Ruin Problems with Model Uncertainty

In this talk I will discuss some ruin problems for insurance models with investment, in a general market framework that contains uncertainties on both market volatility and claim frequency. The problem naturally leads to the evaluation of the ruin probability under the so-called G-expectation, a non-linear expectation developed recently by S. Peng. We prove that the claim process can be naturally described as a class of "G-Compound Poisson process", a special case of the so-called G-Levy process studied by Hu-Peng (2010). Such a relation, however, does not seem to be obvious from the original Denton of a nonlinear expectation. We also establish the relationship between the study of asymptotic behavior of a ruin probability and the sample path large deviation principle (LDP) of a "storage process", defined as a special type of reflected SDE with discontinuous paths, and obtain the asymptotic Lundberg bounds via the rate function of the LDP. Some related issues for the market models under G-expectations will be discussed as well.