Asymptotic expansion of the expected discounted penalty function in a two-scalestochastic volatility risk model.

University essay from Mälardalens högskola/Akademin för utbildning, kultur och kommunikation

Abstract:

In this Master thesis, we use a singular and regular perturbation theory to derive

an analytic approximation formula for the expected discounted penalty function.

Our model is an extension of Cramer–Lundberg extended classical model because

we consider a more general insurance risk model in which the compound Poisson

risk process is perturbed by a Brownian motion multiplied by a stochastic volatility

driven by two factors- which have mean reversion models. Moreover, unlike

the classical model, our model allows a ruin to be caused either by claims or by

surplus’ fluctuation.

We compute explicitly the first terms of the asymptotic expansion and we show

that they satisfy either an integro-differential equation or a Poisson equation. In

addition, we derive the existence and uniqueness conditions of the risk model with

two stochastic volatilities factors.

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