## 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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### References:

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