Probability Theory
Improving Monte Carlo simulations by Dirichlet forms

Presented by: Paul Malliavin

Bouleau, Nicolas1
1 ENPC, ParisTech, 51, rue Gerard, 75013 Paris, France
Comptes Rendus. Mathématique, Volume 341 (2005) no. 5, pp. 303-306.

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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.07.017
Bouleau, Nicolas 1

1 ENPC, ParisTech, 51, rue Gerard, 75013 Paris, France 
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Bouleau, Nicolas. 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://www.numdam.org/articles/10.1016/j.crma.2005.07.017/

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