Two-parameter martingales and their quadratic variation
type: Item
instance of: scholarly article
title: Two-parameter martingales and their quadratic variation
zbMATH Open document ID: 0656.60056
author: P. Imkeller
published in: Lecture Notes in Mathematics
publication date: 1993-06-05T00:00:00Z
review text: Since the publication of the pioneering paper of \textit{R. Cairoli} and \textit{J. B. Walsh} [Acta Math. 134, 111-183 (1975; Zbl 0334.60026)] the theory of two-parameter processes has undergone an active development, especially during the second part of the last decade. The basic survey of \textit{P. A. Meyer} [Théorie élémentaire des processus aléatoires à deux indices, Colloq. E.N.S.T.-C.N.E.T., Paris 1980, Lect. Notes Math. 863, 1-39 (1981; Zbl 0461.60072)] provides an account of the stage of this theory at the beginning of the eighties and may be considered as an introduction to a general theory of two-parameter processes. Some fundamental results such as the regularity properties for square integrable martingales were already presented in this paper of Meyer, but the theory was just at an initial stage. In the last five years stochastic calculus for two-parameter martingales has been developed by several authors, in particular L. Chevalier, P. Imkeller, N. Frangos, D. Nualart, and M. Sanz. These authors have proved different types of Ito formulas for two-parameter processes, but they have considered mainly the case of martingales with continuous paths. One of the basic ingredients in proving these two-parameter versions of the Ito formula is the notion of quadratic variation. In the non-continuous case the situation is much more complicated and, except in some special examples (like the Poisson case), stochastic calculus has not yet been developed. The main objective of this monography is to show the existence of the quadratic variation for a (not necessarily continuous) two-parameter martingale bounded in L \(log^+ L\). To accomplish this goal the author introduces in a clear and detailed way all the necessary notions and results on two-parameter processes which are relevant to the problem. In that sense this work can be regarded as an updated presentation of the fundamental notions of the general theory of two-parameter processes. Chapter I is devoted to pro…
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60G07
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 192999
zbMATH Keywords: general theory of two-parameter processes
zbMATH Keywords: two-parameter martingales
zbMATH Keywords: two-parameter versions of the Ito formula
zbMATH Keywords: decomposition theorems
zbMATH Keywords: martingale with no quadratic variation
reviewed by: David Nualart
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0078096
OpenAlex ID: W1607893001
DOI: 10.1007/BFB0078096
description: scientific article; zbMATH DE number 192999