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Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX

Published online by Cambridge University Press:  15 December 2008

Bernard Roynette  and
Marc Yor
Bernard Roynette
Affiliation:
Institut Elie Cartan, Université Henri Poincaré, BP 239, 54506 Vandoeuvre-lés-Nancy Cedex, France
Marc Yor
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI et VII, 4 place Jussieu, Case 188, 75252 Paris Cedex 05, France Institut Universitaire de France, Paris, France
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Abstract

We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional: $(A_t^{-}:= \int_0^t 1_{X_s < 0}{\rm d}s, t\geq 0)$. On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [B. Roynette, P. Vallois and M. Yor, Studia Sci. Math. Hung.43 (2006) 171–246]).

Keywords

Limit theorems for additive functionalsFeynman-Kac functionalslong Brownian bridges.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 65 - 92
Copyright
© EDP Sciences, SMAI, 2010

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