Lundberg inequalities for a Cox model with a piecewise constant intensity

A Cox risk process with a piecewise constant intensity is considered where the sequence (L_i) of successive levels of the intensity forms a Markov chain. The duration s, of the level L_i is assumed to be only dependent via L_i. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of L_i contains a discrete component, a Cramér-Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).