HJB equations for certain singularly controlled diffusions
| dc.creator | Atar, Rami | |
| dc.creator | Budhiraja, Amarjit | |
| dc.creator | Williams, Ruth J. | |
| dc.date | 2007-11-05 | |
| dc.date.accessioned | 2026-07-07T08:40:43Z | |
| dc.date.available | 2026-07-07T08:40:43Z | |
| dc.description | Given a closed, bounded convex set $\mathcal{W}\subset{\mathbb {R}}^d$ with nonempty interior, we consider a control problem in which the state process $W$ and the control process $U$ satisfy \[W_t= w_0+\int_0^t\vartheta(W_s) ds+\int_0^tσ(W_s) dZ_s+GU_t\in \mathcal{W},\qquad t\ge0,\] where $Z$ is a standard, multi-dimensional Brownian motion, $\vartheta,σ\in C^{0,1}(\mathcal{W})$, $G$ is a fixed matrix, and $w_0\in\mathcal{W}$. The process $U$ is locally of bounded variation and has increments in a given closed convex cone $\mathcal{U}\subset{\mathbb{R}}^p$. Given $g\in C(\mathcal{W})$, $κ\in{\mathbb{R}}^p$, and $α>0$, consider the objective that is to minimize the cost \[J(w_0,U)\doteq\mathbb{E}\biggl[\int_0^{\infty}e^{-αs}g(W_s) ds+\int_{[0,\infty)}e^{-αs} d(κ\cdot U_s)\biggr]\] over the admissible controls $U$. Both $g$ and $κ\cdot u$ ($u\in\mathcal{U}$) 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 $G\mathcal{U}={\mathbb{R}}^d$ and the finiteness of $\mathcal{H}(q)=\sup_{u\in\mathcal{U}_1}\{-Gu\cdot q-κ\cdot u\}$, $q\in {\mathbb{R}}^d$, where $\mathcal{U}_1=\{u\in\mathcal{U}:|Gu|=1\}$, 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 $\inf_{q\in{\mathbb{R}}^d}\mathcal{H}(q)<0$ 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. | |
| dc.description | Published in at http://dx.doi.org/10.1214/07-AAP443 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/0711.0641 | |
| dc.identifier | http://arxiv.org/abs/0711.0641 | |
| dc.identifier | Annals of Applied Probability 2007, Vol. 17, No. 5,6, 1745-1776 | |
| dc.identifier | doi:10.1214/07-AAP443 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/141453 | |
| dc.subject | Probability | |
| dc.subject | 93E20, 60H30, 60J60, 35J60 (Primary) | |
| dc.title | HJB equations for certain singularly controlled diffusions | |
| dc.type | text |