Long time asymptotics for constrained diffusions in polyhedral domains

Budhiraja, Amarjit; Lee, Chihoon

doi:10.1016/j.spa.2006.11.007

Cited by 53 publications

(79 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…393-394]). Similar results as in (1.1) were established for reflected processes driven by standard Brownian motions; indeed, the V -uniform ergodicity result of [3] (see also [11] for an ergodicity result) in view of Theorem 5.3 of [7] implies (1.1) for semimartingale reflecting Brownian motions. In this regard, the results in this paper can be viewed as a significant step towards further time-asymptotic analysis of RFBM with the aim of establishing similar ergodic properties for reflected processes driven by non-Markovian processes.…”

supporting

confidence: 78%

A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant

Lee

2011

Journal of Applied Probability

Self Cite

View full text Add to dashboard Cite

We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R d + , with drift r 0 ∈ R d and Hurst parameter H ∈ ( 1 2 , 1). Under a natural stability condition on the drift vector r 0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chainZ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact setfor an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

show abstract

supporting

confidence: 78%

A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant

Lee

2011

Journal of Applied Probability

Self Cite

View full text Add to dashboard Cite

show abstract

“…The following result follows from [7] and [8]. Some comments on the proof are given in the Appendix.…”

Section: The Sm Is Lipschitz Continuous In the Following Sense Therementioning

confidence: 91%

“…Part (i) is Theorem 3.2 of [7]. Finiteness of S e |x| η b (dx), for some ∈ (0, ∞), for each fixed b ∈ BM(S, Λ), is established in Corollary 5.11 of [8] under the additional assumption that Rb − a is a Lipschitz function. The Lipschitz assumption is only used in order to ensure unique solutions and the proof of exponential integrability of invariant measures goes through unchanged for the more general setting considered here.…”

Section: Appendixmentioning

confidence: 97%

“…The Lipschitz assumption is only used in order to ensure unique solutions and the proof of exponential integrability of invariant measures goes through unchanged for the more general setting considered here. Exponential integrability, uniformly over b ∈ BM(S, Λ), as stated in (ii) is a consequence of the fact that δ, β, b 2 and C 2 in Lemma 5.9 of [8] can be chosen (independent of b ∈ BM(S, Λ)) so that the estimate (5.6) of the cited paper holds uniformly over b ∈ BM(S, Λ). Part (iii) is immediate from the above exponential integrability (cf.…”

Section: Appendixmentioning

confidence: 99%

“…Part (iv) has been established in Theorem 3.4 of [7] for the case where k is bounded. The main reason for this restrictive condition there was the lack of availability of exponential integrability estimates which were later established in [8]. Using (ii), the proof of (iv) for the general setting considered here is carried out exactly as in [7].…”

Section: Appendixmentioning

confidence: 99%

See 2 more Smart Citations

Ergodic Rate Control Problem for Single Class Queueing Networks

Budhiraja¹,

Ghosh²,

Lee³

2011

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

We consider critically loaded single class queueing networks with infinite buffers in which arrival and service rates are state (i.e., queue length) dependent and may be dynamically controlled. An optimal rate control problem for such networks with an ergodic cost criterion is studied. It is shown that the value function (i.e., optimum value of the cost) of the rate control problem for the network converges, under a suitable heavy traffic scaling limit, to that of an ergodic control problem for certain controlled reflected diffusions. Furthermore, we show that near optimal controls for limit diffusion models can be used to construct asymptotically near optimal rate control policies for the underlying physical queueing networks. The expected cost per unit time criterion studied here is given in terms of an unbounded holding cost and a linear control cost ("cost for effort"). Time asymptotics of a related uncontrolled model are studied as well. We establish convergence of invariant measures of scaled queue length processes to that of the limit reflecting diffusions. Our proofs rely on infinite time horizon stability estimates that are uniform in control and the heavy traffic parameter, for the scaled queue length processes. Another key ingredient, and a result of independent interest, in the proof of convergence of value functions is the existence of continuous near optimal feedback controls for the diffusion control model. AMS 2000 subject classifications: primary 93E20; secondary 60H30, 60J25, 60K25, 93E15.

show abstract

Reflected B rownian Motion

Dieker

2011

Wiley Encyclopedia of Operations Research and Management Science

View full text Add to dashboard Cite

Reflected Brownian motion is a continuous‐time continuous‐state Markov process, which is constrained to its finite‐dimensional state space S by a pushing mechanism on the boundary ∂ S of S . In the context of Operations Research and Management Science, the state space S is typically the nonnegative orthant. This article reviews several aspects of reflected Brownian motion on this orthant: its construction, its positive recurrence, and its stationary distribution.

show abstract

Long time asymptotics for constrained diffusions in polyhedral domains

Cited by 53 publications

References 25 publications

A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant

A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant

Ergodic Rate Control Problem for Single Class Queueing Networks

Reflected B rownian Motion

Contact Info

Product

Resources

About