In a number of different contexts, I have wanted to estimate hitting times for a monotonic process $(T_n)$ taking values in the reals (or sometimes a process $(T_n,X_n)$ taking values in $\mathbb R^2$ where the first component is monotonic). I'm assuming the step size is small, and am interested in the first time, $\tau_1$, that $T_n$ exceeds some threshold $a$ (and in the 2d case, possibly the distribution of $X_n$ at this hitting time).

For some concreteness, assume $a=1$, $T_0=0$ and $(T_n)$ is Markov, where the jump size, $T_{n+1}-T_n$ is much smaller than 1. If necessary, we can assume that the jump sizes are i.i.d., although I'd ultimately prefer to have something more flexible where the distribution of $T_{n+1}-T_n$ is in some sense continuously dependent on $T_n$. I would like to obtain reasonably precise information on the distribution of $\tau_1$.

Is there an established machinery that can address questions of this type?

Here is a specific simple (made up) instance that I would be very interested to see a clean answer to (especially, as indicated above, an answer that might be generalizable away from the i.i.d. jump case): Let $T_{n+1}-T_n\sim \exp(\text{Unif}[-2k,-k])$ and $T_0=0$. What can be said about the distribution of $\tau_1$? How does the distribution of $\tau_1$ depend on $k$?

Markov renewal theory; Çinlar wrote a survey of this subject long ago. $\endgroup$ – Mateusz Kwaśnicki Nov 18 at 22:58