# Expected Value of Brownian motion using ito isometry

Find $$E\ \left[\left(\int_{0}^T e^{s+W_s}dW_s \right)^2\right],$$ where $(W_s)$ is a Brownian motion.

I tried to use Ito isometry to solve this question, but still not yet to find the right path. Appreciate if you could shed the light on this question. Thanks.

-

By Ito isometry, this is $\mathsf{E} \intop_0^T e^{2(s+W_s)} ds$. Then use Fubini theorem to interchange $\intop$ and $\mathsf{E}$, and recall the exponential moments of a Gaussian distribution, or calculate them if you have never done it, or otherwise look here.