Electronic Journal of Probability

Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme

Aurélien Alfonsi, Benjamin Jourdain, and Arturo Kohatsu-Higa

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In this paper, we prove that the time supremum of the Wasserstein distance between the time-marginals of a uniformly elliptic multidimensional diffusion with coefficients bounded together with their derivatives up to the order $2$ in the spatial variables and Hölder continuous with exponent $\gamma$ with respect to the time variable and its Euler scheme with $N$ uniform time-steps is smaller than $C(1+\mathbf{1}_{\gamma=1} \sqrt{\ln(N)})N^{-\gamma}$. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio et al. to compute the time derivative of the Wasserstein distance between the time-marginals. We deduce a stability inequality for the Wasserstein distance which finally leads to the desired estimation.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 70, 31 pp.

Accepted: 23 June 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C30: Stochastic differential and integral equations
Secondary: 60H35: Computational methods for stochastic equations [See also 65C30] 49K99: None of the above, but in this section

Euler scheme Wasserstein distance Optimal transport

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Alfonsi, Aurélien; Jourdain, Benjamin; Kohatsu-Higa, Arturo. Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme. Electron. J. Probab. 20 (2015), paper no. 70, 31 pp. doi:10.1214/EJP.v20-4195. https://projecteuclid.org/euclid.ejp/1465067176

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  • Alfonsi, A.; Jourdain, B.; Kohatsu-Higa, A. Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme. Ann. Appl. Probab. 24 (2014), no. 3, 1049–1080.
  • Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe. Gradient flows in metric spaces and in the space of probability measures. Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. x+334 pp. ISBN: 978-3-7643-8721-1
  • Bolley, François; Gentil, Ivan; Guillin, Arnaud. Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations. J. Funct. Anal. 263 (2012), no. 8, 2430–2457.
  • Bolley, François; Gentil, Ivan; Guillin, Arnaud. Uniform convergence to equilibrium for granular media. Arch. Ration. Mech. Anal. 208 (2013), no. 2, 429–445.
  • Cannarsa, Piermarco; Sinestrari, Carlo. Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston, Inc., Boston, MA, 2004. xiv+304 pp. ISBN: 0-8176-4084-3
  • Dudley, R. M. On second derivatives of convex functions. Math. Scand. 41 (1977), no. 1, 159–174.
  • Faure, O. Simulation du mouvement brownien et des diffusions. PhD Thesis of the Ecole Nationale des Ponts et Chaussées, available at texttt http://pastel.archives-ouvertes.fr
  • Figalli, Alessio; Gigli, Nicola. Local semiconvexity of Kantorovich potentials on non-compact manifolds. ESAIM Control Optim. Calc. Var. 17 (2011), no. 3, 648–653.
  • Friedman, Avner. Stochastic differential equations and applications. Vol. 1. Probability and Mathematical Statistics, Vol. 28. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. xiii+231 pp.
  • Gobet, Emmanuel; Labart, Céline. Sharp estimates for the convergence of the density of the Euler scheme in small time. Electron. Commun. Probab. 13 (2008), 352–363.
  • Gyongy, I. Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probab. Theory Relat. Fields 71 (1986), no. 4, 501–516.
  • Kanagawa, Shëya. On the rate of convergence for Maruyama's approximate solutions of stochastic differential equations. Yokohama Math. J. 36 (1988), no. 1, 79–86.
  • Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. xiv+415 pp. ISBN: 3-540-00313-4
  • Rachev, S.T. and Rüschendorf, L. Mass Transportation problems. Springer-Verlag, 1998.
  • Sbai, M. Modélisation de la dépendance et simulation de processus en finance. PhD thesis, Université Paris-Est (2009), http://tel.archives-ouvertes.fr/tel-00451008/en/.
  • Talay, Denis; Tubaro, Luciano. Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8 (1990), no. 4, 483–509 (1991).
  • Villani, Cédric. Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. ISBN: 978-3-540-71049-3
  • Yan, Liqing. The Euler scheme with irregular coefficients. Ann. Probab. 30 (2002), no. 3, 1172–1194.