Let $S_n$ be a Simple Random Walk. What is $E[S_m|S_n]$ if $m < n$?
Let $S_n = W_1 + ... + W_n$ be a simple random walk with $W_i$ IID and $P[W_i = 1] = P[W_i = -1] = 1/2$. Find $E[S_m | S_n]$ when (a) $m > n$ and (b) $m < n$.
For part (a), I get the answer of $S_n$. However, I do not know how to do part (b). How to find the past give the future? Thanks.
Solution 1:
Note that $nE[W_1|S_n] = E[S_n | S_n] = S_n$ so that $E[W_1|S_n] = S_n/n$.
Thus if $m<n$ we have $$E[S_m | S_n] = \sum^m_{i=1} E[W_i | S_n] = \sum^m_{i=1} E[W_1| S_n] = \frac mn S_n$$