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

Let Xt, t ∈ [0,T], be the solution of a stochastic differential equation, and let Xth, t ∈ [0,T], be the Euler approximation with the step h = Tn. It is known that, for a wide class of functions f, the error Ef(XTh) − Ef(XT) is O(h) or, more exactly, C · h + O(h2). 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.