zbMATH — the first resource for mathematics

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

MSC:
 60H05 Stochastic integrals 60G15 Gaussian processes
Full Text:
References:
 [1] Dasgupta, A., 1998. Fractional Brownian motion: its properties and applications to stochastic integration. Ph.D. Thesis, University of North Carolina, 97 p. [2] Huang, S.; Cambanis, S., Stochastic and multiple Wiener integrals for Gaussian processes, Ann. probab., 6, 585-614, (1978) · Zbl 0387.60064 [3] Mishura, Yu.S.; Valkeila, E., An isometric approach to generalized stochastic integrals, J. theoretical probab., 13, 673-693, (1999) · Zbl 0965.60054 [4] Norros, I.; Valkeila, E.; Virtamo, J., An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli, 5, 571-587, (1999) · Zbl 0955.60034 [5] Novikov, A.; Valkeila, E., On some maximal inequalities for fractional Brownian motions, Statist. probab. lett., 44, 47-54, (1999) · Zbl 0947.60033 [6] Pipiras, V., Taqqu, M.S., 2000. Are classes of deterministic integrands for fractional Brownian motion on an interval complete? Preprint. · Zbl 1003.60055 [7] Rogers, L.C.G., Arbitrage with fractional Brownian motion, Math. finance, 7, 95-105, (1997) · Zbl 0884.90045 [8] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications., (1993), Gordon and Breach London · Zbl 0818.26003 [9] Stein, E.M., Singular integrals and differentiability properties of functions., (1971), Princeton University Press Princeton, NJ
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.