Estimating multidimensional density functions for random variables in Wiener space

A. Kohatsu-Higa; Kazuhiro Yasuda

doi:10.1016/j.crma.2008.01.009

Probability Theory/Numerical Analysis

Estimating multidimensional density functions for random variables in Wiener space

Presented by: Paul Malliavin

A. Kohatsu-Higa ¹; Kazuhiro Yasuda ¹

¹ Graduate school of Engineering Science, Osaka University. 1-3 Machikaneyama-cho, Toyonaka City, Osaka 560-8531, Japan

Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 335-338.

Abstract (Original language)
Résumé

The Malliavin–Thalmaier (Springer Finance, Springer-Verlag, Berlin, 2006) formula was introduced for the simulation of high dimensional probability density functions. However, when this integration by parts formula is applied directly in computer simulations, we show that it is unstable. We propose an approximation to the Malliavin–Thalmaier formula. In this Note, we find the order of the bias and the variance of the approximation error, and we apply the Malliavin–Thalmaier formula for the calculation of Greeks in finance.

La formule de Malliavin–Thalmaier a été présentée (Springer Finance, Springer-Verlag, Berlin, 2006) pour la simulation des fonctions de densité multidimensionnelles. Mais quand cette formule d'intégration par parties est appliquée directement pour la simulation sur ordinateur, nous prouvons qu'elle est instable. Nous proposons une approximation à la formule de Malliavin–Thalmaier. Dans cette Note, nous trouvons l'ordre du biais et la variance de l'erreur d'approximation. Nous appliquons la formula de Malliavin–Thalmaier pour le calcul des Grecques en Finance.

Received: 2007-04-19
Accepted: 2008-01-14
Published online: 2008-02-07

DOI: 10.1016/j.crma.2008.01.009

Author's affiliations:

A. Kohatsu-Higa ¹; Kazuhiro Yasuda ¹

¹ Graduate school of Engineering Science, Osaka University. 1-3 Machikaneyama-cho, Toyonaka City, Osaka 560-8531, Japan

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     author = {A. Kohatsu-Higa and Kazuhiro Yasuda},
     title = {Estimating multidimensional density functions for random variables in {Wiener} space},
     journal = {Comptes Rendus. Math\'ematique},
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     year = {2008},
     doi = {10.1016/j.crma.2008.01.009},
     language = {en},
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A. Kohatsu-Higa; Kazuhiro Yasuda. Estimating multidimensional density functions for random variables in Wiener space. Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 335-338. doi : 10.1016/j.crma.2008.01.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.01.009/

References
Cited by

[1] V. Bally, L. Caramellino, Lower bounds for the density of Ito processes under weak regularity assumptions, preprint

[2] E. Fournié; J.-M. Lasry; J. Lebuchoux; P.-L. Lions; N. Touzi Applications of Malliavin calculus to Monte Carlo methods in finance, Finance Stoch., Volume 3 (1999) no. 4, pp. 391-412

[3] A. Kohatsu-Higa, K. Yasuda, Estimating multidimensional density functions using the Malliavin–Thalmaier formula, preprint

[4] P. Malliavin; A. Thalmaier Stochastic Calculus of Variations in Mathematical Finance, Springer Finance, Springer-Verlag, Berlin, 2006

[5] D. Nualart The Malliavin Calculus and Related Topics, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006

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