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

ESAIM: Probability and Statistics (2010)

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

## Access Full Article

top## Abstract

top## How to cite

topRoynette, 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

top- W. Feller, An introduction to probability theory and its applications, volume II. John Wiley & Sons Inc., New York (1966). Zbl0138.10207
- Y. Hariya and M. Yor, Limiting distributions associated with moments of exponential Brownian functionals. Studia Sci. Math. Hung.41 (2004) 193–242. Zbl1115.60005
- T. Jeulin, Semimartingales et grossissement d'une filtration. Lect. Notes Maths 833. Springer (1980). Zbl0444.60002
- T. Jeulin and M. Yor, Eds., Grossissements de filtrations: exemples et applications. Lect. Notes Maths 1118. Springer (1985).
- I. Karatzas and S. Shreve, Brownian motion and Stochastic Calculus. Springer (1991). Zbl0734.60060
- S. Kotani, Asymptotics for expectations of multiplicative functionals of 1-dimensional Brownian motion. Preprint (2006).
- N.N. Lebedev, Special functions and their applications. Dover (1972). Zbl0271.33001
- R. Mansuy and M. Yor, Random Times and Enlargement of Filtrations in a Brownian Setting. Lect. Notes Maths 1873. Springer (2006). Zbl1103.60003
- J. Najnudel, B. Roynette and M. Yor, A global view of Brownian penalisations. MSJ Memoirs, volume 19. Mathematical Society of Japan, Tokyo (2009). Zbl1180.60004
- D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Third edition. Springer (1999). Zbl0917.60006
- B. Roynette and M. Yor, Penalising Brownian paths. Lect. Notes Maths 1969. Springer (2009). Zbl1190.60002
- 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
- B. Roynette, P. Vallois and M. Yor, Some penalisations of the Wiener measure. Japan J. Math.1 (2006) 263–299.
- 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
- 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
- M. Yor, The distribution of Brownian quantiles. J. Appl. Prob.32 (1995) 405–416. Zbl0829.60065