An insurance risk model with stochastic volatility |
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Authors: | Yichun Chi |
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Institution: | a Department of Financial Mathematics, Peking University, Beijing 100871, China b Department of Statistics, University of Toronto, 100 St. George Street, Toronto, Ontario M5S 3G3, Canada |
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Abstract: | In this paper, we extend the Cramér-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 the 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. |
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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 |
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