Asymptotic analysis and optimization of some insurance models

The aim of the paper is to show how one can perform asymptotic analysis of models arising in insurance, finance, and other applications of probability theory and solve optimization problems. To this end, we consider two insurance models (one continuous‐time and one discrete‐time). The first one is a dual Sparre Andersen insurance model with dividends. It describes not only the functioning of a life insurance company dealing with annuities but also a venture capital investment company or the capital of a business engaged in research and development. The main attention is paid to investigation of a new strategy of dividends payment. The second model deals with short‐term credits in discrete‐time case. We focus here on the system optimization in the framework of cost approach. In other words, expected discounted n‐period costs are chosen as objective function. The optimal policy is obtained using dynamic programming. We introduce also the notion of asymptotic optimality and establish the form of asymptotically optimal policy. The model stability with respect to claim distribution perturbations is dealt with as well. Although we study only the simple cases, the methods proposed here can be useful for solving other optimization and stability problems.