# Math Help - Martingales

1. ## Martingales

A

2. Originally Posted by HansG
I guess you know Itô's formula; you should try it (the two-variable version : with $f(t,x)=x^3-3tx$ for the first one)

3. B

4. Originally Posted by HansG
Thanks for the quick answer. Below I have tried to do the calculations. Please see attachment

Are they correct?

Can I get hint on how to show, that $M_3$ and $M_4$ are true martingales?
Once you get $dM_3(t)=H(t) dB(t)$, you are done proving that $M_3$ is a local martingale, whatever the right-continuous adapted process $H(t)$ is. You don't have to specify that $H(t)$ is a martingale or some kind of product of martingales.

As for proving that $M_3,M_4$ are true martingales, you have the following result: if $M$ is a local martingale with $M_0=0$, the following is equivalent :
(i) $M$ is a true square-integrable martingale ( $E[M_t^2]<\infty$)
(ii) for all $t>0$, $E[\langle M\rangle_t]<\infty$.

Remember that $E[\langle \int_0^\cdot H(s)dB(s)\rangle_t]=\int_0^t E[H(s)^2] ds$ so that you can easily check this criterion in your case.