The law of the Euler scheme for stochastic differential equations

Summary We study the approximation problem ofEf(X_T) byEf(X_Tⁿ), where (X_t) is the solution of a stochastic differential equation, (X_Tⁿ) 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_Tⁿ) 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_Tⁿand compare it to the density of the law ofX_T.