I am confronted with the same problem in this thread, which hasn't had a complete answer yet.

In the sequel, we denote by $Y^T$ the stopped process $Y^T_t = Y_{T \wedge t}$. Consider the following process: $$ X_t = \begin{cases} W_{t/(1-t)}^T &\text{for } 0 \le t < 1,\\ -1 &\text{for } 1 \le t < \infty. \end{cases} $$ where $W$ is a Brownian motion and $T = \inf\{t: W_t = -1\}$. It's easy to tell that $X$ is not a martingale. Now on wikipedia, it claims that the sequence of stopping times $\{\tau_k\}$ localizes $X$, where $\tau_k = \inf\{t: X_t = k\} \wedge k$.

I am confused about how to prove such claim. The "details" on the webpage seem a bit obscure to me.

On the other hand I found in the book "Stochastic calculus and applications" (page 133, Example 5.6.9) a similar example. The authors consider the process $X_t+1$ (where $X_t$ is defined as in our problem) and for uniformity in symbols I slightly change their proof. Unlike the example on wikipedia, they explicitly specify the filtration $\{\tilde{\mathscr{F}}_t = \mathscr{F}_{t/(1-t)}\}$ to which, as they assert, $X_t$ is local martingale. They construct the following stopping times: $$ S_n = \frac{n}{n+1}I(T \geq n) + \Big(\frac{T}{T+1}+n\Big)I(T<n) $$ Then, they say that the following equation can be established: $$ X^{S_n}_t = W^{T \wedge n}_{t/(1-t)}, $$ which entails that $X^{S_n}$ is a $\{\tilde{\mathscr{F}_t}\}$-martingale.

To me the second approach isn't clear either. Indeed, I can't see how in their proof $\tilde{\mathscr{F}_t}$ can be defined for $t \geq 1$. Also, $X^{S_n}_t = W^{T \wedge n}_{t/(1-t)}$ seems to hold only for $t<1$. After doing some algebra I get $$ X^{S_n}_t = W^{T \wedge n}_{t/(1-t)} I(t<1) + W_{T \wedge n} I(t \geq 1), $$ instead. The rightmost term in the above equation, namely $W_{T \wedge n} I(t \geq 1)$, frustrates me in attempt to show martingale property of $X^{S_n}$.

I also considered an alternative approach: to show that for any bounded stopping time $S$, $E[X^{S_n}_S] = 0$. Still I failed to complete the proof.

Can anyone help me with this problem?