Maximal upper bounds for the moments of stochastic integrals and solutions of stochastic differential equations containing fractional Brownian motion with Hurst index \(H<1/2\). I. (Ukrainian, English) Zbl 1164.60378

Kozachenko, Yu. V.; Mishura, Yu. S.

Teor. Jmovirn. Mat. Stat. 75, 45-56 (2006); translation in Theory Probab. Math. Stat. 75, 51-64 (2007).

This paper deals with upper bounds for the Wiener integral with respect to a fractional Brownian motion with Hurst index \(H\in(0;1/2)\). Integrands \(f\) with support on \(\mathbb R\) as well as finite \(f\) are considered. Upper moment bounds for the corresponding integrals are obtained in both cases. New inequalities for maxima of Gaussian random variables and processes are derived. Using these results, moment inequalities for suprema of Wiener integrals with respect to a fractional Brownian motion are obtained.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60H05	Stochastic integrals
60G15	Gaussian processes

Keywords:

maximal upper bounds; moments of stochastic integrals with respect to fractional Brownian motion; solutions of stochastic differential equations; fractional Brownian motion with Hurst index \(H<1/2\)