On distributions of exponential functionals of the processes with independent increments

Vostrikova, Lioudmila

doi:10.15559/20-vmsta159

“…of I µ,σ,1/2 T is given by (1). On the other hand, in the case T < ∞, the density function of I µ,σ,1/2 T is characterized as the solution of a Fokker-Planck equation (see Section 1 in Pintoux and Privault [25], and Corollary 3 in Vostrikova [29]). The conditional density function of [31].…”

Section: The Case H =mentioning

confidence: 99%

“…In particular, the exponential functionals of Lévy processes have been investigated by several authors (see e.g. [1] and [29]). However, few results are known for the exponential functional of fBM.…”

Section: Introductionmentioning

confidence: 99%

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

López-Mimbela¹,

Pérez-Suárez²

2023

Preprint

0

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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.

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

López-Mimbela,

Pérez-Suárez

2024

ALEA

2

0

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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.

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Moments of exponential functionals of Lévy processes on a deterministic horizon – identities and explicit expressions

Palmowski,

Sariev,

Savov

2024

Bernoulli

0

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On distributions of exponential functionals of the processes with independent increments

Abstract: The aim of this paper is to study the laws of exponential functionals of the processes X = (X s) s≥0 with independent increments, namely

Cited by 3 publications

References 28 publications

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

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

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

Moments of exponential functionals of Lévy processes on a deterministic horizon – identities and explicit expressions

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