Exact solution of the ruin problem in the Cram\'er--Lundberg model with proportional investment

TL;DR

This paper provides an exact analytical solution to the ruin problem in the Cramér-Lundberg model with exponential claims and proportional investment, using Heun functions.

Contribution

It reduces the survival probability equation to a doubly confluent Heun equation and derives an explicit solution, advancing analytical methods in risk theory.

Findings

Explicit solution in terms of Heun functions

Verification theorem established

Qualitative analysis of ruin probability versus investment share

Abstract

The Cram\'er-Lundberg model with exponential claims and proportional investment is solved exactly: the integro-differential equation for the survival probability reduces to a doubly confluent Heun equation, yielding an explicit solution in terms of Heun functions, a verification theorem, and a qualitative analysis of ruin probability versus investment share.