I'm trying to solve a problem that's now doing my head in a bit. I'll share with you the question and let's see if somebody can shed some light into the matter:
Let B be a standard Brownian Motion started at zero, and let M be a stochastic process defined by: $$ M_t = \int_0^{\log(\sqrt{1 + 2t})} e^{s}\mathrm dB_s\,. $$
- Show that M is a Standard Brownian Motion.
- Calculate $$E\left(\int_0^tM^6_s\mathrm dM_s\right).$$
- Calculate $$E\left(\left(\int_0^tM_s\mathrm dM_s\right)^3\right).$$
Hope you can give me some hints, I have the feeling it's actually not that tricky. I would solve question number one using Lévy's characterization theorem for Brownian Motion, not so sure about questions 2 and 3.
For part 2:
Is a square-integrable continuous local martingale a true martingale?
I think 'lemma 3' in the first answer tells you how to solve question 2. It shows the stochastic integral inside your expectation is a true martingale, which means the expectation is 0.
For part 3:
The integral is $(M_t^2-t)/2$, where $M$ is a Brownian motion. Then you just cube this and use the fact $E[M_t^n] = t^{n/2}(n-1)!$ for $n$ is even, $E[M_t^n]=0$ for $n$ odd. In fact we could have used it for part 2 but this would be too much hassle...