Electronic Journal of Probability

Weak error for the Euler scheme approximation of diffusions with non-smooth coefficients

Valentin Konakov and Stéphane Menozzi

Full-text: Open access

Abstract

We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of Hölder continuous coefficients as well as piecewise smooth drifts with smooth diffusion matrices.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 46, 47 pp.

Dates
Received: 4 April 2016
Accepted: 28 March 2017
First available in Project Euclid: 11 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1494489631

Digital Object Identifier
doi:10.1214/17-EJP53

Mathematical Reviews number (MathSciNet)
MR3661660

Zentralblatt MATH identifier
1365.60065

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 65C30: Stochastic differential and integral equations

Keywords
diffusion processes Euler scheme parametrix Hölder coefficients Piecewise smooth bounded drifts

Rights
Creative Commons Attribution 4.0 International License.

Citation

Konakov, Valentin; Menozzi, Stéphane. Weak error for the Euler scheme approximation of diffusions with non-smooth coefficients. Electron. J. Probab. 22 (2017), paper no. 46, 47 pp. doi:10.1214/17-EJP53. https://projecteuclid.org/euclid.ejp/1494489631


Export citation

References

  • [Alf05] A. Alfonsi. On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl., 11(4):355–384, 2005.
  • [AJKH14] A. Alfonsi, B. Jourdain, and A. Kohatsu-Higa. Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme. Ann. Appl. Probab., 24(3):1049–1080, 2014.
  • [BT96a] V. Bally and D. Talay. The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function. Prob. Th. Rel. Fields, 104–1:43–60, 1996.
  • [BT96b] V. Bally and D. Talay. The law of the Euler scheme for stochastic differential equations, II. Convergence rate of the density. Monte-Carlo methods and Appl., 2:93–128, 1996.
  • [BP09] R.F. Bass and E.A. Perkins. A new technique for proving uniqueness for martingale problems. From Probability to Geometry (I): Volume in Honor of the 60th Birthday of Jean-Michel Bismut, pages 47–53, 2009.
  • [BBD08] Abdel Berkaoui, Mireille Bossy, and Awa Diop. Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence. ESAIM Probab. Stat., 12:1–11 (electronic), 2008.
  • [DM10] F. Delarue and S. Menozzi. Density estimates for a random noise propagating through a chain of differential equations. Journal of Functional Analysis, 259–6:1577–1630, 2010.
  • [Fri64] A. Friedman. Partial Differential Equations of Parabolic Type. Prentice-Hall, 1964.
  • [GT98] D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order, Second edition. Springer Verlag, 1998.
  • [GR14] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. D. Zwillinger and V. Moll (eds.) Eighth edition, 2014.
  • [GR11] I. Gyöngy and M. Rásonyi. A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients. Stochastic Process. Appl., 121(10):2189–2200, 2011.
  • [IKO62] A. M. Il’in, A. S. Kalashnikov, and O. A. Oleinik. Second-order linear equations of parabolic type. Uspehi Mat. Nauk, 17–3(105):3–146, 1962.
  • [KHLY15] A. Kohatsu-Higa, A. Lejay, and K. Yasuda. Weak Approximation Errors for Stochastic Differential Equations with Non-Regular Drifts. hal-00840211, 2015.
  • [KKM16] V. Konakov, A. Kozhina, and S. Menozzi. Stability of Densities for Perturbed Diffusions and Markov Chains. ESAIM Proba. and Stat., 21:88–112, 2017.
  • [KM00] V. Konakov and E. Mammen. Local limit theorems for transition densities of Markov chains converging to diffusions. Prob. Th. Rel. Fields, 117:551–587, 2000.
  • [KM02] V. Konakov and E. Mammen. Edgeworth type expansions for Euler schemes for stochastic differential equations. Monte Carlo Methods Appl., 8–3:271–285, 2002.
  • [KM15] V. Konakov and A. Markova. Linear trend exclusion for models defined with stochastic differential and difference equations. Automation and Remote Control, 76–10:1771–1783, 2015.
  • [KMM10] V. Konakov, S. Menozzi, and S. Molchanov. Explicit parametrix and local limit theorems for some degenerate diffusion processes. Annales de l’Institut Henri Poincaré, Série B, 46–4:908–923, 2010.
  • [Koz16] A. Kozhina. Stability of densities for perturbed degenerate diffusions. arXiv:1602.04770, To appear in Theory of Prob. and Appl., 2016.
  • [LSU68] O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Ural’ceva. Linear and quasi-linear equations of parabolic type. Vol.23 Trans. Math. Monog., AMS, Providence, 1968.
  • [LM10] V. Lemaire and S. Menozzi. On some non asymptotic bounds for the Euler scheme. Electronic Journal of Probability, 15:1645–1681, 2010.
  • [MS67] H. P. McKean and I. M. Singer. Curvature and the eigenvalues of the Laplacian. J. Differential Geometry, 1:43–69, 1967.
  • [Men11] S. Menozzi. Parametrix techniques and martingale problems for some degenerate Kolmogorov equations. Electronic Communications in Probability, 17:234–250, 2011.
  • [Mik12] R. Mikulevicius. On the rate of convergence of simple and jump-adapted weak Euler schemes for Lévy driven SDEs. Stochastic Process. Appl., 122(7):2730–2757, 2012.
  • [MP91] R. Mikulevičius and E. Platen. Rate of convergence of the Euler approximation for diffusion processes. Math. Nachr., 151:233–239, 1991.
  • [MZ15] R. Mikulevičius and C. Zhang. Weak Euler approximation for Itô diffusion and jump processes. Stoch. Anal. Appl., 33(3):549–571, 2015.
  • [She91] S. J. Sheu. Some estimates of the transition density of a nondegenerate diffusion Markov process. Ann. Probab., 19–2:538–561, 1991.
  • [TT90] D. Talay and L. Tubaro. Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. and App., 8–4:94–120, 1990.
  • [Zyg36] A. Zygmund. Trigonometric Series. Cambridge University Press, 1936.