A basic question on martingales and filtrations

Question

It is stated as a Corollary in a book that if $\tau$ is any stopping time and $M_t$ is a martingale with respect to $(\Omega,\mathcal{F}_t,\mathbb{P})$, then so is $M_{t\wedge\tau}$. However, I am mainly concerned with the filtrations associated with the martingales $M_{t\wedge\tau}$. I know that (from the Theorem in the book from which the Corollary is deduced without stating its proof) for any $0\leq s\leq t$ we have $$\mathbb{E}[M_{t\wedge \tau} \mid \mathcal{F}_{s\wedge \tau}] = M_{s\wedge \tau}.$$ But I think the author means that $M_{t\wedge \tau}$ is a martingale associated with the same filtration as that of $M_t$ (which is $\{\mathcal{F}_t\}$), in that case, I really didn't see how one can prove that $$\mathbb{E}[M_{t\wedge \tau} \mid \mathcal{F}_s] = M_{s\wedge \tau}$$ for any $0\leq s \leq t$. Thanks for any help!

Answer 1

It's effectively just a computation:

\begin{align*} \mathbb{E}[M_{t \wedge \tau} | \mathcal F_s] &= \mathbb{E}[M_{t \wedge \tau}1_{\tau \le s} | \mathcal F_s] + \mathbb{E}[M_{t \wedge \tau}1_{\tau > s} | \mathcal F_s] \\ &= M_\tau 1_{\tau \le s} + \mathbb{E}[M_{t \wedge \tau}1_{\tau > s} | \mathcal F_s] \end{align*}

so we just need to show $\mathbb{E}[M_{t \wedge \tau}1_{\tau > s} | \mathcal F_s] = M_s 1_{\tau > s}.$ Let $A \in \mathcal F_s$, and notice $A \cap \{\tau > s \} \in \mathcal F_{\tau \wedge s}$ so we have

\begin{align*} \int_A M_{t \wedge \tau}1_{\tau > s} dP &= \int_{A \cap \{\tau > s \}} M_{t \wedge \tau} dP \\ &= \int_{A \cap \{\tau > s \}} \mathbb{E}[M_{t \wedge \tau}|\mathcal F_{\tau \wedge s}] dP \\ &= \int_{A \cap \{\tau > s \}} M_{s \wedge \tau} dP \\ &= \int_A M_{s \wedge \tau}1_{\tau > s} dP \\ &= \int_A M_{s}1_{\tau > s} dP \end{align*}

and by definition of conditional expectation we conclude $\mathbb{E}[M_{t \wedge \tau}1_{\tau > s} | \mathcal F_s] = M_s 1_{\tau > s}.$ Therefore

\begin{align*} \mathbb{E}[M_{t \wedge \tau} | \mathcal F_s] &= M_\tau 1_{\tau \le s} + M_s 1_{\tau > s} \\ &= M_{s \wedge \tau}. \end{align*}