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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.

91B30 Risk theory, insurance (MSC2010)
60J27 Continuous-time Markov processes on discrete state spaces
60K15 Markov renewal processes, semi-Markov processes
Full Text: DOI
[1] Asmussen, S., 2000. Ruin Probabilities. World Scientific, Singapore. · Zbl 0960.60003
[2] Csőrgő, M., Révész, P., 1981. Strong Approximations in Probability and Statistics. Academic Press/Akadémiai Kiadó, Budapest. · Zbl 0539.60029
[3] Gerber, H.U., The surplus process as a fair game — utilitywise, ASTIN bulletin, 8, 307-322, (1975)
[4] Grandell, J., 1991. Aspects of Risk Theory. Springer, Berlin. · Zbl 0717.62100
[5] Michaud, F., Estimating the probability of ruin for variable premiums by simulation, ASTIN bulletin, 26, 1, 93-105, (1996)
[6] Petersen, S.S., 1990. Calculation of ruin probabilities when the premium depends on the current reserve. Scandinavian Actuarial Journal, 147-159. · Zbl 0711.62097
[7] Reinhard, J.M., On a class of semi-Markov risk models obtained as classical risk models in Markovian environment, ASTIN bulletin, XIV, 23-43, (1984)
[8] Schmidli, H., 1994. Risk theory in an economic environment and Markov processes. Bulletin of the Swiss Association of Actuaries, 51-70. · Zbl 0816.90042
[9] Sundt, B.; Teugels, J.L., Ruin estimates under interest force, Insurance: mathematics and economics, 16, 7-22, (1995) · Zbl 0838.62098
[10] Sundt, B.; Teugels, J.L., The adjustment function in ruin estimates under interest force, Insurance: mathematics and economics, 19, 85-94, (1997) · Zbl 0910.62107
[11] Taylor, G.C., 1980. Probability of ruin with variable premium rate. Scandinavian Actuarial Journal, 57-76. · Zbl 0426.62069
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