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 .
Article PDF
Similar content being viewed by others
References
Bally, V.: On the connection between the Malliavin covariance matrix and Hörmander's condition. J. Funct. Anal.96, 219–255 (1991)
Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations (II): convergence rate of the density (submitted)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North-Holland 1981
Kloeden, P. E., Platen, E.: Numerical solution of stochastic differential equations. Berlin: Springer 1992
Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus, part II. J. Fac. Sci. Univ. Tokyo32, 1–76 (1985)
Milshtein, G. N.: A method of second-order accuracy integration of stochastic differential equations. Theory Probab. Appl.23, 396–401 (1976)
Milshtein, G. N.: Weak approximation of solutions of systems of stochastic differential equations. Theory Probab. Appl.30, 750–766 (1985)
Newton, N. J.: Variance reduction for simulated diffusions. SIAM J. appl. Math. (to appear)
Nualart, D.: Malliavin calculus and related topics, Springer-Verlag 1995
Pardoux, E.: Filtrage non linéaire et équations aux dérivées partielles stochastiques associées. In: Cours à l'Ecole d'Eté de Probabilités de Saint-Flour XIX (Lect. Notes Math., vol. 1464) Berlin: Springer 1991
Protter, P., Talay, D.: The Euler scheme for Lévy driven stochastic differential equations (submitted)
Talay, D.: Efficient numerical schemes for the approximation of expectations of functionals of S.D.E. In: Szpirglas, J., Korezlioglu, H., Mazziotto, G.: Filtering and control of ramdom processes (Lect. Notes Control and Information Sciences, vol. 61, Proc. ENST-CNET Coll., Paris, 1983) Berlin: Springer 1984
Talay, D.: Discrétisation d'une E.D.S. et calcul approché d'espérances de fonctionnelles de la solution. Math. Modelling Numer. Anal.20, 141–179 (1986)
Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl.8, 94–120 (1990)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01303802