Penalising Brownian Paths
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 Bes
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1969
Bernard Roynette · Marc Yor
Penalising Brownian Paths
ABC
Bernard Roynette
Marc Yor
Université Nancy I Institut Elie Cartan Faculté des Sciences Département de Mathématiques B.P. 239 54506 Vandoeuvre-les-Nancy CX France [email protected]
Université Paris VI Laboratoire de Probabilités 175 rue de Chevaleret 75013 Paris France
ISBN: 978-3-540-89698-2 e-ISBN: 978-3-540-89699-9 DOI: 10.1007/978-3-540-89699-9 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008942371 Mathematics Subject Classification (2000): 60J65, 60F99, 60J25, 60G44, 60G30, 60J55 c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com
Contents
0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 The Penalisation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 Asymptotic Study of the Normalisation Factor . . . . . . . . . . . . . . . . . 0.3 From the Family (ν x , x ∈ R) to a Penalisation Theorem . . . . . . . . . 0.4 Penalisation and Conditioning by an Event of Probability 0 . . . . . . 0.5 Penalisation and Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 0.6 Penalisation as a Machine to Construct Martingales . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 3 8 13 23 27 33
1 1.1 1.2 1.3
35 35 36 40
Some Penalisations of the Wiener Measure . . . . . . . . . . . . . . . Introduction : A Rough Idea about Penalisation . . . . . . . . . . . . . . . . Some Meta-Theorems Leading to Penalised Probabilities . . . . . . . . Case I : Γt = St := sup Xs , or Γt = L0t (X) . . . . . . . . . . . . . . . . . . . . . . t s≤t 1.4 Case II : Γt = ds q(Xs ), with (Xs ) : BM(Rd ), d = 1, 2 . . . . . . . . 0
45
(δ)
1.5 Case III : Γt = L0t (Rt ), with R(δ) := BES(δ), 0 < δ < 2 . . . . . . . . 48 (),() 1.6 Case IV : Γt = Σt where Σt := sup (u − gu ), u≤gt
or Σt := sup(u − gu ), or Σt := sup (u − gu ) . . . . . . . . . . . . . . . . . . . . u
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