TY - JOUR
T1 - A geometric drift inequality for a reflected fractional Brownian motion process on the positive orthant
AU - Lee, Chihoon
PY - 2011/9
Y1 - 2011/9
N2 - 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.
AB - 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.
KW - Geometric drift inequality
KW - Heavy traffic theory
KW - Reflected fractional Brownian motion
KW - Return time
UR - http://www.scopus.com/inward/record.url?scp=80054858033&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=80054858033&partnerID=8YFLogxK
U2 - 10.1239/jap/1316796917
DO - 10.1239/jap/1316796917
M3 - Article
AN - SCOPUS:80054858033
SN - 0021-9002
VL - 48
SP - 820
EP - 831
JO - Journal of Applied Probability
JF - Journal of Applied Probability
IS - 3
ER -