Characterization of Moment Generating Function of Hitting Time

Question

Suppose that $x_t$ follows the stochastic differential equation: \begin{align*} dx_t = (a - b x_t) dt + \sigma x_t dB_t \end{align*} Where $B_t$ is a standard one-dimensional Brownian motion, and $a, b>0$ are constant. Let $\bar x > 0$. Let $\tau := \inf \{t: x_t = \bar x \}$ be the first time $x_t$ ``hits'' the barrier $\bar x$. For $x < \bar x$, I want to characterize the moment generating function $f_{\theta} (x) := \mathbb{E}^x \left[ e^{- \theta \tau} \right]$ using an ordinary differential equation with two boundary conditions (in other word, a standard boundary value problem). I would like to argue that the following Kolmogorov-backward equation, in addition to the following two boundary conditions, are sufficient to fully characterize the moment generating function of interest. \begin{align*} 0 = - \theta f_\theta (x) + (a - bx)f_\theta' (x) + \frac{1}{2} x^2 \sigma^2 f_\theta'' (x) \end{align*} The boundary conditions are as follows: \begin{align*} f_\theta(\bar x) &= 1 \\ \lim_{x \rightarrow - \infty} f_\theta(x) &= 0 \end{align*} The ordinary differential equation is elliptic, but presents a singularity at x = 0, meaning that I cannot use standard existence/uniqueness results of boundary value problems to establish that the BVP above fully characterize the function $f_\theta$. Can anyone (a) opine whether the BVP I have truly characterizes $f_\theta$, and (b) recommend papers I should look at re: existence/uniqueness of solutions to this type of BVP with singularity at a finite number of points? Thanks.