Singular Problems for Integro-Differential Equations in Dynamic Insurance Models

A second order linear integro-differential equation with Volterra integral operator and strong singularities at the endpoints (zero and infinity) is considered. Under limit conditions at the singular points, and some natural assumptions, the problem is a singular initial problem with limit normalizing conditions at infinity. An existence and uniqueness theorem is proved and asymptotic representations of the solution are given. A numerical algorithm for evaluating the solution is proposed, calculations and their interpretation are discussed. The main singular problem under study describes the survival (non-ruin) probability of an insurance company on infinite time interval (as a function of initial surplus) in the Cramer-Lundberg dynamic insurance model with an exponential claim size distribution and certain company's strategy at the financial market assuming investment of a fixed part of the surplus (capital) into risky assets (shares) and the rest of it into a risk free asset (bank deposit). Accompanying "degenerate" problems are also considered that have an independent meaning in risk theory