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
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\[W_{t}=w_{0}+\int_{0}^{t}\vartheta(W_{s})\,ds+\int_{0}^{t}\sigma(W_{s})\,dZ_{s}+GU_{t}\in \mathcal{W},\qquad t\ge0,\]
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where Z is a standard, multi-dimensional Brownian motion, $\vartheta,\sigma\in C^{0,1}(\mathcal{W})$ , 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∈ℝp, and α>0, consider the objective that is to minimize the cost
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\[J(w_{0},U)\doteq\mathbb{E}\biggl[\int_{0}^{\infty}e^{-\alpha s}g(W_{s})\,ds+\int_{[0,\infty)}e^{-\alpha s}\,d(\kappa\cdot U_{s})\biggr]\]
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over the admissible controls U. Both g and κ⋅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 $\mathcal{H}(q)=\sup_{u\in\mathcal{U}_{1}}\{-Gu\cdot q-\kappa\cdot u\}$ , q∈ℝd, where $\mathcal{U}_{1}=\{u\in\mathcal{U}\dvtx|Gu|=1\}$ , 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 $\inf_{q\in{\mathbb{R}}^{d}}\mathcal{H}(q)\textless 0$ 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.
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Our results apply to Brownian control problems that represent formal diffusion approximations to control problems associated with stochastic processing networks.