Improving Monte Carlo simulations by Dirichlet forms

Nicolas Bouleau

doi:10.1016/j.crma.2005.07.017

Probability Theory

Improving Monte Carlo simulations by Dirichlet forms

Presented by: Paul Malliavin

Nicolas Bouleau ¹

¹ ENPC, ParisTech, 51, rue Gerard, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 5, pp. 303-306.

Abstract
Résumé

Equipping the probability space with a local Dirichlet form with square field operator Γ and generator A allows us to improve Monte Carlo simulations of expectations and densities as soon as we are able to simulate a random variable X together with $Γ [X]$ and $A [X]$ . We give examples on the Wiener space, on the Poisson space and on the Monte Carlo space. When X is real-valued we give an explicit formula yielding the density at the speed of the law of large numbers.

Nous montrons que, dans les situations où l'espace de probabilité est équipé d'une forme de Dirichlet locale avec carré du champ Γ et générateur A, la possibilité de simuler une variable aléatoire X ainsi que $Γ [X]$ et $A [X]$ permet d'accélérer le calcul de l'espérance de X et de sa densité. Nous donnons des exemples dans les cas de l'espace de Wiener, de l'espace de Poisson et de l'espace de Monte Carlo. Lorsque X est à valeurs réelles nous donnons une formule explicite permettant d'obtenir la densité à la vitesse de la loi des grands nombres.

Received: 2005-07-01
Accepted: 2005-07-20
Published online: 2005-09-01

DOI: 10.1016/j.crma.2005.07.017

Author's affiliations:

Nicolas Bouleau ¹

¹ ENPC, ParisTech, 51, rue Gerard, 75013 Paris, France

@article{CRMATH_2005__341_5_303_0,
     author = {Nicolas Bouleau},
     title = {Improving {Monte} {Carlo} simulations by {Dirichlet} forms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {303--306},
     publisher = {Elsevier},
     volume = {341},
     number = {5},
     year = {2005},
     doi = {10.1016/j.crma.2005.07.017},
     language = {en},
}

TY  - JOUR
AU  - Nicolas Bouleau
TI  - Improving Monte Carlo simulations by Dirichlet forms
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 303
EP  - 306
VL  - 341
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2005.07.017
LA  - en
ID  - CRMATH_2005__341_5_303_0
ER  -

%0 Journal Article
%A Nicolas Bouleau
%T Improving Monte Carlo simulations by Dirichlet forms
%J Comptes Rendus. Mathématique
%D 2005
%P 303-306
%V 341
%N 5
%I Elsevier
%R 10.1016/j.crma.2005.07.017
%G en
%F CRMATH_2005__341_5_303_0

Nicolas Bouleau. Improving Monte Carlo simulations by Dirichlet forms. Comptes Rendus. Mathématique, Volume 341 (2005) no. 5, pp. 303-306. doi : 10.1016/j.crma.2005.07.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.07.017/

References
Cited by

[1] V. Bally; D. Talay The law of the Euler scheme for stochastic differential equations: II. Convergence rate of the density, Monte Carlo Methods and Appl., Volume 104 (1996) no. 1, pp. 43-80

[2] N. Bouleau Error Calculus for Finance and Physics, the Language of Dirichlet Forms, De Gruyter, 2003

[3] N. Bouleau; F. Hirsch Dirichlet Forms and Analysis on Wiener Space, De Gruyter, 1991

[4] A. Kohatsu-Higa; R. Pettersson Variance reduction methods for simulation of densities on Wiener space, SIAM J. Numer. Anal., Volume 40 (2002) no. 2, pp. 431-450

[5] P. Malliavin; A. Thalmaier Numerical error for SDE: asymptotic expansion and hyperdistributions, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003), pp. 851-856

Cited by Sources:

Comments - Policy