Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces
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. …