Let ${X_i}$ be i.i.d. random variables with $P(X_i = 1) = p$ and $P(X_i = −1)= 1−p$ and $0< p <1$. Let $S_n$ be the gambling player's fortune after the $n^{th}$ game, starting from a capital $a$, $$S_n= a + X_1 + X_2 + ... X_n.$$
Suppose that $0 < a < c$, and let $\tau_0 = inf\{n \ge 0:X_n=0\}$ and $\tau_c = inf\{n \ge 0: X_n = c\}$ be the first hitting time of $0$ and $c$, respectively. Let $\psi(c, a)$ be the probability to be ruined before reaching capital $c$, starting with a capital $a$.
I'm interested in the probability $p_r(t)$ of being ruined before time $t$, without ever having a capital $S \ge c$.
I guess this can be modelled as an absorbing Markov Chain where the $S = 0$ and $S = a$ are two absorbing states. And I'm looking for the probability to be in the first/second absorbing state over time.
I think that it's clear that $p_r(t) \rightarrow \psi(c, a)$ as $t$ goes to infinity. The result in this answer (Gambler's ruin model) gives a lower bound on the probability of being in one of the two absorbing state ($P(min\{ \tau_c, \tau_0 \} < n)$). And show that exept with an exponentially small probability in $n$ the capital has hit $0$ or $c$ by time $n$. And so that$p_r(t)$ converges toward $\psi(c, a)$ exponentially fast.
This more or less answer my question but I would like to know if it is possible to derive more explicit expression for the probability to be ruined or to reach a capital $c$ for my specific problem.
More generally, I'd like to know if it's possible to derive such convergence results for absorbing Markov Chains (under some general hypothesis). I'm not looking for extremely general theorems but more standards results that I haven't been able to find.
It was mentionned in the answer to this question (Gambler's ruin model) that it is possible to compute the two-sided hitting time $min\{ \tau_c, \tau_0 \}$ exactly. Unfortunately there's no reference given and I haven't been able to find a good reference deriving this result.
I've also looked at William Feller's book An Introduction to Probability Theory and Its Applications (around p. 350). There's an exact expression for the probability to be ruined at the nth trial. However the expression is quite involved. In principle it should be possible to sum it over all possible n (it's a geometric sum), but I was wondering if there was some more general results that would allow me to conclude directly.
Maths are not my area of research so it's entirely possible that there are some things not clear or not well defined in my question.