Normal deviation of Gamma Processes in Semi-Markov Random Media

Gamma processes are a special case of subordinators (increasing Lévy processes). Their increasing nature makes them particularly well-suited for modeling risk insurance problems of cumulative claims (see, for example, Risk Theory with the Gamma Process (177–192)), monotonic degradation (see, for example, Survival in Dynamic Environments (86–103)), and so on. In the latter reference, the authors use the gamma process to model cumulative claims in risk processes. They also tabulated the probabilities of ruin for the standardized gamma process.

This presentation follows on from article Diffusion Approximation of Gamma Processes in Semi-Markov Random Media. We consider gamma processes in random media (environment) represented by semi-Markovian processes. We approximate these gamma processes using diffusion processes. We first obtain an average result, which will serve as an equilibrium, and then the diffusion approximation. This is a diffusion approximation with equilibrium or normal deviation of the gamma process in semi-Markovian random media. The tools used here differ from those employed in the case of Markov (see a previous article by the first author [Normal Deviation of Gamma Processes in Random Environment With Applications]) because we do not have semigroups. Applications are also discussed.