Optimal dividend payment under time of ruin contraint: Exponential case

TL;DR

This paper addresses the optimal dividend payment problem in a classical risk model with exponential claims, incorporating a ruin time constraint, and provides a duality-based solution approach with numerical illustrations.

Contribution

It introduces a dual formulation for the constrained dividend problem and demonstrates the absence of duality gap, enabling computation of the optimal value function via auxiliary problems.

Findings

Dual problem formulation satisfies slackness conditions

No duality gap exists in the constrained problem

Numerical examples illustrate the solution approach

Abstract

We consider the classical optimal dividends problem under the Cram\'er-Lundberg model with exponential claim sizes subject to a constraint on the time of ruin. We introduce the dual problem and show that the complementary slackness conditions are satisfied, thus there is no duality gap. Therefore the optimal value function can be obtained as the point-wise infimum of auxiliary value functions indexed by Lagrange multipliers. We also present a series of numerical examples.