HJB equations for certain singularly controlled diffusions

• 成果类型: Article
• 署名作者: Atar, Rami; Budhiraja, Amarjit; Williams, Ruth J.
• 署名单位: Technion Israel Institute of Technology; University of California System; University of California San Diego; University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP443
• 发表日期: 2007
• 页码: 1745-1776
• 关键词: differential-equations

摘要

Given a closed, bounded convex set W subset of R-d with nonempty interior, we consider a control problem in which the state process W and the control process U satisfy [Graphics] where Z is a standard, multi-dimensional Brownian motion, nu, sigma epsilon C-0,C-1 (W), G is a fixed matrix, and w(o) epsilon w. The process U is locally of bounded variation and has increments in a given closed convex cone U C RP. Given g epsilon C(W), K epsilon R-P, and alpha > 0, consider the objective that is to minimize the cost [Graphics] over the admissible controls U. Both g and kappa center dot u (u epsilon U) may take positive and negative values. This paper studies the corresponding dynamic programming equation (DPE), a second-order degenerate elliptic partial differential equation of HJB-type with a. state constraint boundary condition. Under the controllability condition GU = R-d and the finiteness of H(q) = sup(u epsilon U1){-Gu center dot q - kappa center dot u}, q epsilon R-d, where U-1 = {u epsilon U : vertical bar Gu vertical bar = 1}, we show that the cost, that involves an improper integral, is well defined. We establish the following: (i) the value function for the control problem satisfies the DPE (in the viscosity sense), and (ii) the condition inf(q epsilon Rd) H(q) < 0 is necessary and sufficient for uniqueness of solutions to the DPE. The existence and uniqueness of solutions are shown to be connected to an intuitive no arbitrage condition. Our results apply to Brownian control problems that represent formal diffusion approximations to control problems associated with stochastic processing networks.