Burkholder\u2013Davis\u2013Gundy Inequalities in UMD Banach Spaces

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.