Time-changed Brownian Motion

Let $B_t$ be a standard Brownian motion and let $\tau_{-1}= \inf \{ t \geq 0: B_t(\omega) = -1\}$.

By the Continuous Time Stopping Theorem we know that \begin{align} Z_t = B_{t \wedge \tau_{-1}} \end{align} is a martingale. Now, we define a time-change $X_t(\omega) = Z_{\frac{t}{1-t}}(\omega)$ for $0 \leq t <1$, where $X_t$ is a martingale as well, with almost sure convergence \begin{align} P\big(\lim_{t \to 1} X_t = -1\big)=1. \end{align} We extend the defintion by $X_t = -1$ for $t \geq 1$.

I want to show that $X_t$ is not a martingale anymore, but a local martingale which can be localized by the localizing sequence $\tau_k(\omega) = \inf \{ t \geq 0: X_t(\omega) = k\}$. Any help would be appreciated.

• You have problems showing X is not a martingale? – Did Jun 19 '16 at 17:57
• Yep, and to show that $X$ is a local martingale. – iJup Jun 19 '16 at 18:14
• Ìf you have no idea to show that $X$ is not a martingale then the rest of the question is rather moot, I would say... Anyway, can you compare $E(X_0)$ and $E(X_1)$? – Did Jun 19 '16 at 18:16