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April 2001 On Positive Recurrence of Constrained Diffusion Processes
Rami Atar, Amarjit Budhiraja, Paul Dupuis
Ann. Probab. 29(2): 979-1000 (April 2001). DOI: 10.1214/aop/1008956699


Let $G \subset \mathbb{R}^k$ be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces $\{G_i, i =1,\ldots, N\}$, where $n_i$ and $d_i$ denote the inward normal and direction of constraint associated with $G_i$, respectively. Stability properties of a class of diffusion processes, constrained to take values in $G$, are studied under the assumption that the Skorokhod problem defined by the data $\{(n_i,d_i),i = 1,\ldots,N\}$ is well posed and the Skorokhod map is Lipschitz continuous. Explicit conditions on the drift coefficient, $b(\cdot)$, of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure. Define $$\mathscr{C}\doteq \left\{ - \sum^{N}_{i=1} a_i d_i;a_i \ge 0, i \in \{1,\ldots,N\} \right\}.$$ Then the key condition for stability is that there exists $\delta \in (0,\infty)$ and a bounded subset $A$ of $G$ such that for all $x \in G\setminus A, b(x) \in \mathscr{C}$ and $\mathrm{dist} (b(x),\partial\mathscr{C}) \ge \delta$, where $\partial\mathscr{C}$denotes the boundary of $\mathscr{C}$.


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Rami Atar. Amarjit Budhiraja. Paul Dupuis. "On Positive Recurrence of Constrained Diffusion Processes." Ann. Probab. 29 (2) 979 - 1000, April 2001.


Published: April 2001
First available in Project Euclid: 21 December 2001

zbMATH: 1018.60081
MathSciNet: MR1872747
Digital Object Identifier: 10.1214/aop/1008956699

Primary: 60J60
Secondary: 34D20 , 60J65 , 60K25

Keywords: constrained ordinary differential equation , constrained processes , Invariant measures , Law of Large Numbers , positive recurrence , queueing systems , Skorokhod problem , stability

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 2 • April 2001
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