The time to ruin for a class of Markov additive risk process with two-sided jumps

Abstract

We consider risk processes that locally behave like Brownian motion with some drift and variance, these both depending on an underlying Markov chain that is also used to generate the claims arrival process. Thus, claims arrive according to a renewal process with waiting times of phase type. Positive claims (downward jumps) are always possible but negative claims (upward jumps) are also allowed. The claims are assumed to form an independent, identically distributed sequence, independent of everything else. As main results, the joint Laplace transform of the time to ruin and the undershoot at ruin, as well as the probability of ruin, are explicitly determined under the assumption that the Laplace transform of the positive claims is a rational function. Both the joint Laplace transform and the ruin probability are decomposed according to the type of ruin: ruin by jump or ruin by continuity. The methods used involve finding certain martingales by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. We also use certain results from complex function theory as important tools.

Exit times for a class of piecewise exponential Markov processes with two-sided jumps

“…Partial eigenfunctions were used systematically in the earlier papers of Jacobsen [5,6], but had been used earlier also by other authors, e.g. Paulsen and Gjessing [15], Theorem 2.1.…”

“…Paulsen and Gjessing [15], Theorem 2.1. The processes studied in [6] were certain types of Markov modulated Lévy processes where essentially the partial eigenfunctions may be expressed as linear combinations of exponential functions. For the OU processes treated here the building blocks for the partial eigenfunctions are more involved, using functions that can be represented as integrals.…”

“…For the OU processes treated here the building blocks for the partial eigenfunctions are more involved, using functions that can be represented as integrals. But for our results it is vital to assume that the jumps arrive according to a Poisson process whereas in [5,6] renewal arrivals with waiting times of phase type were considered.…”

Two-Sided First Exit Problem for Jump Diffusion Distribution Processes Having Jumps with a Mixture of Erlang

“…Consider the following jump diffusion process Recently, one-sided and two-sided first exit problems for processes with two-sided jumps have attracted a lot of attentions in applied probability (see [1][2][3][4][5][6][7]). For example, Perry and Stadje [1] studied two-sided first exit time for processes with two-sided exponential jumps; Kou and Wang [2] studied the one-sided first passage times for a jump diffusion process with exponential positive and negative jumps.…”

On a discrete risk model with two-sided jumps

“…In such models, the downward jumps can be interpreted as the claim sizes and the upward jumps can represent the random gains of the company. For example, Jacobsen [10] considered a Markov additive risk model and studied the joint Laplace transform of two key ruin quantities in ruin theory, i.e., the time to ruin and the undershoot at ruin; Chen et al [11] analyzed the expected discounted penalty function in a jump-diffusion risk model with two-sided jumps; Xing et al [12] considered a perturbed compound Poisson risk model with two-sided jumps and obtained the Laplace transforms of the time to ruin and the deficit at ruin; Yang and Zhang [13] studied the expected discounted penalty function in a compound Poisson risk model under the assumption that the upward jumps are Coxian distributed.…”