## The Annals of Applied Probability

### Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme

#### Abstract

In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its Euler discretization with $N$ steps is smaller than $O(N^{-2/3+\varepsilon})$ where $\varepsilon$ is an arbitrary positive constant. This rate is intermediate between the strong error estimation in $O(N^{-1/2})$ obtained when coupling the stochastic differential equation and the Euler scheme with the same Brownian motion and the weak error estimation $O(N^{-1})$ obtained when comparing the expectations of the same function of the diffusion and of the Euler scheme at the terminal time $T$. We also check that the supremum over $t\in[0,T]$ of the Wasserstein distance on the space of probability measures on the real line between the laws of the diffusion at time $t$ and the Euler scheme at time $t$ behaves like $O(\sqrt{\log(N)}N^{-1})$.

#### Article information

Source
Ann. Appl. Probab., Volume 24, Number 3 (2014), 1049-1080.

Dates
First available in Project Euclid: 23 April 2014

https://projecteuclid.org/euclid.aoap/1398258095

Digital Object Identifier
doi:10.1214/13-AAP941

Mathematical Reviews number (MathSciNet)
MR3199980

Zentralblatt MATH identifier
1296.65010

#### Citation

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. doi:10.1214/13-AAP941. https://projecteuclid.org/euclid.aoap/1398258095

#### References

• [1] Aronson, D. G. (1967). Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. (N.S.) 73 890–896.
• [2] Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density. Monte Carlo Methods Appl. 2 93–128.
• [3] Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields 104 43–60.
• [4] Cruzeiro, A. B., Malliavin, P. and Thalmaier, A. (2004). Geometrization of Monte-Carlo numerical analysis of an elliptic operator: Strong approximation. C. R. Math. Acad. Sci. Paris 338 481–486.
• [5] Fitzsimmons, P., Pitman, J. and Yor, M. (1993). Markovian bridges: Construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992) 101–134. Birkhäuser, Boston, MA.
• [6] Friedman, A. (1975). Stochastic Differential Equations and Applications, Vol. 1. Probability and Mathematical Statistics 28. Academic Press, New York.
• [7] Gīhman, Ĭ. Ī. andSkorohod, A. V. (1972). Stochastic Differential Equations. Ergebnisse der Mathematik und ihrer Grenzgebiete 72. Springer, New York.
• [8] Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 607–617.
• [9] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering: Stochastic Modelling and Applied Probability. Applications of Mathematics (New York) 53. Springer, New York.
• [10] Gobet, E. (2000). Weak approximation of killed diffusion using Euler schemes. Stochastic Process. Appl. 87 167–197.
• [11] Gobet, E. (2001). Euler schemes and half-space approximation for the simulation of diffusion in a domain. ESAIM Probab. Stat. 5 261–297 (electronic).
• [12] Gobet, E. and Labart, C. (2008). Sharp estimates for the convergence of the density of the Euler scheme in small time. Electron. Commun. Probab. 13 352–363.
• [13] Gobet, E. and Menozzi, S. (2004). Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme. Stochastic Process. Appl. 112 201–223.
• [14] Guyon, J. (2006). Euler scheme and tempered distributions. Stochastic Process. Appl. 116 877–904.
• [15] Gyöngy, I. (1986). Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probab. Theory Related Fields 71 501–516.
• [16] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
• [17] Jourdain, B. and Sbai, M. (2013). High order discretization schemes for stochastic volatility models. Journal of Computational Finance. To appear.
• [18] Kanagawa, S. (1988). On the rate of convergence for Maruyama’s approximate solutions of stochastic differential equations. Yokohama Math. J. 36 79–86.
• [19] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
• [20] Kurtz, T. G. and Protter, P. (1991). Wong–Zakai corrections, random evolutions, and simulation schemes for SDEs. In Stochastic Analysis 331–346. Academic Press, Boston, MA.
• [21] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035–1070.
• [22] Lemaire, V. and Menozzi, S. (2010). On some non asymptotic bounds for the Euler scheme. Electron. J. Probab. 15 1645–1681.
• [23] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
• [24] Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems. Springer, Berlin.
• [25] Rogers, L. C. G. (1985). Smooth transition densities for one-dimensional diffusions. Bull. Lond. Math. Soc. 17 157–161.
• [26] Sbai, M. (2009). Modélisation de la dépendance et simulation de processus en finance. Ph.D. thesis, Univ. Paris-Est, available at http://tel.archives-ouvertes.fr/tel-00451008/en/.
• [27] Seumen Tonou, P. (1997). Méthodes numériques probabilistes pour la résolution d’équations du transport et pour l’évaluation d’options exotiques. Ph.D. thesis, Univ. Aix-Marseille 1.
• [28] Talay, D. and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 483–509.
• [29] Temam, E. (2001). Couverture approchée d’options exotiques. Pricing des options asiatiques. Ph.D. thesis, Univ. Paris 6.
• [30] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.