Let $V_t$ be a solution of the SDE
$$dV_t=V_t(rdt+\sigma_t dW_t) $$
where $\sigma_t$ satisfies some other SDE $$d\sigma_t=\alpha(t,\sigma_t)dt+\beta(t,\sigma_t)dW^{\ \prime}_t $$ and $W_t$ and $W^{\ \prime}_t $ are possibly correlated Brownian motions.
I'm interested in the Laplace transform $\mathbb{E}[\exp({aT}\ )]$ of the stopping time $$T:=\inf\{t>0 \ : \ V_t\le \overline V\ \}. $$
If $\sigma$ is a constant, both $V_t$ and $\mathbb{E}[\exp({aT}\ )]$ are explicitly computable and everything is known.
Question: What is known about $\mathbb{E}[\exp({aT}\ )]$ in the general case? Are there any specifications of the coefficients $\alpha$ and $\beta$ that allow an explicit formula?
Thank you!
