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 $s > \tau$.