An extension of bifractional Brownian motion

Abstract

In this paper we introduce and study a self-similar Gaussian process that is the bifractional Brownian motion B H,K with parameters H ∈ (0, 1) and K ∈ (1, 2) such that HK ∈ (0, 1). A remarkable difference between the case K ∈ (0, 1) and our situation is that this process is a semimartingale when 2HK = 1.2000 Mathematics Subject Classification. Primary 60G15; Secondary 60G18.

Bifractional Brownian motion: existence and border cases

“…For reader's convenience, we briefly recall here (and extend) the key arguments from [9] for the case 0 < k < 1, and those from [2,10,14] for the case 1 < k ≤ 2, proving the existence of bfBm.…”

“…Later on, Bardina and Es-Sebaiy [2] enlarged the zone of existence. Using an idea of Lei and Nualart [10], they proved that bfBm exists on R for 0 < h ≤ 1, 0 < k ≤ min 2, 1 h .…”

The Lamperti Transforms of Self-Similar Gaussian Processes and Their Exponentials

“…(B2, by Ref. [4] ) For H ∈ (0, 1) and K ∈ (1, 2) with HK ∈ (0, 1), bfBM B H ,K has the decomposition,…”

Lectures on Gaussian Processes

“…(3.5) Exercise 3.2 (see [11,108]) Prove the following useful relations between bifractional W α,K and fractional W (α K /2) Brownian motions.…”