$B_t$ is a Brownian motion and $Y_t:=e^{aB_t+bt}$ for $t\ge0$. For which $a, b\in\Bbb R$ is $Y_t$ a martingale?
My calculations with the martingale property lead me to the presumption that it is not a martingale except for $a=b=0$. But how can I prove that properly? (My main problem is that I am obtaining expected values that I can't calculate.)