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Suppose $(\Omega,\mathcal F,\{\mathcal F_n\}_n,\mathbb P)$ is a filtered probability space. Let $\tau$ be a stopping time, and define a collection of sets $\mathcal F_\tau:=\{A\in\mathcal F: A\cap\{\tau=n\}\in\mathcal F_n\forall n<\infty\}$. We can easily show that $\mathcal F_\tau$ is a $\sigma$-algebra.

  1. Let $M=\{M_n\}_n$ be a martingale. Show that $\mathbb E[M_n|\mathcal F_\tau]=M_{n\wedge\tau}, n\ge 0$.

  2. Let $M=\{M_n\}_n$ be a uniformly integrable martingale with last element $M_\infty\in\mathcal F_\infty$. Show that $\mathbb E[M_\infty|\mathcal F_\tau]=M_\tau$.


My attempt:

I would use the definition of conditional expectation to show both of the identities.

For any $S\in\mathcal F_\tau$, we want to show that $\mathbb E[M_{n\wedge\tau};S]=\mathbb E[M_n;S]$. But then I got completely stuck for hours. Even after checking a lot of posts dealing with this kind of problem, I am still quite confused with the definition of $\mathcal F_\tau$. Can someone help me out? Thank you!

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1 Answer 1

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First we have to show that $M_{n\wedge \tau}$ is $\mathcal F_{\tau}$ measurable. For this we have to show that $(M_{n\wedge \tau} \leq x)\cap (\tau=k) \in \mathcal F_k$. But $(M_{n\wedge \tau} \leq x)\cap (\tau=k)=(M_{n\wedge k} \leq x)\cap (\tau=k)$ which is an intersection of two sets in $\mathcal F_k$.

Next we have to show that $\int_A M_{n\wedge \tau} dP=\int_A M_n dP$ for $A \in \mathcal F_{\tau}$. It is enough to show that $\int_{A\cap (\tau=k)} M_{n\wedge \tau} dP=\int_{A\cap (\tau=k)} M_n dP$ for each $k$. In other words we have to show $\int_{A\cap (\tau=k)} M_{n\wedge k} dP=\int_{A\cap (\tau=k)} M_n dP$. This follows by martingale property if $n > k$ and trivially holds if $n \leq k$.

The second part is just obtained by taking limit as $n \to \infty$ since a uniformly integrable martingale converges almost surely and in $L^{1}$.

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  • $\begingroup$ Thank you for your answer! But could please explain the intuition of $\mathcal F_\tau$? I am very confused with the definition of $\mathcal F_\tau$. $\endgroup$
    – Bach
    Dec 10, 2020 at 7:43
  • $\begingroup$ It is the sigma algebra of events up to to the random time $\tau$. Since $\mathcal F_n$'s are orignally defined only for fixed time $n$ we have to consider $A \cap (\tau=n)$ to get a meaningful definition of $A \in \mathcal F_{\tau}$. @Bach $\endgroup$ Dec 10, 2020 at 7:47
  • $\begingroup$ It seems to me that saying $A\in\mathcal F_\tau$ is the same as partitioning $A$ so that each part represents $A$ stops at time $i$ where $i$ varies in $1,2,......$. And if we pick up one part from those partitions, then the corresponding value of the stopping time on that part gives enough infomation that whether we stop or not? $\endgroup$
    – Bach
    Dec 10, 2020 at 8:07

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