Singular Initial-Value and Boundary-Value Problems for Integrodifferential Equations in Dynamical Insurance Models with Investments

Abstract

We investigate two insurance mathematical models of the following behavior of an insurance company in the insurance market: the company invests a constant part of the capital in a risk asset (shares) and invests the remaining part in a risk-free asset (a bank account). Changing parameters (characteristics of shares), this strategy is reduced to the case where all the capital is invested in a risk asset. The first model is based on the classical Cramér–Lundberg risk process for the exponential distribution of values of insurance demands (claims). The second one is based on a modification of the classical risk process (the so-called stochastic premium risk process) where both demand values and insurance premium values are assumed to be exponentially distributed. For the infinite-time nonruin probability of an insurance company as a function of its initial capital, singular problems for linear second-order integrodifferential equations arise. These equations are defined on a semiinfinite interval and they have nonintegrable singularities at the origin and at infinity. The first model yields a singular initial-value problem for integrodifferential equations with a Volterra integral operator with constraints. The second one yields more complicated problem for integrodifferential equations with a non-Volterra integral operator with constraints and a nonlocal condition at the origin. We reduce the problems for integrodifferential equations to equivalent singular problems for ordinary differential equations, provide existence and uniqueness theorems for the solutions, describe their properties and long-time behavior, and provide asymptotic representation of solutions in neighborhoods of singular points. We propose efficient algorithms to find numerical solutions and provide the computational results and their economics interpretation.