Data Analysis and Related Applications, Volume 1

4Optimization and Asymptotic Analysis of Insurance Models¹

Insurance is the oldest domain of applied probability. Moreover, the mathematical models arising there can be used in other areas of applied probability. Therefore, the optimization of insurance models performance and their asymptotic analysis are very important. The modern period in actuarial sciences is characterized by the investigation of complex systems and the employment of sophisticated mathematical tools. Discrete-time models became popular since, in many cases, they describe more precisely the real situation. Hence, we study two models (the discrete-time one and the continuous-time one) in the framework of cost approach. Reinsurance, dividends and bank loans are the controls in optimization problems. Models stability with respect to small perturbations of underlying distributions is treated as well using the probability metrics.

4.1. Introduction

The investigation of insurance models is a primary task of actuarial sciences. A keyword in all definitions of actuarial sciences is risk. It is present whenever the outcome is uncertain, whether favorable or unfavorable. Actuarial sciences emerged in the 17th century (see Bernstein (1996)), although methods for transferring or distributing risk were practiced by Chinese and Babylonian traders as long ago as the 3rd and 2nd millennia BC, respectively. Actuarial sciences have an interesting history consisting of four periods (deterministic, stochastic, financial, modern ...