I have a question about an additive functional of Brownian motion.
Let $d \in \mathbb{N}$. Let $b:\mathbb{R}^{d}\to \mathbb{R}$ be a measurable function and $(X_{t})_{t \in [0,\infty[}$ be a $d$-dimensional Brownian motion.
I want to get an upper estimate of the following integral \begin{align*} E \left[ \exp \left(\int_{0}^{t}b(X_{s})^{2}ds \right) \right]. \end{align*}
For example if $b$ is bounded, \begin{align*} E \left[ \exp \left(\int_{0}^{t}b(X_{s})^{2}ds \right) \right] &\leq \exp\left( M^{2}t\right) \\ & \left(\to 0 \quad \mbox{as} \quad t \to0 \right) \end{align*} where $M=\mbox{esssup}|b|$.
My question
For an unbounded or integrable $b$ , can we get an upper estimate or a small time asymptotic of $E \left[ \exp \left(\int_{0}^{t}b(X_{s})^{2}ds \right) \right]$ ?
Thank you for your consideration.