A geometric drift inequality for a reflected fractional Brownian motion process on the positive orthant

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Abstract

We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = Rd+, with drift r 0 ∈ Rd and Hurst parameter H ∈ ( 1/2 , 1). Under a natural stability condition on the drift vector r0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž 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 (Ž(1))]-V (x) ≤ -βV (x)+b 1c (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.

Original languageEnglish
Pages (from-to)820-831
Number of pages12
JournalJournal of Applied Probability
Volume48
Issue number3
DOIs
StatePublished - Sep 2011

Keywords

  • Geometric drift inequality
  • Heavy traffic theory
  • Reflected fractional Brownian motion
  • Return time

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