Let $W$ be a standard one dimensional Brownian motion, and let $\mathcal F\_t$ be its completed natural filtration.


Let $\tau$ be an $\mathcal F\_t$ stopping time with $\tau < T$ almost surely for some $T > 0$. Suppose $\xi$ is an $\mathcal F\_\tau$ measurable $L^2$ random variable.


**Question:** Does there exist some $\mathcal F\_t$ predictable process $H$ such that


$$\xi = \mathbb E[\xi] + \int\_0^\tau H\_s \, dW\_s$$


almost surely?


**Idea:**


I tried to proceed as follows - since $\xi$ is $\mathcal F\_\tau$ measurable and $\tau < T$ a.s., $\xi$ is clearly $\mathcal F\_T$ measurable. Now the standard martingale representation theorem gives some predictable $H$ such that


$$\xi = \mathbb E[\xi] + \int\_0^T H\_s \, dW\_s$$


almost surely. But since $\xi$ is $\mathcal F\_\tau$ measurable, it should follow that $H\_s = 0$, $d\mathbb P \times d\mu$ a.e. whenever $s > \tau$, whence


$$\xi = \mathbb E[\xi] + \int\_0^\tau H\_s \, dW\_s$$


as desired.


However, I am not fully sure how to rigorously show the claim $H\_s = 0$ whenever $s > \tau$.



Martingale representation theorem up to a stopping time
If $\xi\in \mathbb D^{1,2}$ is in the Sobolev-Watanabe space then we can apply Clark-Ocone formula to get that


$$\xi=E[\xi]+\int\_0^T E[D\_s\xi|\mathcal F\_s]dW\_s$$


where $D\_s$ is the Malliavin derivative. For $s\in [0,T]$ we may write $\xi=\xi 1\_{\{\tau > s\}}+\xi 1\_{\{\tau \leq s\}}$. Then


\begin{align\*}
\xi&=E[\xi]+\int\_0^T E[D\_s(\xi 1\_{\{\tau > s\}}+\xi 1\_{\{\tau \leq s\}})|\mathcal F\_s]dW\_s\\
&=E[\xi]+\int\_0^T E[D\_s(\xi 1\_{\{\tau > s\}})|\mathcal F\_s]dW\_s+\int\_0^T E[D\_s(\xi 1\_{\{\tau \leq s\}})|\mathcal F\_s]dW\_s
\end{align\*}


$\xi 1\_{\{\tau \leq s\}}$ is $\mathcal F\_s$-measurable so $D\_s (\xi 1\_{\{\tau \leq s\}})=0$. Also for $s>\tau$ we have that $\xi 1\_{\{\tau > s\}}=0$ so $D\_s (\xi 1\_{\{\tau > s\}})=0$ and for $s<\tau$ we have that $\xi 1\_{\{\tau > s\}}=\xi$. So


$$\xi=E[\xi]+\int\_0^\tau E[D\_s\xi|\mathcal F\_s]dW\_s.$$


