Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. (English) Zbl 0983.60052

Mémin, Jean; Mishura, Yuliya; Valkeila, Esko

Stat. Probab. Lett. 51, No. 2, 197-206 (2001).

Let \(Z\) be a fractional Brownian motion with Hurst parameter \(H\). For a Wiener-type stochastic integral \(\int fdZ\), properly defined, it is proved that for every \(r>0\) there exists a constant \(c(H,r)\) such that \[ E\left |\int_a^bf(u)dZ_u\right |^r \leq c(H,r)||f||^r_{L^{1/H}(a,b)} \] for each \(0\leq a<b<\infty\), provided that \(H>{1\over 2}\). If \(H<{1\over 2}\), then the opposite inequality holds. It is also shown that lower or upper estimates do not hold in the cases \(H>{1\over 2}\) or \(H<{1\over 2}\), respectively.

Reviewer: Tomasz Bojdecki (Warszawa)

MSC:

60H05	Stochastic integrals
60G15	Gaussian processes

Keywords:

fractional Brownian motion; stochastic integration; moment inequalities

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