This paper examines the question of when a two-parameter process $X$ of independent increments will have Levy's sharp Markov property relative to a given domain $D$. This property states intuitively that the values of the process inside $D$ and outside $D$ are conditionally independent given the values of the process on the boundary of $D$. Under mild assumptions, $X$ is the sum of a continuous Gaussian process and an independent jump process. We show that if $X$ satisfies Levy's sharp Markov property, so do both the Gaussian and the jump process. The Gaussian case has been studied in a previous paper by the same authors. Here, we examine the case where $X$ is a jump process. The presence of discontinuities requires a new formulation of the sharp Markov property. The main result is that a jump process satisfies the sharp Markov property for all bounded open sets. This proves a generalization of a conjecture of Carnal and Walsh concerning the Poisson sheet.
Publié le : 1992-04-14
Classification:
Levy process,
Levy sheet,
Brownian sheet,
Poisson sheet,
Markov property,
sharp field,
60G60,
60G55,
60J75,
60J30,
60E07,
35R60,
60H15
@article{1176989793,
author = {Dalang, Robert C. and Walsh, John B.},
title = {The Sharp Markov Property of Levy Sheets},
journal = {Ann. Probab.},
volume = {20},
number = {4},
year = {1992},
pages = { 591-626},
language = {en},
url = {http://dml.mathdoc.fr/item/1176989793}
}
Dalang, Robert C.; Walsh, John B. The Sharp Markov Property of Levy Sheets. Ann. Probab., Tome 20 (1992) no. 4, pp. 591-626. http://gdmltest.u-ga.fr/item/1176989793/