Electronic Journal of Probability

Approximation of Markov semigroups in total variation distance

Vlad Bally and Clément Rey

Full-text: Open access


In this paper, we consider Markov chains of the form $X^{n}_{(k+1)/n}=\psi _{k}(X^{n}_{k/n},Z_{k+1}/\sqrt{n},1/n)$ where the innovation comes from the sequence $Z_{k},k\in \mathbb{N} ^{\ast }$ of independent centered random variables with arbitrary law. Then, we study the convergence $\mathbb{E} [f(X^{n}_t)]\rightarrow \mathbb{E} [f(X_t)]$ where $(X_t)_{t \geqslant 0}$ is a Markov process in continuous time. This may be considered as an invariance principle, which generalizes the classical Central Limit Theorem to Markov chains. Alternatively (and this is the main motivation of our paper), $X^{n}$ may be an approximation scheme used in order to compute $\mathbb{E} [f(X_t)]$ by Monte Carlo methods. Estimates of the error are given for smooth test functions $f$ as well as for measurable and bounded $f.$ In order to prove convergence for measurable test functions we assume that $Z_{k}$ satisfies Doeblin’s condition and we use Malliavin calculus type integration by parts formulas based on the smooth part of the law of $Z_{k}$. As an application, we will give estimates of the error in total variation distance for the Ninomiya Victoir scheme.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 12, 44 pp.

Received: 27 January 2015
Accepted: 27 December 2015
First available in Project Euclid: 17 February 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60H07: Stochastic calculus of variations and the Malliavin calculus 65C40: Computational Markov chains

approximation schemes Markov processes total variation distance invariance principles Malliavin Calculus

Creative Commons Attribution 4.0 International License.


Bally, Vlad; Rey, Clément. Approximation of Markov semigroups in total variation distance. Electron. J. Probab. 21 (2016), paper no. 12, 44 pp. doi:10.1214/16-EJP4079. https://projecteuclid.org/euclid.ejp/1455717196

Export citation


  • [1] A. Alfonsi, High order discretization schemes for the CIR process: application to affine term structure and Heston models, Math. Comp. 79 (2010), no. 269, 209–237.
  • [2] D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 348, Springer, Cham, 2014.
  • [3] V. Bally and L. Caramellino, On the distance between probability density functions, November 2013.
  • [4] V. Bally and L. Caramellino, Asymptotic development for the CLT in total variation distance, ArXiv e-prints (2014).
  • [5] V. Bally and E. Clément, Integration by parts formula and applications to equations with jumps, Probab. Theory Related Fields 151 (2011), no. 3-4, 613–657.
  • [6] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Related Fields 104 (1996), no. 1, 43–60.
  • [7] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density, Monte Carlo Methods Appl. 2 (1996), no. 2, 93–128.
  • [8] Vlad Bally and Lucia Caramellino, Convergence and regularity of probability laws by using an interpolation method, arXiv preprint arXiv:1409.3118 (2014).
  • [9] R. Bhattacharya and R. Rao, Normal approximation and asymptotic expansions, Society for Industrial and Applied Mathematics, 2010.
  • [10] Sergey G. Bobkov, Gennadiy P. Chistyakov, and Friedrich Götze, Berry–Esseen bounds in the entropic central limit theorem, Probab. Theory Related Fields 159 (2014), no. 3-4, 435–478.
  • [11] Sergey G. Bobkov, Gennadiy P. Chistyakov, and Friedrich Götze, Fisher information and the central limit theorem, Probab. Theory Related Fields 159 (2014), no. 1-2, 1–59.
  • [12] M. Bossy, E. Gobet, and D. Talay, A symmetrized Euler scheme for an efficient approximation of reflected diffusions, J. Appl. Probab. 41 (2004), no. 3, 877–889.
  • [13] E. Gobet, Weak approximation of killed diffusion using Euler schemes, Stochastic Process. Appl. 87 (2000), no. 2, 167–197.
  • [14] Emmanuel Gobet and Stéphane Menozzi, Stopped diffusion processes: boundary corrections and overshoot, Stochastic Process. Appl. 120 (2010), no. 2, 130–162.
  • [15] J. Guyon, Euler scheme and tempered distributions, Stochastic Process. Appl. 116 (2006), no. 6, 877–904.
  • [16] J. Jacod, T. G. Kurtz, S. Méléard, and P. Protter, The approximate Euler method for Lévy driven stochastic differential equations, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 3, 523–558.
  • [17] B. Jourdain and A. Kohatsu-Higa, A review of recent results on approximation of solutions of stochastic differential equations, Progress in Probability, vol. 65, Springer, Basel, 2011 (English).
  • [18] P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992.
  • [19] A. Kohatsu-Higa and P. Tankov, Jump-adapted discretization schemes for Lévy-driven SDEs, Stochastic Process. Appl. 120 (2010), no. 11, 2258–2285.
  • [20] V. Konakov and S. Menozzi, Weak error for stable driven stochastic differential equations: expansion of the densities, J. Theoret. Probab. 24 (2011), no. 2, 454–478.
  • [21] V. Konakov, S. Menozzi, and S. Molchanov, Explicit parametrix and local limit theorems for some degenerate diffusion processes, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 4, 908—923.
  • [22] S. Kusuoka, Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus, Advances in Mathematical Economics. Vol. 6, Adv. Math. Econ., vol. 6, Springer, Tokyo, 2004, pp. 69–83.
  • [23] S. Kusuoka, Gaussian K-scheme: justification for KLNV method, Advances in Mathematical Economics. Vol. 17, Adv. Math. Econ., vol. 17, Springer, Tokyo, 2013, pp. 71–120.
  • [24] M. Ledoux, I. Nourdin, and G. Peccati, Stein’s method, logarithmic Sobolev and transport inequalities, ArXiv e-prints (2014).
  • [25] E. Löcherbach and D. Loukianova, On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions, Stochastic Process. Appl. 118 (2008), no. 8, 1301–1321.
  • [26] T. Lyons and N. Victoir, Cubature on Wiener space, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2041, 169–198, Stochastic analysis with applications to mathematical finance.
  • [27] G.N. Milstein, Weak approximation of solutions of systems of stochastic differential equations, Numerical Integration of Stochastic Differential Equations, Mathematics and Its Applications, vol. 313, Springer, Netherlands, 1995, pp. 101–134 (English).
  • [28] S. Ninomiya and N. Victoir, Weak approximation of stochastic differential equations and application to derivative pricing, Appl. Math. Finance 15 (2008), no. 1-2, 107–121.
  • [29] I. Nourdin, G. Peccati, and Y. Swan, Entropy and the fourth moment phenomenon, J. Funct. Anal. 266 (2014), no. 5, 3170–3207.
  • [30] I. Nourdin and G. Poly, An invariance principle under the total variation distance, 15 pages, October 2013.
  • [31] E. Nummelin, A splitting technique for Harris recurrent Markov chains, Z. Wahrsch. Verw. Gebiete 43 (1978), no. 4, 309–318.
  • [32] Yu.V. Prokhorov, A local theorem for densities, Doklady Akad. Nauk SSSR (N.S) in Russian 83 (1952), 797–800.
  • [33] P. Protter and D. Talay, The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab. 25 (1997), no. 1, 393–423.
  • [34] D. Talay and L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl. 8 (1990), no. 4, 483–509 (1991).
  • [35] A. Yu. Zaĭtsev, Approximation of convolutions of probability distributions by infinitely divisible laws under weakened moment constraints, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 194 (1992), no. Problemy Teorii Veroyatnost. Raspred. 12, 79–90, 177–178.