Derivative for the intersection local time of two independent fractional Brownian motions. Issue 3 (3rd April 2022)

• Record Type: Journal Article
• Authors: Yan, Litan; Sun, Xichao
• Is Part Of: Stochastics. Volume 94:Issue 3(2022)
• Page Start: 459
• Page End: 492
• Publication Date: 2022-04-03
• Subjects: Fractional Brownian motion -- intersection local time -- occupation formula -- hybrid quadratic covariation -- Malliavin calculus; 60G15 -- 60G18 -- 60F25; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/17442508.2021.1959584
• Languages: English
• ISSNs: 1744-2508

Abstract

Abstract : Let B H 1 and B ~ H 2 be two independent fractional Brownian motions with respective indices H 1, H 2 ∈ ( 0, 1 ) and H 1 ≤ H 2 . In this paper, we consider the derivative of their intersection local time ℓ ( t, a ) . We show that the derivative exists in L p for all p ∈ [ 1, + ∞ ) and it is joint Hölder continuous in space and time if H 1 + H 2 > 3 H 1 H 2 . In particular, we show that ℓ ( t, a ) is differentiable in the spatial variable a under the condition, and we introduce the so-called hybrid fractional quadratic covariation [ f ( B H 1 − B ~ H 2 ), B H 1 ] ( H C ) . When 0 < H 1 ≤ 1 2, we construct a Banach space H of measurable functions such that the quadratic covariation exists in L 2 ( Ω ) and 1 f ( B H 1 − B ~ H 2 ), B H 1 ] t ( H C ) = − ∫ R f ( a ) ℓ ( t, d a ) for all f ∈ H . When H 1 > 1 2 a similar result is proved to hold for all Hölder functions f of order ν > 2 H 1 − 1 H 1 .