Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T16:44:47.840Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Chihoon Lee
Chihoon Lee*
Affiliation:
Colorado State University
*
∗ Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA. Email address: chihoon@stat.colostate.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 ∈ (½, 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 chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= E x [V(Ž̆(1))] − V(x) ≤ −βV(x) + b 1 C (x), x ∈ S, for 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.

Keywords

Reflected fractional Brownian motionheavy traffic theorygeometric drift inequalityreturn time

MSC classification

Secondary: 60G22: Fractional processes, including fractional Brownian motion 90B18: Communication networks 60G15: Gaussian processes 60G18: Self-similar processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 820 - 831
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Atar, R., Budhiraja, A. and Dupuis, P. (2001). On positive recurrence of constrained diffusion processes. Ann. Prob. 29, 979–1000.Google Scholar
[2] Bernard, A. and el Kharroubi, A. (1991). Régulations déterministes et stochastiques dans le premier “orthant” de R n . Stoch. Stoch. Reports 34, 149–167.Google Scholar
[3] Budhiraja, A. and Lee, C. (2007). Long time asymptotics for constrained diffusions in polyhedral domains. Stoch. Process. Appl. 117, 1014–1036.Google Scholar
[4] Delgado, R. (2007). A reflected fBm limit for fluid models with ON/{OFF} sources under heavy traffic. Stoch. Process. Appl. 117, 188–201.Google Scholar
[5] Delgado, R. (2008). State space collapse for asymptotically critical multi-class fluid networks. Queueing Systems 59, 157–184.Google Scholar
[6] Delgado, R. (2010). On the reflected fractional Brownian motion process on the positive orthant: asymptotics for a maximum with application to queueing networks. Stoch. Models 26, 272–294.Google Scholar
[7] Down, D., Meyn, S. P. and Tweedie, R. L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Prob. 23, 1671–1691.Google Scholar
[8] Dupuis, P. and Ishii, H. (1991). On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stoch. Stoch. Reports 35, 31–62.Google Scholar
[9] Dupuis, P. and Ramanan, K. (1999). Convex duality and the Skorokhod problem. I. Prob. Theory Relat. Fields 115, 153–195.Google Scholar
[10] Dupuis, P. and Ramanan, K. (1999). Convex duality and the Skorokhod problem. II. Prob. Theory Relat. Fields 115, 197–236.Google Scholar
[11] Dupuis, P. and Williams, R. J. (1994). Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Prob. 22, 680–702.Google Scholar
[12] Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.Google Scholar
[13] Harrison, J. M. and Williams, R. J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115.Google Scholar
[14] Konstantopoulos, T. and Lin, S.-J. (1996). Fractional Brownian approximations of queueing networks. In Stochastic networks (New York, 1995; Lecture Notes Statist. 117), Springer, New York, pp. 257–273.Google Scholar
[15] Lee, C. (2011). On the return time for a reflected fractional Brownian motion process on the positive orthant. J. Appl. Prob. 48, 145–153.Google Scholar
[16] Majewski, K. (2003). Large deviations for multidimensional reflected fractional Brownian motion. Stoch. Stoch. Reports 75, 233–257.Google Scholar
[17] Majewski, K. (2005). Fractional Brownian heavy traffic approximations of multiclass feedforward queueing networks. Queueing Systems 50, 199–230.Google Scholar
[18] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437.Google Scholar
[19] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster-Lyapunov criteria for continous-time processes. Adv. Appl. Prob. 25, 518–548.Google Scholar
[20] Meyn, S. P. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.Google Scholar
[21] Ruzmaikina, A. A. (2000). Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100, 1049–1069.Google Scholar
[22] Williams, R. J. (1998). An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems 30, 5–25.Google Scholar
[23] Williams, R. J. (1998). Reflecting diffusions and queueing networks. In Proceedings of the International Congress of Mathematicians (Berlin, 1998), Vol. III, pp. 321–330.Google Scholar