Hitting time of a diffusion process

Question

Consider a diffusion process (SDE)

$ dX(t) = \mu (X(t)) dt + \sigma(X(t)) dW(t), X(0) = x. $

I am interested in a first hitting time of this process to a point $y < x$, more particularly I wonder if there are any explicit formulas for the expectation of the hitting time $\mathbb{E}[\tau]$, where $\tau = \inf \{t\geq 0: X(t) \leq y\}.$ I would assume that the answer should use scale function or speed measure, but I cannot write the expectation explicitly and surpisingly I almost cannot find anything related in classical books. There are some related expressions in Karatzas and Shreeve BMSC (see Chapter 5.5, in particular 5.5.C), but they don't provide a satisfying answer when $X(t)$ is defined on the whole real line.

I will be grateful for the answer or any references where I can find related questions.

Answer 1

As they mention, in Shreve-Karatzas 5.5c, they work on the general setting that includes the real line since they allow $\ell=-\infty$ and $r=+\infty$. So the answer is exactly the one you mentioned at 5.59 in Shreve-Karatzas 5.5c. We simply take $a=y$ $b=+\infty$, then

$$E_{x}[T_{y,\infty}]=M_{y,+\infty}(x),$$

for

scale function $p$, and speed measure $m$ which are in terms of $\mu,\sigma$.