# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.