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February 2018 The Bismut–Elworthy–Li formula for mean-field stochastic differential equations
David Baños
Ann. Inst. H. Poincaré Probab. Statist. 54(1): 220-233 (February 2018). DOI: 10.1214/16-AIHP801


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.

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.


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David Baños. "The Bismut–Elworthy–Li formula for mean-field stochastic differential equations." Ann. Inst. H. Poincaré Probab. Statist. 54 (1) 220 - 233, February 2018.


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

zbMATH: 06880052
MathSciNet: MR3765887
Digital Object Identifier: 10.1214/16-AIHP801

Primary: 60H07 , 60H10 , 60J60 , 65C05

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

Rights: Copyright © 2018 Institut Henri Poincaré


Vol.54 • No. 1 • February 2018
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