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The Euler scheme for stochastic differential equations: error analysis with Malliavin calculus

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  • Bally, Vlad
  • Talay, Denis

Abstract

We study the approximation problem of Ef(XT) by Ef(XTn), where (Xt) is the solution of a stochastic differential equation, (Xtn) is defined by the Euler discretization scheme with step Tn, and f is a given function. For smooth f's, Talay and Tubaro had shown that the error Ef(XT) − Ef(XTn) can be expanded in powers of Tn, which permits to construct Romberg extrapolation procedures to accelerate the convergence rate. Here, we present our following recent result: the expansion exists also when f is only supposed measurable and bounded, under a nondegeneracy condition (essentially, the Hörmander condition for the infinitesimal generator of (Xt)): this is obtained with Malliavin's calculus. We also get an estimate on the difference between the density of the law of XT and the density of the law of XTn.

Suggested Citation

  • Bally, Vlad & Talay, Denis, 1995. "The Euler scheme for stochastic differential equations: error analysis with Malliavin calculus," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 38(1), pages 35-41.
    • Handle: RePEc:eee:matcom:v:38:y:1995:i:1:p:35-41
    DOI: 10.1016/0378-4754(93)E0064-C
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    Citations

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    Cited by:

    1. Brandt, Michael W. & Santa-Clara, Pedro, 2002. "Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets," Journal of Financial Economics, Elsevier, vol. 63(2), pages 161-210, February.
    2. Denis Belomestny & Tigran Nagapetyan, 2014. "Multilevel path simulation for weak approximation schemes," Papers 1406.2581, arXiv.org, revised Oct 2014.
    3. J.-P. Morillon, 2015. "Solving Wentzell-Dirichlet Boundary Value Problem with Superabundant Data Using Reflecting Random Walk Simulation," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 697-719, September.
    4. Wu, Shujin & Han, Dong, 2007. "Algorithmic analysis of Euler scheme for a class of stochastic differential equations with jumps," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 211-219, January.
    5. Protter, Philip & Qiu, Lisha & Martin, Jaime San, 2020. "Asymptotic error distribution for the Euler scheme with locally Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2296-2311.
    6. Madalina Deaconu & Samuel Herrmann, 2023. "Strong Approximation of Bessel Processes," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-24, March.
    7. BALLY Vlad & TALAY Denis, 1996. "The Law of the Euler Scheme for Stochastic Differential Equations: II. Convergence Rate of the Density," Monte Carlo Methods and Applications, De Gruyter, vol. 2(2), pages 93-128, December.
    8. Givon, Dror & Kupferman, Raz, 2004. "White noise limits for discrete dynamical systems driven by fast deterministic dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 335(3), pages 385-412.

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