An Insurance Risk Model with Stochastic Volatility
China Institute for Actuarial Science, Central University of Finance and Economics
University of Toronto - Department of Statistics
X. Sheldon Lin
University of Toronto
December 15, 2008
Insurance: Mathematics and Economics, Vol. 46, No. 1, pp. 52-66
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.
Number of Pages in PDF File: 31
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
Date posted: December 15, 2008 ; Last revised: February 16, 2010
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