Let $\{W_t\}_{t \ge 0}$ be a Brownian motion. Determine whether $Y_t=e^{W_t^2 - t}$ is a martingale.
Attempt:
For $0\le s<t$, \begin{align*} E[Y_t|F_s] &= E[e^{W_t^2-t}|F_s] \\ &= E[e^{(W_t-W_s+W_s)^2 - t + s - s]}|F_s] \\ &= e^{-(t-s)} E[e^{(W_t-W_s)^2 + 2(W_t-W_s)W_s + W_s^2 - s}|F_s] \\ &= e^{-(t-s)} e^{W_s^2 - s} E[e^{(W_t-W_s)^2 + 2(W_t-W_s)W_s}| F_s] \\ &= e^{-(t-s)} e^{W_s^2 - s} e^{W_s} e^{2(t-s)} E[e^{(W_t-W_s)^2} | F_s] \\ &= e^{t-s} e^{W_s^2-s} e^{W_s} E[e^{(W_t-W_s)^2}|F_s]. \end{align*}
I got stucked here. What's next should I do? Thanks in advanced.