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Ruin in the perturbed compound Poisson risk process under interest force

Part of: Markov processes Mathematical economics

Published online by Cambridge University Press:  01 July 2016

Jun Cai  and
Hailiang Yang
Jun Cai*
Affiliation:
University of Waterloo
Hailiang Yang*
Affiliation:
The University of Hong Kong
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Email address: [email protected]
∗∗ Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. Email address: [email protected]
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Abstract

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In this paper, we study ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, we show that the ruin probability in the perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. We decompose the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived when the interest force is constant. When the claim sizes are exponentially distributed, explicit solutions of the ruin probabilities are derived from the integrodifferential equations. Numerical examples are given to illustrate the effects of diffusion volatility and interest force on the ruin probabilities.

Keywords

Compound Poisson risk processdiffusionBrownian motionjump diffusion processruin probabilityHamilton-Jacobi-Bellman equationviscosity solutionconfluent hypergeometric functionKummer's confluent hypergeometric equation

MSC classification

Primary: 60J75: Jump processes
Secondary: 60J65: Brownian motion 91B30: Risk theory, insurance
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 819 - 835
Copyright
Copyright © Applied Probability Trust 2005 

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