Question: Is $X_t = e^{\int_0^t B_sdB_s - \frac{1}{2}\int_0^t B_s^2 dB_s}$ a martingale with respect to the filtration generated by $B_t$?
In order to determine whether the above expression is a martingale, I thought that it'd be a good first step to calculate each of the integrals in the exponent.
Thus, I first found that $\displaystyle \int_0^tB_sdB_s=\frac{B_t^2}{2}-\frac{t}{2}$ by using Ito's lemma on $B_s^2$ and manipulating the resulting equation.
However, I am not sure how to solve for the second integral, $ \displaystyle \int_0^tB_s^2dB_s$. I can't think of an expression on which to utilize Ito's lemma to be able to isolate for the given integral.
Am I even going about this question the right way? Do I need to solve for the integrals? Any guidance would be appreciated.