Stable Dividends under Linear-Quadratic Optimization

TL;DR

This paper develops a framework for optimizing stable dividends in risky businesses using linear-quadratic criteria, bridging actuarial risk models and diffusion models by deriving explicit optimal affine strategies for general Lévy processes.

Contribution

It introduces a novel approach to dividend optimization that explicitly incorporates stability demands within a linear-quadratic framework for Lévy processes.

Findings

Derived explicit optimal affine dividend strategies.

Characterized the value function via Hamilton-Jacobi-Bellman equation.

Compared stability-focused strategies with classical expected value maximization.

Abstract

The optimization criterion for dividends from a risky business is most often formalized in terms of the expected present value of future dividends. That criterion disregards a potential, explicit demand for stability of dividends. In particular, within actuarial risk theory, maximization of future dividends have been intensively studied as the so-called de Finetti problem. However, there the optimal strategies typically become so-called barrier strategies. These are far from stable and suboptimal affine dividend strategies have therefore received attention recently. In contrast, in the class of linear-quadratic problems a demand for stability if explicitly stressed. These have most often been studied in diffusion models different from the actuarial risk models. We bridge the gap between these patterns of thinking by deriving optimal affine dividend strategies under a linear-quadratic criterion for a general L\'evy process. We characterize the value function by the Hamilton-Jacobi-Bellman equation, solve it, and compare the objective and the optimal controls to the classical objective of maximizing expected present value of future dividends. Thereby we provide a framework within which stability of dividends from a risky business, as e.g. in classical risk theory, is explicitly demanded and explicitly obtained.