Perfect cocycles through stochastic differential equations
type: Item
instance of: scholarly article
title: Perfect cocycles through stochastic differential equations
zbMATH Open document ID: 0821.60061
DOI: 10.1007/BF01192196
author: Michael Scheutzow
author: Ludwig Arnold
publication date: 1995-09-18T00:00:00Z
review text: Suppose \(\varphi\) is a continuous random dynamical system on \(\mathbb{R}^ d\). This is essentially a skew product flow consisting of random diffeomorphisms over \(\mathbb{R}^ d\), continuous in time and space, over a measurable base flow. The paper has two main results. First it is shown that \(\varphi\) is the solution of a stochastic differential equation with a vector field valued semimartingale with stationary increments as driving process, if and only if \(t \mapsto \varphi (t, \omega) x\) is a semimartingale for every \(x \in \mathbb{R}^ d\). This is worked out for two- sided time (where usually stochastic analysis is concerned with nonnegative time \(\mathbb{R}^ +\) only). Then it is proved that a (very general) crude cocycle with a locally compact group as `time', taking values in a countably generated Hausdorff group, can be perfected.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 93E03
zbMATH DE Number: 739104
zbMATH Keywords: continuous random dynamical system
zbMATH Keywords: random diffeomorphisms
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: countably generated Hausdorff group
reviewed by: Hans Crauel
MaRDI profile type: Publication
cites work: Q4847944
cites work: Q3338999
cites work: Mécanique aléatoire
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 739104