On martingale inequalities in non-commutative stochastic analysis
type: Item
instance of: scholarly article
title: On martingale inequalities in non-commutative stochastic analysis
zbMATH Open document ID: 0935.46054
published in: Journal of Functional Analysis
publication date: 2000-04-24T00:00:00Z
review text: The authors give \(L^p\)-estimates for stochastic integrals of the \textit{C. Barnett, R. F. Streater} and \textit{I. F. Wilde} type [J. Funct. Anal. 48, 172-212 (1982; Zbl 0492.46051)]. These inequalities enable the extension of the stochastic integrals to all \(L^p\)-spaces for \(1\leq p < \infty\). The integrals are taken with respect to a pair \(P, Q\) of independent free fermionic fields. Let, for a simple adapted integrand \(G\), \(\langle G, G\rangle=\sum_{j}G_{j}^*G_{j}(t_{j}-s_{j})\), and let \(\langle{\mathbb G}, {\mathbb G}\rangle\)= \(\langle G_P, G_P\rangle + \langle G_Q, G_Q \rangle\). Then, for \(1\leq p \leq 2\), there is a constant \(K_p\) depending only on \(p\) such that \[ \|G_P dP\|_p + \|G_Q dQ\|_p \leq K_p\min\{\|\langle {\mathbb G},{\mathbb G}\rangle^{1/2}\|_p, \|\langle {\mathbb G^*},{\mathbb G^*}\rangle^{1/2}\|_p\} \] for any pair \(G_P, G_Q\) of simple adapted processes. Moreover, an \(L^p\)-martingale with \(1<p\leq 2\) can be written as a sum of stochastic integrals with respect to \(dP\) and \(dQ\), and the corresponding integrands \(G_P,G_Q\) are given by an explicit formula. To this end, a differential calculus for the Clifford algebra has been developed. This work should be compared with the paper by \textit{G. Pisier} and \textit{Q. Xu} [Commun. Math. Phys. 189, No. 3, 667-698 (1997; Zbl 0898.46056)].
reviewed by: Stanisław Goldstein
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 1215588
zbMATH Keywords: martingale inequalities
zbMATH Keywords: Clifford stochastic integral
zbMATH Keywords: noncommutative martingales
zbMATH Keywords: free fermionic fields
zbMATH Keywords: Clifford algebra
author: Eric Carlen
author: Paul Kree
MaRDI profile type: Publication
full work available at URL: jfan.1998.3299
OpenAlex ID: W2000229301
cites work: Dirichlet forms and Markov semigroups on \(C^*\)-algebras
cites work: An inequality for Hilbert-Schmidt norm
cites work: Stopping non-commutative processes
cites work: The Ito-Clifford integral
cites work: Quasi-free quantum stochastic integrals for the CAR and CCR
cites work: Q3285650
cites work: Distribution function inequalities for martingales
cites work: Optimal hypercontractivity for Fermi fields and related non-commutative integration inequalities
cites work: Q4895892
cites work: Formes linéaires sur un anneau d'opérateurs
cites work: Existence and uniqueness of physical ground states
cites work: Hypercontractivity and logarithmic Sobolev inequalities for the Clifford- Dirichlet form
cites work: The strong Markov property for canonical Wiener processes
cites work: Quantum Ito's formula and stochastic evolutions
cites work: Q4731678
cites work: Inequalities for some operator and matrix functions
cites work: Non commutative Khintchine and Paley inequalities
cites work: Generalized Schur functions
cites work: Quantum probability for probabilists.
cites work: The free Markoff field
cites work: Notes on non-commutative integration
cites work: Q5680837
cites work: Q4003958
cites work: Q4315893
cites work: Inequalities for traces on von Neumann algebras
cites work: A non-commutative extension of abstract integration
cites work: Tensor algebras over Hilbert spaces. II
cites work: Construction of nonlinear local quantum processes. I
cites work: Q4768187
cites work: Conditional expectation in an operator algebra
cites work: Q3694393
cites work: Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations
DOI: 10.1006/JFAN.1998.3299
Recommended article: Q4717459
Recommended article: Non-commutative martingale inequalities
Recommended article: Noncommutative martingale concentration inequalities
Recommended article: Q4640660
Recommended article: \(\Phi\)-inequalities of noncommutative martingales
Recommended article: Distributional inequalities for noncommutative martingales
Recommended article: A noncommutative martingale convexity inequality
Recommended article: Inequalities for noncommutative weakly dominated martingales and applications
Recommended article: Noncommutative martingale deviation and Poincaré type inequalities with applications
description: scientific article; zbMATH DE number 1215588