# expected value of a function [duplicate]

Possible Duplicate:
sub martingales and more

Can someone help me compute the expected value of $X_{n+1}$, that is : $E[X_{n+1}| X_0,\dots,X_n]$?

Given : $X_n = X_0 e^ {\mu S_n}$, $X_0 > 0$

where $S_n$ is a symmetric random walk and $\mu$ is greater than zero.

I am aware that the expected value of a given function is the mean. But i would like to know a method to compute the above. What is the right approach to get started on such problems on expected value computation.

Update:

I understand that $X_{n+1} = X_n \cdot e^{\mu (S_{n+1} - S_n)}$ ? How do I proceed with computing the expectation?

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## marked as duplicate by Ilya, Guess who it is., Asaf Karagila, Nate Eldredge, Zev ChonolesFeb 8 '12 at 14:11

Hint: $X_n = e^{\mu Y_n} X_{n-1}$ where $Y_n = S_n - S_{n-1}$ is the $n$'th step of the random walk.
Since $Y_n$ and $X_{n-1}$ are independent, $E[X_n] = E[e^{\mu Y_n}] E[X_{n-1}]$. If this is a simple random walk ($Y_n = \pm 1$, each with probability $1/2$), $E[e^{\mu Y_n}] = (e^\mu + e^{-\mu})/2 = \cosh(\mu)$. –  Robert Israel Feb 7 '12 at 16:59