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

Summary: We consider risk processes {R t } t0 with the property that the rate β 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 {Z t } t0 such that β=β 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:
62P05Applications of statistics to actuarial sciences and financial mathematics
60J99Markov processes
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)
60J70Applications of Brownian motions and diffusion theory