Optimal dividend problem: Asymptotic analysis

We revisit 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 Bäuerle [Math. Finance, 14 (2004), pp. 99-113], but we obtain sharper results because we use a different technique for obtaining them. Specifically, Bäuerle 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.