Logarithmic estimates for submartingales and their differential subordinates
type: Item
instance of: scholarly article
title: Logarithmic estimates for submartingales and their differential subordinates
zbMATH Open document ID: 1234.60045
published in: Journal of Theoretical Probability
publication date: 2011-09-20T00:00:00Z
review text: The author establishes several sharp inequalities for (sub)martin\-gale sequences \(f=(f_n)_{n\geq 0}\) and sequences \(g=(g_n)_{n\geq 0}\) that are \(\alpha\)-subordinate (or differentially subordinate) to \(f\). The first main result states that, if \(g\) is \(\alpha\)-subordinate to a nonnegative submartingale \(f\), \(\alpha\in [0,1]\), then, for any \(K>1\), \[ \|g\|_1\leq K \sup_n \operatorname{E}f_n\log^+ f_n +L(K,\alpha), \] with an explicitly determined constant \(L(K,\alpha)\) which is best possible. Furthermore, if \(K\leq 1\), the inequality does not hold in general with any universal \(L(K,\alpha)<\infty\). Related sharp logarithmic inequalities are derived for \(g\) and its one-sided maximal function in the case when both \(f\) and \(g\) are martingales and \(g\) is differentially subordinate to \(f\). The results are proved in a more general continuous-time settings for \(X\) and \(Y\) real-valued càdlàg semimartingales. As an application, the author derives logarithmic estimates for smooth functions on Euclidean domains.
reviewed by: Zoran Vondraček
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 31B05
zbMATH DE Number: 5948562
zbMATH Keywords: martingale
zbMATH Keywords: submartingale
zbMATH Keywords: subharmonic function
zbMATH Keywords: maximal function
zbMATH Keywords: differential subordination
zbMATH Keywords: strong differential subordination
zbMATH Keywords: LlogL class
zbMATH Keywords: boundary value problem
author: Adam Osȩkowski
MaRDI profile type: Publication
full work available at URL: s10959-009-0268-9
OpenAlex ID: W2045216194
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DOI: 10.1007/S10959-009-0268-9
description: scientific article; zbMATH DE number 5948562