✦ LIBER ✦

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

✍ Scribed by Vlad Bally; Denis Talay

Publisher: Elsevier Science
Year: 1995
Tongue: English
Weight: 304 KB
Volume: 38
Category: Article
ISSN: 0378-4754
DOI: 10.1016/0378-4754(93)e0064-c

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✦ Synopsis

We study the approximation problem of Ef(Xr) by Ef(X~.), where (Xt) is the solution of a stochastic differential equation, (X~) is defined by the Euler discretization scheme with step T/n, and f is a given function. For smooth f's, Talay and Tubaro had shown that the error Ef(Xr) -Ef(X~) can be expanded in powers of T/n, 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 H6rmander 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 Xr and the density of the law of X~.

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