Consider the SDEs \begin{align} dX_t&=\alpha X_tdt+\sqrt{v_t}X_tdB_t \\ dv_t&=\eta(\theta-v_t)dt+\xi \sqrt{v_t}dW_t \end{align} where $\alpha,\eta,\theta,\xi$ are constants and $\rho$ is the correlation between the standard Brownian motions $B_t$ and $W_t$.
For $a>0$, let $\tau_a=\inf\limits_{t\geq0}\{X_t=a\}$ be the first hitting/passage time.
Question: Is the Laplace transform of the density function of $\tau_a$ known analytically: $$\mathbb{E}[e^{-s\tau_a}]=?$$
I guess it may be known `semi-analytically,' involving the characteristic function of $\ln(X_t)$?
For the special case $\eta=\xi=0$, the process $X_t$ is a geometric Brownian motion and we know that \begin{align*} \mathbb{E}[e^{-s\tau_a}]=\begin{cases} \left(\frac{X_0}{a}\right)^{\beta_1} &\text{if } X_0\leq a \\ \left(\frac{X_0}{a}\right)^{\beta_2} &\text{if } X_0\geq a \end{cases} \end{align*} where $\beta_{1,2}$ are roots to a quadratic polynomial derived from the infinitesimal generator of the geometric Brownian motion.
