BOUNDS OF RUIN PROBABILITIES FOR INSURANCE COMPANIES IN THE PRESENCE OF STOCHASTIC VOLATILITY ON INVESTMENTS

In this work we consider a model of an insurance company where the insurer has to face a claims process which follows a Compound Poisson process with finite exponential moments. The insurer is allowed to invest in a bank account and in a risky asset described by Geometric Brownian motion with stochastic volatility that depends on an external factor modelled as a diffusion process. By using exponential martingale techniques we obtain upper and lower bounds for the ruin probabilities, that recover the known bounds for constant volatility models. Finally we apply the results to a truncated Scott model. Résumé. Dans ce travail, nous considérons un modèle d’une compagnie d’assurance où l’assureur doit faire face a un processus de sinistres qui suit un processus de Poisson Composé avec des moments exponentiels finis. L’assureur est autorisé à investir dans un compte bancaire et un actif à risque décrit par le mouvement Brownien Géométrique à volatilité stochastique qui dépend d’un facteur externe modélisé comme un processus de diffusion. En utilisant des techniques de martingale exponentielle, nous obtenons une borne supérieur et une borne inférieur pour les probabilités de ruine, qui récupèrent les bornes connues pour les modèles de volatilité constante. Enfin, nous appliquons les résultats au modèle de Scott tronqué. Introduction Since the classical papers of Cramér and Lundberg which first considered the ruin problem of an insurance company in conditions of uncertainty, risk theory has attracted a lot of attention and many works has been done in this area. When the claims admit a Laplace transform, and satisfy the safety loading condition, the ruin probability ψ(x) for the classical Cramér-Lundberg Model satisfies: Ce−νx ≤ ψ(x) ≤ e−νx, where ν is the Lundberg coefficient and x is the initial wealth, (see for example [1,12]). An additional feature is considered in [3, 6, 13], the insurer is allowed to invest in a risky asset modelled by the Geometric Brownian motion. Via the Hamilton-Jacobi-Bellman equation Hipp and Plum in [6] prove the existence of an optimal strategy that minimizes the ruin probabilities and Gaier et al. in [3] obtain the exponential bounds: ∗ This work was partially supported by grants PAPIIT-DGAPA-UNAM IN103606, IN117109 and DGEP-UNAM. ∗∗ This work was partially supported by grant PAPIIT-DGAPA-UNAM IN103606, IN117109. 1 Escuela Superior de Ingenieŕıa Mecánica y Eléctrica, Unidad Zacatenco, IPN. Gustavo A. Madero, 07738 México, e-mail: mbadaoui@ipn.mx 2 Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM), Coyoacan 04510, México, e-mail: bff@fc.unam. mx c © EDP Sciences, SMAI 2011 Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi. org/10.1051/proc/2011003 ESAIM: PROCEEDINGS 41 Ce−θx ≤ ψ(x) ≤ e−θx, where θ is the root of the equation λ(MY (θ)− 1) = cθ + μ 2σ2 , and MY (θ) = E[e ]. The upper bound is obtained by considering a constant strategy given by K = μ θσ2 , (1) and using maximal exponential martingales inequalities. By using control techniques Hipp and Schmidli in [7] showed that this strategy is asymptotically optimal as the initial wealth goes to infinity. Observe that θ ties in some way the return of the risky asset and the wealth of the Cramér-Lundberg process. Empirical observations of financial market show that some indicators of market volatility behave in a highly erratic manner, which make unrealistic to assume μ, σ and r constants over long time periods. This fact has motivated several authors to study the so called stochastic volatility models (see between others [5, 14,15]). In this paper we consider an insurance company whose wealth is given by the classical Cramér-Lundberg process