Given a closed, bounded convex set 
with nonempty interior, we consider a control problem in which the state process W and the control process U satisfy
![\[W_{t}=w_{0}+\int_{0}^{t}\vartheta(W_{s})\,ds+\int_{0}^{t}\sigma(W_{s})\,dZ_{s}+GU_{t}\in \mathcal{W},\qquad t\ge0,\]](/DPubS?service=Repository&version=1.0&verb=Disseminate&view=body&handle=euclid.aoap/1191419182&content-type=gif_2)
where Z is a standard, multi-dimensional Brownian motion, 
, G is a fixed matrix, and 
. The process U is locally of bounded variation and has increments in a given closed convex cone 
. Given 
, κ∈ℝp, and α>0, consider the objective that is to minimize the cost
![\[J(w_{0},U)\doteq\mathbb{E}\biggl[\int_{0}^{\infty}e^{-\alpha s}g(W_{s})\,ds+\int_{[0,\infty)}e^{-\alpha s}\,d(\kappa\cdot U_{s})\biggr]\]](/DPubS?service=Repository&version=1.0&verb=Disseminate&view=body&handle=euclid.aoap/1191419182&content-type=gif_7)
over the admissible controls U. Both g and κ⋅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 
and the finiteness of 
, q∈ℝd, where 
, 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 
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.
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.
Read more about accessing full-text
References
Atar, R. and Budhiraja, A. (2006). Singular control with state constraints on unbounded domain. Ann. Probab. 34 1864--1909.
Bass, R. (1995). Probabilistic Techniques in Analysis. Springer, Berlin.
Crandall, M. G. Ishii, H. and Lions, P. L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1--67.
Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Univ. Press.
Graves, L. M. (1956). The Theory of Functions of Real Variables. McGraw-Hill, New York.
Harrison, J. M. (2003). A broader view of Brownian networks. Ann. Appl. Probab. 13 1119--1150.
Harrison, J. M. and Williams, R. J. (2005). Workload reduction of a generalized Brownian network. Ann. Appl. Probab. 15 2255--2295.
Lions, P. L. and Sznitman, A. S. (1984). Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 511--537.
Soner, H. M. (1986). Optimal control with state-space constraint. I. SIAM J. Control Optim. 24 552--561.
Tanaka, H. (1979). Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 163--177.
