 
Summary: Martingalevalued Measures,
OrnsteinUhlenbeck Processes with Jumps
and Operator SelfDecomposability in Hilbert Space
David Applebaum
Probability and Statistics Department, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield,
England, S3 7RH, email: D.Applebaum@sheffield.ac.uk
Dedicated to the memory of PaulAndrŽe Meyer
Summary. We investigate a class of Hilbert space valued martingalevalued 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
martingalevalued measures. As an application, we investigate the stochastic convolution of a C0semigroup
with a LŽevy process and the associated OrnsteinUhlenbeck process. We give an infinite dimensional gen
eralisation of the concept of operator selfdecomposability and find conditions for random variables of this
type to be embedded into a stationary OrnsteinUhlenbeck process.
Key Words and Phrases: martingalevalued measure, positive operator valued mea
sure, trace class operator, nuclear, decomposable, LŽevy process, C0semigroup, sto
chastic convolution, OrnsteinUhlenbeck process, operator selfdecomposability, expo
nentially stable semigroup.
1 Introduction
The aim of this paper is to introduce some new concepts into stochastic analysis of
