THE ORDINARY DIFFERENTIAL-EQUATION APPROACH TO ASYMPTOTICALLY EFFICIENT SCHEMES FOR SOLUTION OF STOCHASTIC DIFFERENTIAL-EQUATIONS

We consider the numerical approximation to strong solutions of stochas tic differential equations (SDE's) using a fixed time step and given o nly the increments of tho Brownian path over each time step. Using the approach generalised by Ben Arous, Castell and Hu, of approximating t he solution to an SDE over small time by the solution to a time inhomo geneous ordinary differential equation (ODE), we obtain ODE's which, a s the number of time steps increases, yield an asymptotically efficien t sequence of approximations to the solution of an SDE, where the conc ept of asymptotic efficiency is that of Clark and Newton, We distingui sh between the two cases of an SDE driven by a one-dimensional Brownia n path or satisfying the commutativity condition on the one hand and a n SDE driven by a multi-dimensional Brownian path and with a non-commu tative Lie algebra on the other hand. When the ODE`s presented are sol ved numerically, the property of asymptotic efficiency is preserved as long as the solution is accurate enough, The methods of this paper re present an alternative and easily generalisable way of looking at the approximation of strong solutions to SDE's.