Evaluate integral of a stochastic process

Consider the stochastic process

$$X_t=\exp(-0.5b^2t+bW_t),$$

where $W_t$ is Standard Wiener process defined on a probability space $(\Omega,\mathcal{F}, \mathbb{P})$ and $b\neq0$. I want to evaluate

$$\underset{t\rightarrow \infty}{\lim}\int_{\underline{b}}^{\overline{b}}X_tdb,$$

where $-\infty<\underline{b}<\overline{b}<\infty$. The strong law of large numbers for a Wiener process states $W_t/t \rightarrow 0$ almost surely as $t\rightarrow \infty$ (see e.g. Karatzas-Shreve section 2.9). A direct implication is that $X_t \rightarrow 0$ almost surely. Then if we could interchange the limit and integral, the result would be that the integral approaches 0 almost surely. However, I am not sure if I can do that. Any hints?

• Is $\underline{b}>-\infty$ and $\bar{b}<\infty$? – saz Apr 30 '17 at 17:27
• @saz yes, I'll add that – fesman Apr 30 '17 at 17:32
• Hmh, is $\underline{b}>0$ ....? – saz Apr 30 '17 at 17:49
• @saz Not necessarily – fesman Apr 30 '17 at 17:51
• @saz. One can evaluate the integral directly, but the result is quite messy and it is not obvious how to evaluate the limit after that. Therefore it would be very convenient, if one could take the limit inside the integral somehow. – fesman Apr 30 '17 at 17:58