?
Singular problems for integro-differential equations in dynamic insurance models
P. 27–44.
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 Crame´r– 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.
Language:
English