Viscosity solutions of the integro-differential equation for the Cram\'er--Lundberg model with annuity payments and investments

TL;DR

This paper proves the existence and regularity of viscosity solutions for an integro-differential equation related to the ruin problem in annuity payment models, showing these solutions are classical.

Contribution

It establishes the existence and regularity of viscosity solutions for the integro-differential equation in the Cramér--Lundberg model with annuities and investments, extending prior work.

Findings

Existence of viscosity solutions is proved.

Solutions are shown to be classical due to regularity.

The work extends the mathematical understanding of ruin problems with annuities.

Abstract

This note is an addendum to the work initiated by Promyslov on the integro-differential equation arising in the ruin problem for annuity payment models. First, the existence of viscosity solutions is proved. Then the regularity of these solutions is established, showing that they are indeed classical solutions.