Let \(B_t\) be a standard one dimensional Brownian

Motion. Find the function \(F:\mathbf{R}^5 \rightarrow \mathbf R\) so that

\begin{align*}

B_t^3 – F\Big(t,B_t,B_t^2,\int_0^t B_s ds, \int_0^t B_s^2 ds\Big)

\end{align*}

is a Martingale.

Hint: It might be useful to introduce the processes

\[X_t=B_t^2\qquad Y_t=\int_0^t B_s ds \qquad Z_t=\int_0^t B_s^2 ds\]