Here we study
$$M_{t}=B_{A_{t}}\stackrel{d}{=}\int_{0}^{t}\sqrt{1+e^{W_{s}}}dW_{s}.$$
First, as mentioned Martingale Convergence here
Theorem 4 Let X be a continuous martingale. Then, almost surely, one of the following is satisfied
- $X_\infty=\lim_{t\rightarrow\infty}X_t$ exists and is finite.
- $\limsup_{t\rightarrow\infty}X_t=\infty$ and $\liminf_{t\rightarrow\infty}X_t=-\infty$. In this case, the process hits every value in $\mathbb R$ at arbitrarily large times.
By Itô isometry we have
$$E[M^{2}_{t}]=\int^t 1+Ee^{W_{s}}ds=\int^t 1+e^{\frac{s}{2}}ds \to +\infty,$$
and so we have the second case.