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limit theorems for additive functionals

ESAIM: Probability and Statistics (1)

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Local limit theorems for Brownian additive functionalsand penalisation of Brownian paths, IX
Bernard Roynette, Marc Yor
Journal:

ESAIM: Probability and Statistics / Volume 14 / 2010

Published online by Cambridge University Press:

15 December 2008, pp. 65-92

Print publication:

2010
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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]).