Consider a simple random walk; that is, a sequence $S_n=\sum _1 ^n X_i$ where $X_i=1$ wp $\frac{1}{2}$ and $X_i=-1$ wp $\frac{1}{2}$. Let $\tau$ be the stopping time of the walk; that is, for $a,b>0$, we let $\tau = \{\min n | S_{n}=a \text{ or } S_{n}=-b\}$.
I want to calculate $E[\tau S_\tau]$. Now, it can be shown that $Z_n = S_n^3-3nS_n$ is a martingale without too much work. Thus, if we verify the conditions of the Optional Stopping Theorem, we would have $E[\tau S_\tau] = E[S_\tau ^3]$. Then, it is easy to calculate $E[S_\tau ^3]$, since it is easy to show $P(S_\tau =a)=1-P(S_\tau=-b)=\frac{b}{a+b}$.
But I cannot show that the conditions of the optional stopping theorem hold! From that link, it seems that conditions (a) and (c) do not hold, but I cannot show that (b) holds-I know that $E[\tau]=ab<\infty$ but cannot show that $E[|Z_n-Z_{n-1}|]$ is bounded!