Risk processes analyzed as fluid queues. (English) Zbl 1092.91037

This paper deals with the risk model based on the Markovian arrival process which makes it fairly general. On the one hand, this model includes both the classical and most Sparre Andersen models, so long as claim amounts are modelled as phase-type random variables. On the other hand, Markovian arrival processes allow for correlated arrival processes, and can be used to model situations where environmental factors change radically. The proposed approach is based on the explicit recognition of the similarity between the evolution of the risk process \(R(t)\) and that of a fluid queue defined as follows. Take a Markov process \(\{\phi(x)\}_{x\geq0}\) on a finite state space \(S\) and associate a real number \(r_{i}\) to each state \(i\in S\): \(r_{i}\) is the rate at which some fluid increases or decreases when the Markov process is in the state \(i\). Define the piecewise linear function \(F(x)=u+\int_{0}^{x}r_{\phi(v)}\,dv\), where \(u\) is the fluid level at the time zero. The fluid queue is defined as \(\Phi(x)=F(x)-\min(0,\widetilde F(x))\), where \(\widetilde F(x)=\min_{0\leq v\leq x}F(v)\). The Laplace-Stieltjes transform of the time of ruin is derived.


91B30 Risk theory, insurance (MSC2010)
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