A regularity property of fractional Brownian sheets

2024/01/27 by Philippe Bouafia, Bouafia, Philippe, Thierry De Pauw +1 · 1 citation
Economics, Econometrics and Finance · #26A45 #60G17 #60G22 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2401.15427

openalex publication_date 2024/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A function f defined on [0, 1]^d is called strongly chargeable if there is a continuous vector-field v such that f(x₁, …,x_d) equals the flux of v through the rectangle [0, x₁] × ⋯ × [0, x_d] for all (x₁, …, x_d) ∈ [0, 1]^d. In other words, f is the primitive of the divergence of a continuous vector-field. We prove that the sample paths of the Brownian sheet with d ≥ 2 parameters are almost surely not strongly chargeable. On the other hand, those of the fractional Brownian sheet of Hurst parameter (H₁, …, H_d) are shown to be almost surely strongly chargeable whenever (H₁ + ⋯ + H_d)/(d) gt; (d - 1)/(d).

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