Sensitive Discount Optimality in Controlled One-Dimensional Diffusions

Abstract

In this paper we consider the problem of optimally controlling a diffusion process on a compact interval in one-dimensional Euclidean Space. Under the assumptions that the action space is finite and the cost rate, drift and diffusion coefficients are piecewise analytic, we present a constructive proof that there exist piecewise constant $n$-discount optimal controls for all finite $n \geqq 1$ and measurable $\infty$-discount optimal controls. In addition we present a sequence of second order differential equations that characterize the coefficients of the Laurent series of the expected discounted cost of an $n$-discount optimal control.