I am trying to determine for what values of a, b the process $X_t=e^{aW_t+bt}, t \ge 0$ is a martingale with respect to $F_t^{W}$. Here $W_t$ is a brownian motion.
I know I need to show that $\mathbb E(X_t|F_s^{W})=X_s$, but I am not sure how to compute $\mathbb E(e^{aW_t+bt}|F_s^{W}).$ Any help would be appreciated.