Sample path large deviations for order statistics

In reality insurance claims may be delayed for several reasons and risk models with this feature have been discussed for some years. In this paper we present a sample path large deviation principle for the delayed claims risk model presented in (Yuen K.C., Guo J., and Ng K.W., 2005, On ultimate ruin in a delayed-claims model, Journal of Applied Probability, 42, 163–174). Roughly speaking each main claim induces another type of claim called by-claim; any by-claim occurs later than its main claim and the time of delay is random. Successively we use Gärtner Ellis Theorem to prove a large deviation principle for a more general version of this model, in which the following items depend on the evolution of an irreducible (continuous time) Markov chain with finite state space: the intensity of the Poisson claim number process, the distribution of the claim sizes and the distribution of the random times of delay. Finally we present the Lundberg's estimate for the ruin probabilities; in the fashion of large deviations this estimate provides the exponential decay of the ruin probability as the initial capital goes to infinity.