The sharp Markov property of Lévy sheets
type: Item
instance of: scholarly article
title: The sharp Markov property of Lévy sheets
zbMATH Open document ID: 0783.60049
author: Robert C. Dalang
author: John B. Walsh
published in: The Annals of Probability
publication date: 1993-01-16T00:00:00Z
review text: This article extends the work of the authors on the strong Markov property (SMP) of the Brownian sheet [Acta Math. 168, No. 3/4, 153-218 (1992; Zbl 0759.60056)] to two-parameter Lévy processes \(X\). They decompose \(X\) into a Gaussian and a jump part and show that the second part satisfies the SMP with respect to all open sets which are bounded or have bounded complement, where the sharp field (i.e. the field generated by \(X)\) has to be replaced by the uniform sharp field if the sign of the jumps is not constant. This result extends the SMP of the Poisson sheet proved by \textit{E. Carnal} and \textit{J. B. Walsh} [Stochastic analysis, Proc. Conf. Honor Moske Zakai 65th Birthday, Haifa/Isr. 1991, 91-110 (1991; Zbl 0734.60052)] and shows clearly that the SMP is a less restrictive property for jump processes than for continuous processes. Moreover the authors prove that the SMP of \(X\) implies the SMP of both components in the decomposition. They also consider the minimal splitting field.
reviewed by: M.Dozzi
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 89777
zbMATH Keywords: Brownian sheet
zbMATH Keywords: two-parameter Lévy processes
zbMATH Keywords: sharp field
zbMATH Keywords: Poisson sheet
zbMATH Keywords: splitting field
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989793
OpenAlex ID: W2041318105
DOI: 10.1214/AOP/1176989793
description: scientific article; zbMATH DE number 89777