# A problem regarding the stopped $\sigma$-field

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!

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 $$L^{1}$$.

• Thank you for your answer! But could please explain the intuition of $\mathcal F_\tau$? I am very confused with the definition of $\mathcal F_\tau$.
– Bach
Dec 10, 2020 at 7:43
• 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 Dec 10, 2020 at 7:47
• 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?
– Bach
Dec 10, 2020 at 8:07