 Solution

The increments have expected value $p-q<0$. The strong law of large numbers for independent and identically distributed random variables with finite absolute mean givesHence $S_n\to -\infty$. Since $q/p>1$,Its expected value nevertheless stays equal to one. Thus this martingale is not uniformly integrable, and it does not converge to zero in $L^1$. This distinction will be avoided in part (ii) by stopping at an upper level, which makes the stopped process bounded.