Approximations to ultimate ruin probabilities with a Wienner process perturbation

Abstract

In this paper, we adapt the classic Cram\'er-Lundberg collective risk theory model to a perturbed model by adding a Wiener process to the compound Poisson process, which can be used to incorporate premium income uncertainty, interest rate fluctuations and changes in the number of policyholders. Our study is part of a Master dissertation, our aim is to make a short overview and present additionally some new approximation methods for the infinite time ruin probabilities for the perturbed risk model. We present four different approximation methods for the perturbed risk model. The first method is based on iterative upper and lower approximations to the maximal aggregate loss distribution. The second method relies on a four-moment exponential De Vylder approximation. The third method is based on the first-order Pad\'e approximation of the Renyi and De Vylder approximations. The last method is the second order Pad\'e-Ramsay approximation. These are generated by fitting one, two, three or four moments of the claim amount distribution, which greatly generalizes the approximations. We test the precision of approximations using a combination of light and heavy tailed distributions for the individual claim amount. We assess the ultimate ruin probability and present numerical results for the exponential, gamma, and mixed exponential claim distributions, demonstrating the high accuracy of these four methods. Analytical and numerical methods are used to highlight the practical implications of our findings.