Hitting probabilities for fractional Brownian motion with deterministic drift

2021/12/03 by Hakiki, Youssef, Erraoui, Mohamed
#28A78 #60G17 #60J45 #62134 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2112.02085

Abstract

Let B^H be a d-dimensional fractional Brownian motion with Hurst index H∈(0,1), f:[0,1]\longrightarrowℝ^d a Borel function, and E⊂[0,1], F⊂ℝ^d are given Borel sets. The focus of this paper is on hitting probabilities of the fractional Brownian motion B^H with the deterministic drift f. It aims to highlight the role of the regularity properties of the drift f as well as that of the dimension of E in determining the upper and lower bounds of ℙ\(B^H+f)(E)∩ F≠ ∅ \ for F a subset of ℝ^d and also for F a singleton.

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