Let $(B_t)_{t\geq 0}$ be a Standard Brownian Motion and $(M_t)_{t\geq 0}$ the martingale defined by $$ M_t = B_t + e^{\sqrt{2}B_t - t},\quad t\geq 0. $$ I have to compute $E(M_\sigma)$ and $E(e^{-\sigma})$ where $$ \sigma = \inf\{t\geq 0: B_t = 1\} $$
My idea is to use Optional Sampling Theorem for the first one, but I do not know if what I done to see integrability is okay (I said that $E|M_{t\wedge \sigma}| \leq 1 + e^{\sqrt{2}}$... it sounds weird. Maybe Doob's inequality is fine?). On the other hand, I have no intuition to calculating $Ee^{-\sigma}$. Any hint, please?