Minimal asymptotic error for one-point approximation of SDEs with time-irregular coefficients

Abstract

We consider strong one-point approximation of solutions of scalar stochastic differential equations (SDEs) with irregular coefficients. The drift coefficient a:[0,T]×R→R is assumed to be Lipschitz continuous with respect to the space variable but only measurable with respect to the time variable. For the diffusion coefficient b:[0,T]→R we assume that it is only piecewise Hölder continuous with Hölder exponent ϱ∈(0,1]. We show that, roughly speaking, the error of any algorithm, which uses n values of the diffusion coefficient, cannot converge to zero faster than n−min{ϱ,1/2} as n→+∞. This best speed of convergence is achieved by the randomized Euler scheme.