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A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach all the states. We consider a process $\{R_t\}$ which is $\mathcal{F}_t$ supermartingale, therefore for any two stopping times $ T \geq S$, $E[R_T|\mathcal{F_t}] \leq E[R_S|\mathcal{F_t}]$ (due to optional stopping theorem). Now, let $g$ be a real valued bounded function of $\mathbb{R}$.

Intuitively, I seems obvious to me that:

$$\sup_{t \leq \tau \leq \infty} E[R_{\tau} g(X_{\tau})| \mathcal{F}_t] = \max_{k\in {1,\dots,N}}E[R_{\tau_k}g(X_{\tau_k})|\mathcal{F}]$$ where

$\tau_k= \inf\{s \geq t, X_s= k\}$.

I am having trouble showing it though, the 'math' behind my intuition is this : take the supremum over the partition $\cup_k \tau(\{\omega: X_{\tau} =k \})$ and then the supremum will become

$\max(\sup_{\tau( \{\omega: X_{\tau} =1 \})}E[R_\tau g(1)|\mathcal{F}_t], \cdots, \sup_{\tau( \{\omega: X_{\tau} =N \})} E[R_\tau g(N)|\mathcal{F}_t]) = \max_{k\in {1,..,N}}E[R_{\tau_k}g(X_{\tau_k})|\mathcal{F}]$

as $\sup_{\tau( \{\omega: X_{\tau} =k \})} E[R_\tau g(k)|\mathcal{F}_t]= E[R_{\tau_k}g(X_{\tau_k})|\mathcal{F}]$.

I am not sure about my first step though i.e the supremum over the union.

I appreciate any suggestion. Thanks.

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  • $\begingroup$ There are very simple counterexamples to your claim. e.g. if $g\equiv-1$, $R_t=-t$, then the optimizer is $\tau=\infty$. Even if you assume $g, R \geq 0$ it does not hold, for instance taking $R_t=1$ for $t=0,1$ and $R_t=0$ for $t \geq 2$. Then the optimizer $\tau$ will be $0$ or $1$ but it will not be in general the hitting time of a singleton. $\endgroup$ – pgassiat Sep 4 '14 at 8:18
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Consider the sequence of stopping times $\tau_{1,k} = \inf\{s \geq t, X_s = k\}$, $\tau_{2,k} = \inf\{s \geq \tau_{1,k}, X_s = k\}$, $\tau_{3,k} = \inf\{s \geq \tau_{2,k}, X_s = k\}$, etc. Then $\cup_{i \geq t, k} \{\tau_{i,k}\}$ is all the natural numbers bigger than $t$, and also $E[R_{\tau_{i,k}} g(X_{\tau_{i,k}}) | \mathcal{F}_t]$ is decreasing in $i$ for each $k$. I think these give you your result, i.e. $$ \sup_{t \leq \tau < \infty} E[R_\tau g(X_\tau) | \mathcal{F}_t] = \sup_{i \geq t, k} E[R_{\tau_{i,k}} g(X_{\tau_{i,k}}) | \mathcal{F}_t] = \max_k E[R_{\tau_{1,k}} g(X_{\tau_{1,k}}) | \mathcal{F}_t]. $$

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