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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.

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  • 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. 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.
    2. 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.
    3. Madalina Deaconu & Samuel Herrmann, 2023. "Strong Approximation of Bessel Processes," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-24, March.
    4. 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.
    5. Denis Belomestny & Tigran Nagapetyan, 2014. "Multilevel path simulation for weak approximation schemes," Papers 1406.2581, arXiv.org, revised Oct 2014.
    6. 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.
    7. 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.
    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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