Derandomization of the Euler scheme for scalar stochastic differential equations

Abstract

Consider a scalar stochastic differential equation with solution process X. We present a deterministic algorithm to approximate the marginal distribution of X at t=1 by a discrete distribution, and hereby we get a deterministic quadrature rule for expectations E(f(X(1))). The construction of the algorithm is based on derandomization of the Euler scheme. We provide a worst case analysis for the computational cost and the error, assuming that the coefficients of the equation have bounded derivatives up to order four and that the derivatives of f are polynomially bounded up to order four. In terms of the computational cost the error is almost of the order 2/3, if the diffusion coefficient is bounded away from zero, and in general we almost achieve the order 1/2.