Abstract: A Markovian risk process is considered in this paper, which is the generalization of the classical risk model.It is proper that a risk process with large claims is modelled as the Markovian risk model.In such a model, the occurrence of claims is described by a point process {N(t)}_t>0 with N(t) being the number of jumps during the interval (0,t] for a Markov jump process. The ruin probability Ψ(u) of a company facing such a risk model is mainly studied. An integral equation satisfied by the ruin probability function Ψ(u) is obtained and the bounds for the convergence rate of the ruin probability Ψ(u) are given by using a generalized renewal technique developed in the paper.

Key words: risk process, ruin probability, Markov jump process