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

$$E[M^{2}_{t}]=\int^t 1+Ee^{W_{s}}ds=\int^t 1+e^{\frac{s}{2}}ds \to +\infty,$$

which also shows that $<M>_{\infty}=\infty$. In general for continuous martingales we have that

"$X_\infty=\lim_{t\rightarrow\infty}X_t$ exists and is finite" iff $<X>_{\infty}<\infty.$

The left-right direction that is relevant here is as follows: we use the stopping time $\tau_{n}:=\inf\{t\geq 0: |X_{t}|\geq n\}$ and the bound $$E[<X>_{t\wedge \tau_{n}}]=E[X^{2}_{t\wedge \tau_{n}}]\leq n^{2}$$

to get that the event $\{\sup|X_t|<\infty\}=\bigcup_{n}\{\sup|X_t|<n\}$ implies that the finiteness $<X>_{\infty}<\infty$. (For the other direction we can use instead $\tau_{n}:=\inf\{t\geq 0: <X>_{t}\geq n\}$.)

So we can't have the first case in the above theorem, because it would give a contradiction and so we have the second case.

For the "in probability" part, we will use a reverse tail inequality: when $\mathbb {E} [X]=0,\,\mathbb {E} [X^{2}]=1$ then

$$\Pr(X\geq 0)\geq {\frac {2{\sqrt {3}}-3}{\mathbb {E} [X^{4}]}},$$

from Berger's "The Fourth Moment Method. By symmetry of Brownian motion we write

$$P[M_{t}\leq 0]=P[0\leq \tilde{M}_{t}:=\int_{0}^{t}\sqrt{1+e^{-W_{s}}}dW_{s}].$$

We still have $E[\tilde{M}_{t}]=0$ and we will need the fourth moment bound

$$E\left(\int f dW\right)^{4}\leq c_{2}\left(\int Ef^{2} ds\right)^{2},$$

from Corollary 4 in "Notes on the Itô Calculus" by Steven P. Lalley. Therefore, for $X_{t}:=\tilde{M}_{t}\frac{1}{\sqrt{E[\tilde{M}_{t}^{2}]}}$ and using the above bound we get

$$P[\tilde{M}_{t}\geq 0]\geq {\frac {2{\sqrt {3}}-3}{\mathbb {E} [X^{4}]}}\geq \frac {2{\sqrt {3}}-3}{c_{2}}>0$$

for all $t>0$.

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