| Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time II (2007) | |||||||||||||||
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| LIMITING LAWS ASSOCIATED WITH BROWNIAN MOTION PERTURBED ITS MAXIMUM MINIMUM AND LOCAL TIME Bernard ROYNETTE Pierre VALLOIS and Marc YOR October ccsd version Oct Universit Henri Poincar Institut Math matiques Elie Cartan F Vand uvree s Nancy Cedex Laboratoire Probabilit Mod les atoires Universit Paris VII Place Jussieu e Case Paris Cedex Institut Universitaire France Abstract Let denote the Wiener measure ned the canonical space R Ft and resp be the one sided maximum resp minimum the local time and the number down crossings from with Let Borel function and process chosen within the set St Lt It t which consists elements prove penalization result under some suitable assumptions there exists positive martingale Mtf starting such that lim E E s and determine the law under the de ned For the and elements the set prove rst that and more generally where sup A with the convention sup Secondly split the trajectory two parts g and and describe their laws under conditionally For the and elements similar result holds replacing resp I Key words and phrases penalization enlargement ltration maximum minimum local time down crossings AMS subject classi cations G J Introduction Let the family Wiener measures ned the canonical space R Ft adapted non negative process such that for any R associate the probability measure ned follows QF Ft t A priori the family not consistent may erent from t for and fact easy see that consistent and only QF well ned since Ft Fs and t previous study have consider | |||||||||||||||
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