A Sharp and Strict Lp-Inequality for Stochastic Integrals

Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

Let (Ω, F, P) be a complete probability space and (F t )t ≥ 0 a nondecreasing right-continuous family of sub-σ-fields of F where F 0 contains all AF with P(A) = 0. Suppose that M = (M t )t ≥ 0 is a real martingale adapted to (F t )t ≥ 0 such that almost all of the paths of M are right-continuous on [0, ∞) and have left limits on (0, ∞). Let V = (V t )t ≥ 0 be a predictable process with values in [−1, 1] and denote by N= VM the stochastic integral of V with respect to M: N is an adapted right-continuous process with left limits on (0, ∞) such that
$${N_t} = \int_{\left[ {0,\,t} \right]} {{V_s}\,d{M_s}} \,a.s.$$

Keywords

Convolution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    BURKHOLDER, D. L. (1984). Boundary value problems and sharp inelualities for martingale transforms. Ann. Probab. 12647-702.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    BURKHOLDER, D. L. (1985). An elementary proof of an inequality of R. E. A. C. Paley. Bull. London Math. Soc. 17474-478.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dellacherie, C. and Meyer, P. A. (1980). Probabilités et Potentiel: Théorie des Martingales, rev. ed., Chap. V-VIII. Hermann, Paris. Also published in an English translation by North-Holland, Amsterdam (1982).Google Scholar
  4. 4.
    IkedaN. and WatanabeS. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.MATHGoogle Scholar
  5. 5.
    KunitaH. and WatanabeS. (1967). On square integrable martingales. Nagoya Math. J. 30209-245.MATHMathSciNetGoogle Scholar
  6. 6.
    MeyerP. A. (1976). Un cours sur les intégrales stochastiques. Séminaire de Probabilités X. Lecture Notes in Math. 511245-400.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations