Penalising a process is to modify its distribution with a limiting procedure, thus defining a new process whose properties differ somewhat from those of the original one. We are presenting a number of examples of such penalisations in the Brownian and Bessel processes framework. The Martingale theory plays a crucial role. A general principle for penalisation emerges from these examples. In particular, it is shown in the Brownian framework that a positive sigma-finite measure takes a large class of penalisations into account.
Some penalisations of theWiener measure.- Feynman-Kac penalisations for Brownian motion.- Penalisations of a Bessel process with dimension d(0 d 2) by a function of the ranked lengths of its excursions.- A general principle and some questions about penalisations. From the reviews: "In this book the authors give a systematic study of penalisation. The book is divided into 5 chapters. ... This book is very useful for graduate students and researchers interested in learning penalisations." (Ren Ming Song, Mathematical Reviews, Issue 2010 e)
