Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by : 
$$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$

and for $t\geq 0$, we define the following stopping time:
$$\sigma_t := \inf \{s\geq 0: \ A_s > t\}$$

Notice that a.s: 
$$\forall \ t\geq 0: \quad \sigma_t \leq t$$

Now, define the martingale $M$ as a time changed Brownian motion, i.e. $M := \left(W_{\sigma_t}\right)_{t\geq 0}$. Can we prove that $M$ goes to infinity in probability ? Thanks.