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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.

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    $\begingroup$ Have you tried decomposing $W_t = W_s + (W_t-W_s)$? It's the only trick I know, but it works. $\endgroup$ – Michał Miśkiewicz Apr 19 '17 at 22:58
  • $\begingroup$ How do I deal with the expectation of the exponential though? $\endgroup$ – user75514 Apr 19 '17 at 23:00
  • $\begingroup$ The brownian motion $W_t$ has a normal distribution. Have you done any exercise about deriving the moment generating function of normal before? $\endgroup$ – BGM Apr 20 '17 at 4:27
  • $\begingroup$ Yes, I am familiar with MGFs. So I can substitute the MGF for $W_t$ for the expression inside the conditional expectation, but I am not sure how to tease the rest out. $\endgroup$ – user75514 Apr 20 '17 at 20:57

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