Singular Problems for integro-differential equations in the dynamic
insurance models
On the nonnegative semi-axis, a second order linear
integro-differential equation with a Volterra integral operator and
strong singularities at zero and infinity is considered. Limit
conditions at singular points are posed. Under some natural
assumptions, it 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 given. The
singular problem under study describes the survival (non-ruin)
probability of an insurance company on an infinite time interval
(as a function of initial surplus) in the Cram'er-Lundberg dynamic
insurance model with exponential distributions of claims and
certain company's strategy at the financial market assuming
investment of a fixed part of a surplus (capital) into risky assets
(shares) and the rest of it into a risk-free asset (bank deposit).
