Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces

In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X . Assuming that M 0 = 0 , we show that the following two-sided inequality holds for all 1 ≤ p < ∞ : Here γ ( [ [ M ] ] t ) is the L 2 -norm of the unique Gaussian measure on X...

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Bibliographic Details
Published in:Communications in mathematical physics Vol. 379; no. 2; pp. 417 - 459
Main Author: Yaroslavtsev, Ivan
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2020
Springer Nature B.V
Subjects:
Banach spaces
Brownian motion
Classical and Quantum Gravitation
Complex Systems
Covariance
Inequalities
Integrals
Isomorphism
Martingales
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theoretical
ISSN:0010-3616, 1432-0916
Online Access:Get full text
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Summary:In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X . Assuming that M 0 = 0 , we show that the following two-sided inequality holds for all 1 ≤ p < ∞ : Here γ ( [ [ M ] ] t ) is the L 2 -norm of the unique Gaussian measure on X having [ [ M ] ] t ( x ∗ , y ∗ ) : = [ ⟨ M , x ∗ ⟩ , ⟨ M , y ∗ ⟩ ] t as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of ( ⋆ ) was proved for UMD Banach functions spaces X . We show that for continuous martingales, ( ⋆ ) holds for all 0 < p < ∞ , and that for purely discontinuous martingales the right-hand side of ( ⋆ ) can be expressed more explicitly in terms of the jumps of M . For martingales with independent increments, ( ⋆ ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of ( ⋆ ) for arbitrary martingales implies the UMD property for X . As an application we prove various Itô isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide Itô isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.
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ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-020-03845-7