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 $1\leq p<\infty$: \begin{align}\label{eq:main}\tag{{$\star$}} \mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E \gamma([\![M]\!]_t)^p ,\;\;\; t\geq 0. \end{align} 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 \eqref{eq:main} was proved for UMD Banach functions spaces $X$. We show that for continuous martingales, \eqref{eq:main} holds for all $0<p<\infty$, and that for purely discontinuous martingales the right-hand side of \eqref{eq:main} can be expressed more explicitly in terms of the jumps of $M$. For martingales with independent increments, \eqref{eq:main} is shown to hold more generally in reflexive Banach spaces $X$ with finite cotype. In the converse direction, we show that the validity of \eqref{eq:main} for arbitrary martingales implies the UMD property for $X$. As an application we prove various It\^o 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 It\^o isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.