The Ruin Problem for Finite Markov Chains

Abstract

We derive an asymptotic approximation of the joint distribution $\operatorname{prob}(N(u) - n \in A, S_{N(u)} - u \in B)$ as $n$ and $u \rightarrow \infty$. Here $N(u) = \min\{n; S_n > u\}$ denotes the first passage time for a random walk of the form $S_n = \sum^n_{k = 1}U_k(\xi_{k - 1},\xi_k)$, where $\xi_0,\xi_1,\ldots$ is a finite Markov chain and where $\{U_k(i,j)\}^\infty_{k = 1}$ is a sequence of independent random variables. The approximation holds for all sets $B$ and a fairly large class of sets $A$.