Suppose we have a simple random walk $S_n$ starting at 20 – i.e. $S_n = X_0+…+X_n$ where $P(X_i = 1)=P(X_i=-1) =1/2$ for all $i$, and $S_0 = 20$. Then $S_n$ is a martingale. Let $\tau$ be the first time that $S_n$ reaches 0 or 100. Then $\tau$ is a stopping time and Wald's equation gives
$$E(S_{\tau})=E(\tau)E(X_0)$$
but we know that $E(S_{\tau}) = E(S_0)=20$, we know that the expectation of the stopping time is finite from other methods, and we know $E(X_0)=0$! These observations aren't consistent with Wald's equation – could anyone point to the mistake here?
Best Answer
If you are starting at $S_0 = 20$, then you cannot say that $S_n = X_0 + \dots + X_n$ where $X_i$ is randomly chosen from $\{-1,1\}$. You could, instead, say that $S_n = 20 + X_1 + \dots + X_n$.
Then Wald's equation gives us $\mathbb E[S_\tau - 20] = \mathbb E[\tau] \mathbb E[X_1]$. Here, it is perfectly reasonable to use $\mathbb E[X_1] = 0$ to conclude $\mathbb E[S_\tau - 20] = 0$: this implies that $S_{\tau} - 20$ is $-20$ with probability $\frac45$ and $80$ with probability $\frac15$.