On the return time for a reflected fractional Brownian motion process on the positive orthant

Abstract

We consider a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R₊^d, with drift r⁰ ∈ R^d and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r⁰ and reflection directions, we establish a return time result for the RFBM process Z; that is, for some δ, κ > 0, sup_x∈BE_x[τ_B(δ)] < ∞, where B = {x ∈ S : |x| ≤ κ} and τ_B(δ) = inf{t ≥ δ : Z(t) ∈ B}. Similar results are known for reflected processes driven by standard Brownian motions, and our result can be viewed as their FBM counterpart. Our 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.