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Probability Theory and Related Fields

, Volume 104, Issue 1, pp 43–60 | Cite as

The law of the Euler scheme for stochastic differential equations

I. Convergence rate of the distribution function
  • V. Bally
  • D. Talay
Article

Summary

We study the approximation problem ofEf(X T ) byEf(X T n ), where (X t ) is the solution of a stochastic differential equation, (X T n ) 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 errorEf(X T ) −f(X T n ) 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 T n and compare it to the density of the law ofX T .

Mathematics Subject Classification

60H07 60H10 60J60 65C05 

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • V. Bally
    • 1
  • D. Talay
    • 2
  1. 1.Université du Maine and Laboratoire de ProbabilitésUniversité Paris 6Paris Cédex 05France
  2. 2.INRIAValbonne CedexFrance

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