Finding a regular process which is a martingale

Question

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...

Answer 1

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:

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