# Optimal stopping for random walks

Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $Z_k= (N+1-k)S_k^2$. The goal is to get $\tau^*=\operatorname{argsup}_{\tau} E[Z_{\tau}]$, where $\tau$ is any stopping time. How to get the explicit form of $\tau^*$ or the Snell envelope of $Z_k$? Thanks for any hints, discussion and help.

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