \right) ,\infty }\right) \) and satisfies the Hamilton-Jacobi-Bellman equation of dynamic programming,\n... | Proof. Equations (9.16), (9.29), and (9.40) imply\n\n\[ {V}_{\infty }^{\prime }\left( x\right) = {\mathcal{Y}}_{\infty }\left( x\right) ,\;{V}_{\infty }^{\prime \prime }\left( x\right) = {\mathcal{Y}}_{\infty }^{\prime }\left( x\right) ,\;x > {\mathcal{X}}_{\infty }\left( \infty \right) . \]\n\nWe may thus rewrite the ... | Yes |
In a continuous-time capital asset pricing model with an underlying \( N \) -dimensional Markov state process, the risk premia of assets can be computed theoretically from their covariances with a set of \( N + 1 \) mutual funds. | Breeden (1979) shows that rather than using the set of all covariances, one can in principle compute risk premia from the covariance of assets with the consumption process of an optimally behaving investor. Like the simple mean-variance capital asset pricing model, this consumption-based capital asset pricing model doe... | No |
Theorem 6.3 (Existence of an equilibrium market): Choose \( {\underset{ \sim }{\Lambda }}^{ * } \in \) \( {}^{ * }\lbrack 0,\infty {)}^{K} \) to satisfy (5.17) and (6.15). Define \( r\left( \cdot \right) ,\theta \left( \cdot \right) \), and \( A\left( \cdot \right) \) by (6.20)-(6.22). Let \( \sigma \left( t\right) = {... | Proof. Because of Corollary 5.4, we need only verify that \( \mathcal{M} \) is a standard, complete financial market satisfying Condition 3.1. (Recall that we are omitting dividends from the markets in this chapter.) Condition 2.1(iii), the integrability of \( \beta \left( \cdot \right) \), and the boundedness of \( \n... | Yes |
Theorem 6.4 (Uniqueness of the equilibrium market): Assume that (6.4) holds. Then the equilibrium money market process \( {S}_{0}\left( \cdot \right) \), the state price density process \( {H}_{0}\left( \cdot \right) \), and the market price of risk process \( \theta \left( \cdot \right) \), are uniquely determined, as... | Proof. The uniqueness of \( {H}_{0}\left( \cdot \right) \) follows from Corollary 5.4, Theorem 6.1, and the initial condition \( {H}_{0}\left( 0\right) = 1 \) . The uniqueness of \( {\widehat{c}}_{1}\left( \cdot \right) ,\ldots ,{\widehat{c}}_{K}\left( \cdot \right) \) also follows from Theorem 6.1. The semimartingale ... | Yes |
Example 7.1 (Logarithmic utility with subsistence consumption): Let \( {U}_{k}\left( c\right) = \log \left( {c - {\bar{c}}_{k}}\right) \), for \( c > {c}_{k}, k = 1,\ldots, K \), where each \( {\bar{c}}_{k} \) is a nonnegative constant. Then | \[ {U}^{\prime }\left( {c;\Lambda }\right) = \mathcal{H}\left( {c;\Lambda }\right) = \frac{1}{c - \bar{c}}\mathop{\sum }\limits_{{k = 1}}^{K}{\lambda }_{k},\;c > \bar{c}. \] We normalize \( \Lambda \) by setting \( \mathop{\sum }\limits_{{k = 1}}^{K}{\lambda }_{k} = \epsilon \left( 0\right) - \bar{c} \), a strictly pos... | Yes |
Example 7.2 (Power utility with subsistence consumption): Let \( {U}_{k}\left( c\right) = \) \( \frac{1}{p}{\left( c - {\bar{c}}_{k}\right) }^{p} \) for \( c > {\bar{c}}_{k}, k = 1,\ldots, K \), where \( p < 1, p \neq 0 \), and each \( {\bar{c}}_{k} \) is a nonnegative constant. Then | \[ {U}^{\prime }\left( {c;\underset{ \sim }{\Lambda }}\right) = \mathcal{H}\left( {c;\underset{ \sim }{\Lambda }}\right) = {\left\lbrack \frac{\mathop{\sum }\limits_{{k = 1}}^{K}{\lambda }_{k}^{\frac{1}{1 - p}}}{c - \bar{c}}\right\rbrack }^{1 - p},\;c > \bar{c}. \] We normalize \( \Lambda \) by setting \( \mathop{\sum ... | Yes |
Example 7.5 (Constant aggregate endowment): If the aggregate endowment \( \epsilon > \bar{c} \) is constant, then the unique vector \( \Lambda \) satisfying the normalization \( \mathcal{H}\left( {\epsilon ;\Lambda }\right) = 1 \) is \[ \underset{ \sim }{\Lambda } = \left( {\frac{1}{{U}_{1}^{\prime }\left( {\widehat{c}... | Constant aggregate endowment implies \( \nu \left( \cdot \right) \equiv 0,\xi \left( \cdot \right) \equiv 0,\rho \left( \cdot \right) \equiv 0 \) in (2.2), and the local time of \( \epsilon \left( \cdot \right) \) at every point is zero. Therefore, the equilibrium market coefficients (6.20)-(6.22) are \[ r\left( t\righ... | No |
Example 7.6 \( \\left( {K = 2,{U}_{1}\\left( c\\right) = \\log c,{U}_{2}\\left( c\\right) = \\sqrt{c}\\text{.}}\\right) : \) In this case, we have\n\n\[ \n{U}^{\\prime }\\left( {c;\\Lambda }\\right) = \\mathcal{H}\\left( {c;\\Lambda }\\right) = \\frac{{\\lambda }_{1}}{2c}\\left\\lbrack {1 + \\sqrt{1 + c{\\left( \\frac{... | The positive constants \( {\\lambda }_{1} \) and \( {\\lambda }_{2} \) are uniquely determined by (5.17) with \( k = 1 \) :\n\n\[ \n2{\\int }_{0}^{T}{e}^{-{\\int }_{0}^{t}\\beta \\left( u\\right) {du}}{dt} \n\]\n\n\[ \n= E{\\int }_{0}^{T}{e}^{-{\\int }_{0}^{t}\\beta \\left( u\\right) {du}}\\left\\lbrack {1 + \\sqrt{1 +... | Yes |
Example 7.8 (Ergodic aggregate endowment): Let us suppose that each agent \( k \) has utility function \( {U}_{k} \) with \( {\bar{c}}_{k} = 0 \) and \( {U}_{k}^{\prime }\left( 0\right) = \infty \), so \( \bar{c} = 0 \) . Let us further suppose that the aggregate endowment process \( \epsilon \left( \cdot \right) \) is... | Then the diffusion process \( \epsilon \left( \cdot \right) \) is ergodic with invariant measure \( m\left( {dc}\right) /m\left( \mathcal{I}\right) \) (cf. Proposition 5.5.22 and Exercise 5.5.40 in Karatzas and Shreve (1991)). | Yes |
Example 7.3 (European call option): We consider one stock \( S\left( \cdot \right) = {S}_{1}\left( \cdot \right) \) driven by a single Brownian motion, we assume (7.1)-(7.3), and we denote \( {\sigma }_{11} \) by \( \sigma \) . A European call option corresponds to \( \varphi \left( x\right) = {\left( x - q\right) }^{ ... | \[ \widehat{\varphi }\left( x\right) = \left\{ \begin{array}{ll} {\left( \frac{\beta - 1}{q}\right) }^{\beta - 1}{\left( \frac{x}{\beta }\right) }^{\beta }, & \text{ if }0 < x \leq \frac{\beta q}{\beta - 1}, \\ x - q, & \text{ if }x \geq \frac{\beta q}{\beta - 1}. \end{array}\right. \] (7.25) The function \( \widehat{\... | Yes |
Example 7.4 (European put option): We assume again (7.1)-(7.3) and consider one stock. A European put option corresponds to \( \varphi \left( x\right) = {\left( q - x\right) }^{ + } \) , where \( q \geq 0 \) . We consider again \( K = \left\lbrack {\alpha ,\beta }\right\rbrack \) with \( - \infty \leq \alpha \leq 0 \le... | \[ \widehat{\varphi }\left( x\right) = \left\{ \begin{array}{ll} q - x, & \text{ if }0 < x \leq \frac{\alpha q}{\alpha - 1}, \\ {\left( \frac{\left| \alpha - 1\right| }{q}\right) }^{\alpha - 1}{\left( \frac{x}{\left| \alpha \right| }\right) }^{\alpha }, & \text{ if }x \geq \frac{\alpha q}{\alpha - 1} \end{array}\right.... | Yes |
Example 8.8 (Incomplete market): Consider the case \( K = \{ p \in \) \( \\left. {{\\mathbb{R}}^{N};{p}_{M + 1} = \\cdots = {p}_{N} = 0}\\right\} \) of Example 4.1(iii), where there are only \( M \) stocks available for investment, but these are driven by the \( N \) -dimensional Brownian motion \( W\\left( \\cdot \\ri... | \[ \\zeta \\left( \\nu \\right) + {p}^{\\prime }\\nu = 0,\\;\\forall p \\in K,\\;\\nu \\in \\widetilde{K}. \] | Yes |
Example 10.2 (Prohibition of short-selling): We consider a market with constant coefficients and one stock, i.e., \( N = 1 \) . When short-selling is prohibited (Example 9.7(ii), \( K = \left\lbrack {0,\infty ),{K}_{ - } = ( - \infty ,0}\right\rbrack \) ), we have \( \widetilde{K} = \lbrack 0,\infty ) \) , \( \zeta \le... | For a European call, we have\n\n\[ \check{\varphi }\left( x\right) = \mathop{\inf }\limits_{{\nu \geq 0}}{\left( x{e}^{-\nu } - q\right) }^{ + } = 0,\;\forall x > 0, \]\n\nand the lower hedging value is zero. This is also the conclusion of Proposition 9.14(ii) in a more general context. For a European put option,\n\n\[... | Yes |
Example 10.3 (Prohibition of borrowing): We consider again a market with constant coefficients and one stock, i.e., \( N = 1 \) . When borrowing from the money market is prohibited (Example 9.7(vi), \( K = ( - \infty ,1\rbrack ,{K}_{ - } = \) \( \lbrack 1,\infty )) \), we have \( \widetilde{K} = ( - \infty ,0\rbrack ,\... | \[ \check{\varphi }\left( x\right) = \mathop{\inf }\limits_{{\nu \leq 0}}\left\lbrack {{e}^{\nu }\varphi \left( {x{e}^{-\nu }}\right) }\right\rbrack ,\;\forall x > 0. \] For a European call option, we have \[ \check{\varphi }\left( x\right) = \mathop{\inf }\limits_{{\nu \leq 0}}{\left( x - q{e}^{\nu }\right) }^{ + } = ... | Yes |
Proposition 5.1 (Weak Duality): Suppose (5.2) and (5.3) hold. Then, for any given \( y \in \left( {0,\infty }\right) \) and with \( x = {\mathcal{X}}_{{\nu }_{y}}\left( y\right) \) , (i) there exists an optimal consumption and portfolio-proportion process pair \( \left( {\widehat{c},\widehat{p}}\right) \in {\mathcal{A}... | Proof. First, let us note that (5.3), (3.10), and (3.27) imply \[ V\left( {x;K}\right) \leq \widetilde{V}\left( y\right) + {xy},\forall y > 0,\forall x > 0. \] (5.5) Now fix \( y \in \left( {0,\infty }\right) \), let \( x = {\mathcal{X}}_{{\nu }_{y}}\left( y\right) \), and note that the assumption \( {\widetilde{V}}_{{... | Yes |
Example 7.2 (Logarithmic utility, incomplete market): \( {U}_{1}\left( {t, x}\right) = \) \( {U}_{2}\left( x\right) = \log x \) for every \( \left( {t, x}\right) \in \left\lbrack {0, T}\right\rbrack \times \left( {0,\infty }\right) \). | This is Example 4.2, specialized to the case of an incomplete market; i.e., \( K \) given by (7.1). The expression in (4.23) to be minimized over \( \xi \in {\mathbb{R}}^{L} \) is\n\n\[ \frac{1}{2}{\begin{Vmatrix}\widetilde{\theta }\left( t\right) + {\rho }^{\prime }\left( t\right) \left( a\left( t\right) + \xi - r\lef... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.