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!