Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T15:07:50.524Z Has data issue: false hasContentIssue false
  • English
  • Français

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: jcai@math.uwaterloo.ca
∗∗ Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. Email address: hlyang@hkusua.hku.hk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.Google Scholar
Cai, J. (2004). Ruin probabilities and penalty functions with stochastic rates of interest. Stoch. Process. Appl. 112, 53–78.Google Scholar
Cai, J. and Dickson, D. C. M. (2002). On the expected discounted penalty function at ruin of a surplus process with interest. Insurance Math. Econom. 30, 389–404.Google Scholar
Crandall, M. G. and Lions, P. L. (1983). Viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc. 277, 1–42.Google Scholar
Crandall, M. G., Ishii, H. and Lions, P.-L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27, 1–67.Google Scholar
Dufresne, F. and Gerber, H. U. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math. Econom. 10, 51–59.Google Scholar
Fleming, W. H. and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York.Google Scholar
Gerber, H. U. and Landry, B. (1998). On the discounted penalty at ruin in a Jump-diffusion and the perpetual put option. Insurance Math. Econom. 22, 263–276.Google Scholar
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.Google Scholar
Hipp, C. and Plum, M. (2003). Optimal investment for investors with state dependent income, and for insurers. Finance Stoch. 7, 299–321.CrossRefGoogle Scholar
Paulsen, J. and Gjessing, H. K. (1997). Ruin theory with stochastic economic environment. Adv. Appl. Prob. 29, 965–985.Google Scholar
Seaborn, J. B. (1991). Hypergeometric Functions and Their Applications. Springer, New York.Google Scholar
Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press.Google Scholar
Sundt, B. and Teugels, J. L. (1995). Ruin estimates under interest force. Insurance Math. Econom. 16, 7–22.Google Scholar
Wang, G. (2001). A decomposition of the ruin probability for the risk process perturbed by diffusion. Insurance Math. Econom. 28, 49–59.Google Scholar
Wang, G. and Wu, R. (2001). Distributions for the risk process with a stochastic return on investments. Stoch. Process. Appl. 95, 329–341.Google Scholar
Yang, H. L. and Zhang, L. (2001). Spectrally negative Lévy processes with applications in risk theory. Adv. Appl. Prob. 33, 281–291.CrossRefGoogle Scholar