Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The Bismut–Elworthy–Li formula for mean-field stochastic differential equations

David Baños

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Abstract

We generalise the so-called Bismut–Elworthy–Li formula to a class of stochastic differential equations whose coefficients might depend on the law of the solution. We give some examples of where this formula can be applied to in the context of finance and the computation of Greeks and provide a simple but rather illustrative simulation experiment showing that the use of the Bismut–Elworthy–Li formula, also known as Malliavin method, is more efficient compared to the finite difference method.

Résumé

Nous généralisons la formule dite Bismut–Elworthy–Li à une classe d’équations différentielles stochastiques dont les coefficients pourrait dépendre de la loi de la solution. Nous donnons quelques exemples où cette formule peut être appliquée dans le contexte de la finance et le calcul des Grecs et de fournir une expérience de simulation simple mais significative montrant que l’utilisation de la formule Bismut–Elworthy–Li, également connu comme méthode de Malliavin, est plus efficace que la méthode des différences finies.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 220-233.

Dates
Received: 26 October 2015
Revised: 5 October 2016
Accepted: 11 October 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030826

Digital Object Identifier
doi:10.1214/16-AIHP801

Mathematical Reviews number (MathSciNet)
MR3765887

Zentralblatt MATH identifier
06880052

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65] 65C05: Monte Carlo methods

Keywords
Bismut–Elworthy–Li formula Malliavin calculus Monte Carlo methods Stochastic differential equations Integration by parts formulas

Citation

Baños, David. The Bismut–Elworthy–Li formula for mean-field stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 220--233. doi:10.1214/16-AIHP801. https://projecteuclid.org/euclid.aihp/1519030826


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