Suboptimal Control of Dividends under Exponential Utility

TL;DR

This paper investigates the optimal dividend payout strategy for an insurance company modeled by a Brownian motion with drift, focusing on exponential utility maximization with bounded dividend rates, and introduces a new numerical method to evaluate suboptimal strategies.

Contribution

It proposes a novel numerical approach to estimate the distance to the value function and assesses common suboptimal strategies in a complex, non-linear utility setting.

Findings

Optimal strategy likely involves a non-linear barrier.

Standard numerical methods fail in certain parameter regimes.

The new method effectively evaluates suboptimal strategies.

Abstract

We consider an insurance company modelling its surplus process by a Brownian motion with drift. Our target is to maximise the expected exponential utility of discounted dividend payments, given that the dividend rates are bounded by some constant. The utility function destroys the linearity and the time homogeneity of the considered problem. The value function depends not only on the surplus, but also on time. Numerical considerations suggest that the optimal strategy, if it exists, is of a barrier type with a non-linear barrier. In the related article by granditz et al., it has been observed that standard numerical methods break down in certain parameter cases and no close form solution has been found. For these reasons, we offer a new method allowing to estimate the distance of an arbitrary smooth enough function to the value function. Applying this method, we investigate the goodness of the most obvious suboptimal strategies - payout on the maximal rate, and constant barrier strategies - by measuring the distance of its performance function to the value function.