Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

David Baños

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • [1] D. Baños and T. Nilssen. Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation. Stochastics 88 (4) (2016) 540–566.
  • [2] J. M. Bismut. Large Deviation and Malliavin Calculus. Progress in Mathematics 45. Birkhäuser, 1984.
  • [3] R. Buckdahn, J. Li, P. Peng and C. Rainer. Mean-field stochastic differential equations and associated PDEs. Available at arXiv:1407.1215 [math.PR].
  • [4] G. Di Nunno, B. Øksendal and F. Proske. Malliavin Calculus for Lévy Processes with Applications to Finance. Springer, 2008.
  • [5] K. D. Elworthy and X.-M. Li. Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125 (1994) 252–286.
  • [6] E. Fournié, J.-M. Lasry, J. Lebuchoux, P.-L. Lions and N. Touzi Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch. 3 (4) (1999) 391–412.
  • [7] J. E. Hutton and P. I. Nelson. Interchanging the order of differentiation and stochastic integration. Stochastic Process. Appl. 18 (1984) 371–377.
  • [8] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. I. In Stochastic Analysis (Katata/Kyoto, 1982) 271–306. North-Holland Math. Library 32. North-Holland, Amsterdam, 1984.
  • [9] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. II. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 32 (1) (1985) 1–76.
  • [10] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. III. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 34 (2) (1987) 391–442.
  • [11] P. Malliavin. Stochastic calculus of variations and hypoelliptic operators. In Proc. Inter. Symp. on Stoch. Diff. Equations (Kyoto, 1976) 195–263. Wiley, 1978.
  • [12] P. Malliavin. Stochastic Analysis. Springer, 1997.
  • [13] X. Mao. Stochastic Differential Equations and Applications. Woodhead Publishing, Glasgow, 2007.
  • [14] E. McMurray. Regularity of McKean-Vlasov Stochastic Differential Equations and Applications. Ph.D. thesis, Imperial College, London.
  • [15] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Springer, 2010.
  • [16] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Applications of Mathematics (New York) 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004.