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Probability of ruin with variable premium rate in a Markovian environment. (English) Zbl 0999.91048
A risk reserve model has been discussed in which the claim number process \(\{N(t): t\geq 0\}\) is a Cox process with an intensity process \(\{\lambda(t): t>0\}\) modeled as a homogeneous \(n\)-state Markov process. The successive claims \(X_1, X_2,\dots\) are assumed to be i.i.d. and independent of the claim number process. The premiums are received at a differentiable rate \(c(r)\) depending on the current reserve \(R(t)=r\), where \(R(t)\) is the risk reserve at time \(t\), i.e. \[ R(t)= R(0)+ \int_0^t c(R(s)) ds- \sum_{i=1}^{N(t)} x_i, \quad t\geq 0. \] The author’s main result provides an integral equation for the conditional probability of ruin given \(\lambda(0)= \lambda_i\) and \(R(0)=u\), from which the total probability of ruin \(\Psi(u)\) is immediate.
For a special premium plan, taking a fixed interest investment of the reserve into account, the Laplace transforms of the corresponding ruin probabilities can be determined via a system of differential equations. More explicit forms of the latter are given in case of exponential claims and a two-state intensity process.

MSC:
91B30 Risk theory, insurance (MSC2010)
60J27 Continuous-time Markov processes on discrete state spaces
60K15 Markov renewal processes, semi-Markov processes
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