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Risk theory in a Markovian environment. (English) Zbl 0684.62073

Summary: We consider risk processes ${\left\{{R}_{t}\right\}}_{t\ge 0}$ with the property that the rate $\beta$ of the Poisson arrival process and the distribution of B of the claim sizes are not fixed in time but depend on the state of an underlying Markov jump process ${\left\{{Z}_{t}\right\}}_{t\ge 0}$ such that $\beta ={\beta }_{i}$ and $B={B}_{i}$ when ${Z}_{t}=i·$

A variety of methods, including approximations, simulation and numerical methods, for assessing the values of the ruin probabilities are studied and in particular we look at the Cramér-Lundberg approximation and diffusion approximations with correction terms. The mathematical framework is Markov-modulated random walks in discrete and continuous time, and in particular Wiener-Hopf factorisation problems and conjugate distributions (Esscher transforms) are involved.

MSC:
 62P05 Applications of statistics to actuarial sciences and financial mathematics 60J99 Markov processes 65C99 Probabilistic methods, simulation and stochastic differential equations (numerical analysis) 60J70 Applications of Brownian motions and diffusion theory