Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities
type: Item
instance of: scholarly article
title: Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities
zbMATH Open document ID: 0832.60055
author: Gang Wang
published in: The Annals of Probability
publication date: 1996-02-13T00:00:00Z
review text: If \(X\) and \(Y\) are a pair of discrete martingales, suppose the difference sequence of \(X\) satisfies a pointwise domination with the corresponding sequence of \(Y\). Then one says that \(X\) is differentially subordinate to \(Y\). In the continuous parameter case the concept is in terms of their quadratic variations. Thus if \(X = \{X_t, {\mathcal F}_t, t \geq 0\}\) and \(Y = \{Y_t, {\mathcal F}_t, t \geq 0\}\) with values in a (separable) Hilbert space \({\mathcal H}\) are two adapted càdlàg martingales, one says that \(Y\) is differentially subordinate to \(X\) if \([X,X]_t - [Y,Y]_t \geq 0\) and is increasing. These concepts and several of the sharp inequalities on such processes have been the subject of D. L. Burkholder's work (with \({\mathcal H} = \mathbb{R})\) from the late 1970's in several papers. Based on the methodology of this work, the author extends some of the results, of which the following is representative. Theorem. Let \(X\) and \(Y\) be a pair of adapted càdlàg \({\mathcal H}\)- valued martingales with \(Y\) being differentially subordinate to \(X\). Then for each \(1 < p < \infty\), one has \(|Y |_p \leq (p^* - 1) |X |_p\), where the constant \((p^* - 1)\) is the best possible and there is strict inequality if \(p \neq 2\) and \(|X |_p \neq 0\). Here \(p^* = \max (p,p/(p - 1))\). The extension is nontrivial. The author remarks finally: ``When working on sharp martingale inequalities the finite-dimensional discrete-time case is more challenging and more interesting.'' With such extensions, it is now natural and possible to consider the corresponding theory in the context of Orlicz spaces in lieu of the present Lebesgue spaces.
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 796509
zbMATH Keywords: sharp inequalities
zbMATH Keywords: martingale inequalities
zbMATH Keywords: Orlicz spaces
zbMATH Keywords: Lebesgue spaces
reviewed by: Malempati M. Rao
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988278
OpenAlex ID: W1980491615
DOI: 10.1214/AOP/1176988278
description: scientific article; zbMATH DE number 796509