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For $x\ge m$, $X_n=x$ implies $M_n\ge m$. For $x<m$, reflect every step after the first visit to $m$. The reflection principle for simple symmetric random walk bijects such paths ending at $x$ with unrestricted paths ending at $2m-x$. Therefore

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.