Burkholder\u2013Davis\u2013Gundy Inequalities in UMD Banach Spaces

Ivan Yaroslavtsev

doi:10.1007/s00220-020-03845-7

Abstract

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 1le p<infty : Here gamma ([![M]!]_t) is the L^2-norm of the unique Gaussian measure on X having [![M]!]_t(x^*,y^*):= [langle M,x^*rangle , langle M,y^*rangle ]_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 (star ) was proved for UMD Banach functions spaces X. We show that for continuous martingales, (star ) holds for all 0<p<infty , and that for purely discontinuous martingales the right-hand side of (star ) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, (star ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of (star ) 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.

Highlights

In the celebrated paper [12] Burkholder, Davis, and Gundy proved that if M = (Mt )t≥0 is a real-valued martingale satisfying M0 = 0, for all 1 ≤ p < ∞ and t ≥ 0 one has the two-sided inequality pE sup |Ms |p p E[M]t2, (1.1)where [M] is the quadratic variation of M, i.e., N [ M ]t := P−lim mesh(π )→0 n=1 |M − M|2
Is there an analogue of (1.3) for a general X -valued local martingale M and how should the right-hand side of (1.3) look like? In the current article we present the following complete solution to this problem for local martingales M with values in a UMD Banach space X
In Subsection 7.1 we develop a theory of vector-valued stochastic integration

Summary

Introduction

(1.6) follows from a decoupling inequality due to Garling [22] and a martingale transform inequality due to Burkholder [8] (each of which holds if and only if X has the UMD property) together with the equivalence of Rademacher and Gaussian random sums with values in spaces with finite cotype due to Maurey and Pisier (see [53]). In Subsection 7.4 we prove the following martingale domination inequality: for all local martingales M and N with values in a UMD Banach space X such that. This problem might be resolved via using recently discovered decoupled tangent martingales, see [83]

Burkholder–Davis–Gundy Inequalities

Gaussian Characteristics

Itô isomorphism

UMD Banach Function Spaces