  We study the ruin problem over a risk process described by a discrete-time
Markov model. In contrast to previous studies that focused on the asymptotic
behaviour of ruin probabilities for large values of the initial capital, we
provide a new technique to compute the quantity of interest for any initial
value, and with any given precision. Rather than focusing on a particular model
for risk processes, we give a general characterization of the ruin probability
by providing corresponding recursions and fixpoint equations. Since such
equations for the ruin probability are ill-posed in the sense that they do not
allow for unique solutions, we approximate the ruin probability by a
two-barrier ruin probability, for which fixpoint equations are well-posed. We
also show how good the introduced approximation is by providing an explicit
bound on the error and by characterizing the cases when the error converges to
zero. The presented technique and results are supported by two computational
examples over models known in the literature, one of which is extremely
heavy-tailed.
