I am struggling the following exercise:
Processes $X$ and $Y$ are defined as: $$X=\Bigg( \int_{0}^{t}e^{W_{s}^{2}}\mathrm{d}W_{s}\Bigg)_{t\ge 0},$$ $$Y=\Bigg(\int_{0}^{t}W_{s}^{5}\mathrm{d}X_{s} \Bigg)_{t\ge 0}.$$ Prove that $\ (Y_{t})_{t\in[0,\epsilon]} \ $ is a martingale for some $\epsilon>0$.
As I understand, it is straight forward to check, that $X$ is a local martingale. Thus, applying integration by substitution, we get $$Y_{s}=\int_{0}^{t}W_{s}^{5}e^{W_{s}^{2}}\mathrm{d}W_{s}.$$ Just from construction of isometric integral, I see that $Y$ is a local martingale too. But I do not know how to handle it from here. Only one way I can see, would be proving that $Y$ is somehow bounded for some small $\epsilon$. But I am not so sure if that is really the case here.