An insurance risk model with stochastic volatility

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.