An Insurance Risk Model with Stochastic Volatility

31 Pages Posted: 15 Dec 2008 Last revised: 16 Feb 2010

Yichun Chi

China Institute for Actuarial Science, Central University of Finance and Economics

Sebastian Jaimungal

University of Toronto - Department of Statistics

X. Sheldon Lin

Department of Statistical Sciences, University of Toronto

Date Written: December 15, 2008

Abstract

In this paper, we extend the Cramer-Lundberg insurance risk model perturbed by diffusion to incorporate stochastic volatility and study the resulting Gerber-Shiu expected discounted penalty (EDP) function. Under the assumption that volatility is driven by an underlying Ornstein-Uhlenbeck (OU) process, we derive the integro-differential equation which the EDP function satisfies. Not surprisingly, no closed-form solution exists; however, assuming the driving OU process is fast mean-reverting, we apply singular perturbation theory to obtain an asymptotic expansion of the solution. Two integro-differential equations for the first two terms in this expansion are obtained and explicitly solved. When the claim size distribution is of phase-type, the asymptotic results simplify even further and we succeed in estimating the error of the approximation. Hyper-exponential and mixed-Erlang distributed claims are considered in some detail.

Keywords: Gerber-Shiu expected discounted penalty function, Integro-differential equation, Singular perturbation theory, Stochastic volatility, Perturbed compound Poisson risk process, Phase-type distribution, Ornstein-Uhlenbeck process

Suggested Citation

Chi, Yichun and Jaimungal, Sebastian and Lin, X. Sheldon, An Insurance Risk Model with Stochastic Volatility (December 15, 2008). Insurance: Mathematics and Economics, Vol. 46, No. 1, pp. 52-66. Available at SSRN: https://ssrn.com/abstract=1316223

Yichun Chi

China Institute for Actuarial Science, Central University of Finance and Economics ( email )

Beijing, 100081
China

Sebastian Jaimungal (Contact Author)

University of Toronto - Department of Statistics ( email )

100 St. George St.
Toronto, Ontario M5S 3G3
Canada

HOME PAGE: http://www.utstat.utoronto.ca/sjaimung

Xiaodong Sheldon Lin

Department of Statistical Sciences, University of Toronto ( email )

Department of Statistical Sciences
100 St George Street
Toronto, Ontario M5S 3G3
Canada

Paper statistics

Downloads
429
Rank
54,628
Abstract Views
1,901