Source: Adv. in Appl. Probab. Volume 37, Number 3
(2005), 819-835.
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.
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription.
Read more about accessing full-text
References
Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.
Cai, J. (2004). Ruin probabilities and penalty functions with stochastic rates of interest. Stoch. Process. Appl. 112, 53--78.
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.
Crandall, M. G. and Lions, P. L. (1983). Viscosity solutions of Hamilton--Jacobi equations. Trans. Amer. Math. Soc. 277, 1--42.
Mathematical Reviews (MathSciNet):
MR690039
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.
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.
Fleming, W. H. and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York.
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.
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.
Hipp, C. and Plum, M. (2003). Optimal investment for investors with state dependent income, and for insurers. Finance Stoch. 7, 299--321.
Paulsen, J. and Gjessing, H. K. (1997). Ruin theory with stochastic economic environment. Adv. Appl. Prob. 29, 965--985.
Seaborn, J. B. (1991). Hypergeometric Functions and Their Applications. Springer, New York.
Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press.
Mathematical Reviews (MathSciNet):
MR107026
Sundt, B. and Teugels, J. L. (1995). Ruin estimates under interest force. Insurance Math. Econom. 16, 7--22.
Wang, G. (2001). A decomposition of the ruin probability for the risk process perturbed by diffusion. Insurance Math. Econom. 28, 49--59.
Wang, G. and Wu, R. (2001). Distributions for the risk process with a stochastic return on investments. Stoch. Process. Appl. 95, 329--341.
Yang, H. L. and Zhang, L. (2001). Spectrally negative Lévy processes with applications in risk theory. Adv. Appl. Prob. 33, 281--291.
