Measuring the suboptimality of dividend controls in a Brownian risk model

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 problem considered. 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 nonlinear barrier. In the related article of Grandits et al. (Scand. Actuarial J. 2, 2007), it has been observed that standard numerical methods break down in certain parameter cases, and no closed-form solution has been found. For these reasons, we offer a new method allowing one to estimate the distance from 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 from their performance functions to the value function.