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Martingale-valued Measures, Ornstein-Uhlenbeck Processes with Jumps

Summary: Martingale-valued Measures,
Ornstein-Uhlenbeck Processes with Jumps
and Operator Self-Decomposability in Hilbert Space
David Applebaum
Probability and Statistics Department, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield,
England, S3 7RH, e-mail: D.Applebaum@sheffield.ac.uk
Dedicated to the memory of Paul-AndrŽe Meyer
Summary. We investigate a class of Hilbert space valued martingale-valued measures whose covariance
structure is determined by a trace class positive operator valued measure. The paradigm example is the mar-
tingale part of a LŽevy process. We develop both weak and strong stochastic integration with respect to such
martingale-valued measures. As an application, we investigate the stochastic convolution of a C0-semigroup
with a LŽevy process and the associated Ornstein-Uhlenbeck process. We give an infinite dimensional gen-
eralisation of the concept of operator self-decomposability and find conditions for random variables of this
type to be embedded into a stationary Ornstein-Uhlenbeck process.
Key Words and Phrases:- martingale-valued measure, positive operator valued mea-
sure, trace class operator, nuclear, decomposable, LŽevy process, C0-semigroup, sto-
chastic convolution, Ornstein-Uhlenbeck process, operator self-decomposability, expo-
nentially stable semigroup.
1 Introduction
The aim of this paper is to introduce some new concepts into stochastic analysis of


Source: Applebaum, David - Department of Probability and Statistics, University of Sheffield


Collections: Mathematics