Inverse local times of fractional Brownian motion

Abstract

In this paper, we study the inverse local time of a d -dimensional fractional Brownian motion. We obtain that if L − 1 ( t ) is the inverse local time of fractional Brownian motion of Hurst index 0 < β < 1 with β d < 1 , then ∫ 0 ∞ E ( exp ( − λ L − 1 ( t ) ) ) d t = Γ ( 1 − β d ) / λ 1 − β d ( 2 π ) d / 2 . This result raises the question whether the inverse local time of a fractional Brownian motion has a stable law of parameter 1 − β d as it is the case for the standard one-dimensional Brownian motion.