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

TL;DR

This paper derives explicit bounds and estimates for the distribution and moments of exponential functionals of fractional Brownian motion, providing tools for their analysis and understanding their dependence on the Hurst parameter.

Contribution

It introduces explicit computable bounds for the distribution and moments of exponential functionals of fractional Brownian motion, including continuity results in the Hurst parameter.

Findings

Explicit upper and lower bounds for the c.d.f. of exponential functionals.

Estimates for the p-th order moments and moment-generating functions.

Continuity in law of the exponential functional with respect to the Hurst parameter.

Abstract

Our aim in this article is to provide explicit computable estimates for the cumulative distribution function (c.d.f.) and the $p$-th order moment of the exponential functional of a fractional Brownian motion (fBM) with drift. Using elementary techniques, we prove general upper bounds for the c.d.f. of exponential functionals of continuous Gaussian processes. On the other hand, by applying classical results for extremes of Gaussian processes, we derive general lower bounds. We also find estimates for the $p$-th order moment and the moment-generating function of such functionals. As a consequence, we obtain explicit lower and upper bounds for the c.d.f. and the $p$-th order moment of the exponential functionals of a fBM, and of a series of independent fBMs. In addition, we show the continuity in law of the exponential functional of a fBM with respect to the Hurst parameter.