Existence of a classical solution to the integro-differential equation arising in the Cram\'er--Lundberg non-life insurance model with proportional investment

TL;DR

This paper proves that the survival probability in a non-life insurance model with proportional investment is a smooth classical solution to the related integro-differential equation, under minimal assumptions on claim size distribution.

Contribution

It demonstrates existence of a classical $C^2$-solution to the integro-differential equation with minimal moment conditions on claim sizes.

Findings

Survival probability is a $C^2$-solution of the integro-differential equation.

Minimal moment conditions on claim size distribution are sufficient.

The result applies to continuous claim size distributions with finite positive moments.

Abstract

This paper establishes that the survival probability in the non-life Cram\'{e}r--Lundberg insurance model with proportional investment is a classical $C^2$-solution of the associated integro-differential equation under minimal moment conditions: it suffices that the claim size distribution is continuous and possesses a finite moment of some positive order.