Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.