| We study the ruin problem for insurance models that involve investments. Our risk reserve process is an extension of the classical Cramer-Lundberg model, which will contain stochastic interest rates, reserve-dependent expense loading, as well as diffusion perturbed models as special cases. By introducing a new type of exponential martingale parameterized by a general rate function, we put various Cramer-Lundberg type estimations into a unified framework. We show that many existing Lundberg-type bounds for ruin probabilities can be recovered by appropriately choosing the rate function. We also prove that in this general case the ruin probability can still be expressed in terms of a special type of storage process, characterized by a generalized reflected stochastic differential equation with discontinuous paths. We apply this general storage process with large deviation techniques to analyze the limiting behavior of ruin probabilities for perturbed risk models. We provide asymptotic estimations of two types of ruin and show that in certain situations the adjustment coefficient can be derived. |
