= Solution
Define
$$
H_n=
\begin{cases}
1,&n\leq T_a,\\
-1,&n>T_a.
\end{cases}
$$
The event $\{n\leq T_a\}$ 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 gives
$$
\widehat X_n-\widehat X_{n-1}
=H_n(X_n-X_{n-1}),
$$
and therefore
$$
\widehat X_n=\sum_{k=1}^nH_k(X_k-X_{k-1}).
$$
Part (c) shows that $\widehat X$ is a martingale. Its increments still take values in $\{-1,1\}$, so part (b) shows that they are IID symmetric signs. Consequently $\widehat X$ is a simple symmetric random walk. This is the path transformation behind the <reflection principle for simple symmetric random walk>.
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