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The law of the Euler scheme for stochastic differential equations
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  • Published: March 1996

The law of the Euler scheme for stochastic differential equations

I. Convergence rate of the distribution function

  • V. Bally1 &
  • D. Talay2 

Probability Theory and Related Fields volume 104, pages 43–60 (1996)Cite this article

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Summary

We study the approximation problem ofE f(X T ) byE f(X n T ), where (X t ) is the solution of a stochastic differential equation, (X n T ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE f(X T ) −f(X n T ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf is only supposed measurable and bounded, under an additional nondegeneracy condition of Hörmander type for the infinitesimal generator of (X t ): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law ofX n T and compare it to the density of the law ofX T .

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References

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Authors and Affiliations

  1. Université du Maine and Laboratoire de Probabilités, Université Paris 6, 4 Place Jussieu, F-75252, Paris Cédex 05, France

    V. Bally

  2. INRIA, 2004 Route des Lucioles, Sophia-Antipolis, B.P. 109, F-06561, Valbonne Cedex, France

    D. Talay

Authors
  1. V. Bally
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  2. D. Talay
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Bally, V., Talay, D. The law of the Euler scheme for stochastic differential equations. Probab. Th. Rel. Fields 104, 43–60 (1996). https://doi.org/10.1007/BF01303802

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  • Received: 05 May 1994

  • Revised: 24 February 1995

  • Issue Date: March 1996

  • DOI: https://doi.org/10.1007/BF01303802

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Mathematics Subject Classification

  • 60H07
  • 60H10
  • 60J60
  • 65C05
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