Expectation of stochastic integral

Question

I'm not able to understand why $$E\bigg[\int_0^tB_s^3dBs\bigg]=0$$ it would be true if $B_s^3 \in \Lambda^2$ but it seems to me that $B_s^3\in \Lambda^2_{loc}$

Answer 1

Let $f(x):=x^4/4$. Then, using Itô's lemma, $$ f(B_t)=\int_0^t f'(B_s)\,dB_s+\frac{1}{2}\int_0^t f''(B_s)ds=\int_0^t B_s^3 \,dB_s+\frac{3}{2}\int_0^t B_s^2\,ds, $$ and, therefore, $$ \mathsf{E}\left[\int_0^t B_s^3 \,dB_s\right]=\frac{3}{4}t^2-\frac{3}{4}t^2 =0.$$