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Zbl 0684.62073
Asmussen, Søren
Risk theory in a Markovian environment.
(English)
[J] Scand. Actuarial J. 1989, No.2, 69-100 (1989). ISSN 0346-1238

Summary: We consider risk processes $\{R\sb t\}\sb{t\ge 0}$ 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 $\{Z\sb t\}\sb{t\ge 0}$ such that $\beta =\beta\sb i$ and $B=B\sb i$ when $Z\sb t=i.$ \par 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 2000:
*62P05 Appl. of statistics to actuarial sciences and financial mathematics
60J99 Markov processes
65C99 Numerical simulation
60J70 Appl. of diffusion theory

Keywords: risk processes; Poisson arrival process; Markov jump process; approximations; simulation; ruin probabilities; Cramér-Lundberg approximation; diffusion approximations; correction terms; Markov- modulated random walks; Wiener-Hopf factorisation problems; conjugate distributions; Esscher transforms

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