Optimal Dividend Problem: Asymptotic Analysis

TL;DR

This paper analyzes the classical optimal dividend payment problem, comparing the risk model to its diffusion approximation, and provides sharper convergence results using differential equations rather than probabilistic methods.

Contribution

It introduces a novel approach using differential equations to obtain sharper convergence rates for the value functions in the optimal dividend problem.

Findings

Diffusion approximation closely estimates the classical risk model.

The new method yields faster convergence rate results.

Comparison results improve understanding of approximation accuracy.

Abstract

We re-visit the classical problem of optimal payment of dividends and determine the degree to which the diffusion approximation serves as a valid approximation of the classical risk model for this problem. Our results parallel some of those in B\"auerle (2004), but we obtain sharper results because we use a different technique for obtaining them. Specifically, B\"auerle (2004) uses probabilistic techniques and relies on convergence in distribution of the underlying processes. By contrast, we use comparison results from the theory of differential equations, and these methods allow us to determine the rate of convergence of the value functions in question.