Convergence rate of Euler scheme for stochastic differential equations: Functionals of solutions

Abstract

Let X t , t ∈ [0, T], be the solution of a stochastic differential equation, and let X t h , t ∈ [0, T], be the Euler approximation with the step h = T n . It is known that, for a wide class of functions f, the error E f( X T h ) − E f( X T ) is O( h) or, more exactly, C · h + O( h 2). We propose an extension of these results to a class of functionals f depending on the trajectories of the solution on the whole time interval [0, T]. The functionals are defined on an appropriate semi-martingale space.