Nearly Optimal Singular Controls for Wideband Noise Driven Systems

Abstract

Singular stochastic control problems arise in many applications, for example, in storage, inventory, finite fuel, consumption and investment and limits of impulsive control problems. Here, the increment of the control effort is not of the usual form $u(t)dt$, but is the differential of a nondecreasing and suitably adapted process. The diffusion process models used are only approximations to some “physical” process—which might be a “wideband” noise driven system or a suitably scaled discrete parameter process. Since the optimal controls for the “physical” processes are usually impossible to obtain, it is of interest to know whether “nearly” optimal controls for the diffusion model are “nearly” optimum when applied to the physical problem. It is shown that this is true under broad conditions, for discounted and average cost per unit time problems. The usual weak convergence analysis via the Skorokhod topology on $D[0,\infty ]$ is not appropriate here, due to the nature of the singular controls, and it is necessary to use a combination of the Skorokhod and “pseudopath” topology.