Аннотация:This work relates to the problem of the identifying of some
solutions to linear integro-differential equations as the probability of survival
(non-ruin) in the corresponding collective risk models involving
investments. The equations for the probability of non-ruin as a function
of the initial reserve are generated by the infinitesimal operators of
corresponding dynamic reserve processes. The direct derivation of such
equations is usually accompanied by some significant difficulties, such
as the need to prove a sufficient smoothness of the survival probability.
We propose an approach that does not require a priori proof of the
smoothness. It is based on previously proven facts for a certain class of
insurance models with investments: firstly, under certain assumptions,
the survival probability is at least a viscosity solution to the corresponding
integro-differential equation, and secondly, any two viscosity solutions
with coinciding boundary conditions are equivalent. We apply this
approach, allowing us to justify rigorously the form of the survival probability,
to the collective life insurance model with investments.