Compute the expectation of this stochastic process

Question

Consider the following stochastic process

$X_t = (1-t)\int_{0}^{t} \frac{1}{1-s} dW_s$ and let $ \bar{X}_t = \begin{cases} X_t &\mbox{if } 0\leq t <1 \\ 0 & \mbox{if } t=1 \end{cases} $

Compute $E[\bar{X}_t]$

My attempt: $E[(1-t)\int_{0}^{t} \frac{1}{1-s} dW_s] = (1-t)E[\int_{0}^{t}\frac{1}{1-s} dW_s] = 0 $

since $E[\int_{0}^{t} \frac{1}{1-s} dW_s] = 0 $

I'm unsure whether my working is correct here or if im missing something in computing its expectations.

Answer 1

Your answer is correct, as phrased. If $t = 1$ then $E[\bar{X}_t] = E[0] = 0$. If $t < 1$, then recognize $\int_0^1 \frac{1}{1 - s} dW_s$ as a martingale and thus $E[\bar{X}_t] = E[\bar{X}_0] = 0$.