Maximal inequalities for moments of Wiener integrals with respect to fractional Brownian motion. (English. Ukrainian original) Zbl 0985.60032

Krvavych, Yu. V.; Mishura, Yu. S.

Theory Probab. Math. Stat. 61, 75-86 (2000); translation from Teor. Jmovirn. Mat. Stat. 61, 72-83 (2000).

This paper deals with the Wiener integrals \(I_{\tau}=\int_{0}^{\tau}f(t) dB^{H}_{t}\), where \(B^{H}_{t}\) is a fractional Brownian motion with Hurst index \(H\in (1/2,1)\) and \(f\in L_{2}^{\Gamma}\), the space of equivalent classes of measurable functions \(f\) such that \(\langle f,f\rangle_{\Gamma}<\infty,\) \(\langle f,f\rangle_{\Gamma}=H(2H-1)\int_{0}^{\infty}\int_{0}^{\infty}f(t)f(s)|s-t|^{2H-2}dsdt.\) The upper and the lower estimations for the moments of the supremum of the Wiener integrals \(I_{\tau}\) of fractional Brownian motion are obtained both for deterministic and for random \(\tau.\) Estimations on the deterministic intervals are obtained with the help of Gaussian \(I_{\tau},\) and on the random intervals with the help of representation of fractional Brownian motion by the so-called Molchan martingale [see, for example, G. M. Molchan and Ju. I. Golosov, Sov. Math., Dokl. 10, 134-137 (1969); translation from Dokl. Akad. Nauk SSSR 184, 546-549 (1969; Zbl 0181.20704)]. These estimations essentially depend on the properties of function \(f\).

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60G15	Gaussian processes
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44	Martingales with continuous parameter

Keywords:

fractional Brownian motion; Wiener integrals; maximal inequalities

Citations:

Zbl 0181.20704