The law of the Euler scheme for stochastic differential equations. I: Convergence rate of the distribution function
We study the approximation problem of \(\mathbb{E} f(X_T)\) by \(\mathbb{E} f(X^n_T)\), where \((X_t)\) is the solution of a stochastic differential equation, \((X^n_t)\) is defined by the Euler discretization scheme with step \(T/n\), and \(f\) is a given function. For smooth \(f\)'s, Talay and Tubaro have shown that the error \(\mathbb{E} f(X_T) - f(X^n_T)\) can be expanded in powers of \(1/n\), 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 variation calculus. In the second part of this work, we will consider the density of the law of \(X^n_T\) and compare it to the density of the law of \(X_T\).
scientific article; zbMATH DE number 849088
scientific article