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Find a regular process $A_t$, such that

$$M_t=tB_t^2-A_t$$

is a martingale. $B_t$ is Brownian motion.

I'm totally lost. I have an idea that I should apply Ito formula here, but I don't know where to start...

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    $\begingroup$ Apply Itô's formula to $t B_t^2$ $\endgroup$
    – saz
    Feb 9, 2020 at 19:04

1 Answer 1

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I will post the solution using the suggestion by saz here for those who are interested.

Applying Ito formula to function $F(x,t)=tx^2$ we get

$$tB_t^2=2\int_0^tsB_sdB_s+\int_0^tB^2_sds+\int_0^tsds.$$

It can be proven that integral $\int_0^tsB_sdB_s$ is a martingale. Since $A_t$ is a regular process, we get:

$$A_t=\int_0^t\left(B^2_s+s\right)ds.$$

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