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Phase-type distributions and risk processes with state-dependent premiums. (English) Zbl 0876.62089
Summary: Consider a risk reserve process with initial reserve $$u$$, Poisson arrivals, premium rule $$p(r)$$ depending on the current reserve $$r$$ and claim size distribution which is phase-type in the sense of M. F. Neuts [see “Matrix-geometric solutions in stochastic models. An algorithmic approach” (1981; Zbl 0469.60002)]. It is shown that the ruin probabilities $$\psi(u)$$ can be expressed as the solution of a finite set of differential equations, and similar results are obtained for the case where the process evolves in a Markovian environment (e.g., a numerical example of a stochastic interest rate is presented). Further, an explicit formula for $$\psi(u)$$ is presented for the case where $$p(r)$$ is a two-step function. By duality, the results apply also to the stationary distribution of storage processes with the same input and release rate $$p(r)$$ at content $$r$$.

##### MSC:
 62P05 Applications of statistics to actuarial sciences and financial mathematics 91B30 Risk theory, insurance (MSC2010) 62M99 Inference from stochastic processes
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