 Solution

DefineThe event $\{n\le T_a\}$ means that level $a$ has not been reached by time $n-1$, so it belongs to ${\mathcal{F}}_{n-1}$. Hence $H$ is a bounded predictable process. A direct check across the hitting time givesand thereforePart (c) shows that $\hat{X}$ is a martingale. Its increments still take values in $\{-1,1\}$, so part (b) shows that they are IID symmetric signs. Consequently $\hat{X}$ is a simple symmetric random walk. This is the path transformation behind the reflection principle for simple symmetric random walk.