# Martingales

A

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#### Laurent

MHF Hall of Honor
I guess you know Itô's formula; you should try it (the two-variable version : with $$\displaystyle f(t,x)=x^3-3tx$$ for the first one)

B

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#### Laurent

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

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

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

HansG