The Law of the Euler scheme for stochastic differential equations : I. convergence rate of the distribution function
Abstract
We study the approximation problem of $\ee f(X_T)$ by $\ee f(X_T^n)$, where $(X_t)$ is the solution of a stochastic differential equation, $(X^n_t)$ is defined by the Euler discretization scheme with step $\fracTn$,and $f$ is a given function. For smooth $f$'s, Talay and Tubaro have shown that the error $\ee f(X_T)-f(X_T^n)$ can be expanded in powers of $\frac1n$, which permits to construct Romberg extrapolation procedures to accelerate the convergence rate. Here, we prove that the expansion exists also when $f$ 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 of $X_T^n$ and compare it to the density of the law of $X_T$. \noindent\bf AMS(MOS) classification: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05
Domains
Other [cs.OH]
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