Density estimates for the exponential functionals of fractional Brownian motion

Abstract

In this note, we investigate the density of the exponential functional of the fractional Brownian motion. Based on the techniques of Malliavin's calculus, we provide a log-normal upper bound for the density.

Estimates for exponential functionals of continuous Gaussian processes with emphasis on fractional Brownian motion

“…The log-normal estimate (1.2) is consistent with the upper bound proved in Dung et al (2022). In contrast to the estimates obtained in Dung (2019b) and Dung et al (2022), our bounds (1.2) and (1.3) are valid for any µ ∈ R and H ∈ (0, 1], and do not depend on unknown constant parameters. Also, they are obtained by elementary arguments.…”

“…Our present results do not address the density of I µ,σ,H T . Nevertheless, both the upper bound obtained in Dung et al (2022) and our estimates yield that the c.d.f. of I µ,σ,H T is upper bounded by a log-normal c.d.f.…”

“…The Kolmogorov distance between I µ,σ,H 1 ∞ and I µ,σ,H 2 ∞ has not been investigated in the literature. Finally, in Dung et al (2022) a log-normal upper bound for the density function of I µ,σ,H T was proved for H ∈ 1 2 , 1 and T < ∞. This was achieved by exploiting the fact that I µ,σ,H T is Malliavin differentiable for T < ∞.…”

“…of I µ,σ,H ∞ was proved for any µ < 0, σ > 0 and H ∈ (0, 1). In a recent paper, Dung et al (2022) established a log-normal upper bound for the density function of I µ,σ,H T for any µ ∈ R, σ > 0, T < ∞ and H ∈ (1/2, 1).…”