Computing expectation of a stochastic integral

Question

I need to compute the expectation

$$E\left[\int_0^tu \, dB_u \int_0^s u \, dB_u \right].$$

Being that is my first question, how can I initialize MathJax if I have it on my hard drive.

Answer 1

No idea about MathJax but your integral is $E[M_tM_s]$ where $M_t=\int\limits_0^tu \, \mathrm dB_u$. The process $(M_r)_{r\geqslant0}$ is a square-integrable martingale with $M_0=0$ hence, for every $t\geqslant s$, $E[M_tM_s]=E[M_s^2]=E[\langle M\rangle_s]$. Now, $\langle M\rangle_s=\int\limits_0^su^2 \, \mathrm d\langle B\rangle_u$ hence $\langle M\rangle_s$ is $____$ and...