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

Bernard Roynette; Marc Yor

ESAIM: Probability and Statistics (2010)

  • Volume: 14, page 65-92
  • ISSN: 1292-8100

Abstract

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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 - : = ∫ 0 t 1 X s < 0 d s , t ≥ 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]).

How to cite

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Roynette, Bernard, and Yor, Marc. "Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX." ESAIM: Probability and Statistics 14 (2010): 65-92. <http://eudml.org/doc/252258>.

@article{Roynette2010,
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]). },
author = {Roynette, Bernard, Yor, Marc},
journal = {ESAIM: Probability and Statistics},
keywords = {Limit theorems for additive functionals; Feynman-Kac functionals; long Brownian bridges.; local limit theorem; Brownian additive functional; Feynman-Kac functional; penalisation problem; long Brownian bridge},
language = {eng},
month = {3},
pages = {65-92},
publisher = {EDP Sciences},
title = {Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX},
url = {http://eudml.org/doc/252258},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Roynette, Bernard
AU - Yor, Marc
TI - Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 14
SP - 65
EP - 92
AB - 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]).
LA - eng
KW - Limit theorems for additive functionals; Feynman-Kac functionals; long Brownian bridges.; local limit theorem; Brownian additive functional; Feynman-Kac functional; penalisation problem; long Brownian bridge
UR - http://eudml.org/doc/252258
ER -

References

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  10. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Third edition. Springer (1999).  Zbl0917.60006
  11. B. Roynette and M. Yor, Penalising Brownian paths. Lect. Notes Maths 1969. Springer (2009).  Zbl1190.60002
  12. B. Roynette, P. Vallois and M. Yor, Penalisation of a Brownian motion with drift by a function of its one-sided maximum and its position, III. Periodica Math. Hung.50 (2005) 247–280.  Zbl1150.60308
  13. B. Roynette, P. Vallois and M. Yor, Some penalisations of the Wiener measure. Japan J. Math.1 (2006) 263–299.  
  14. B. Roynette, P. Vallois and M. Yor, Limiting laws associated with Brownian motion perturbed by normalized exponential weights. Studia Sci. Math. Hung.43 (2006) 171–246.  Zbl1121.60027
  15. B. Roynette, P. Vallois and M. Yor, Limiting laws associated with Brownian motions perturbed by its maximum, minimum, and local time, II. Studia Sci. Math. Hung.43 (2006) 295–360.  Zbl1121.60004
  16. M. Yor, The distribution of Brownian quantiles. J. Appl. Prob.32 (1995) 405–416.  Zbl0829.60065

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