An efficient approximation method for stochastic differential equations by means of the exponential Lie series

We describe a method of approximation of strong solutions to Stratonovich differential equations, that depends only on the Brownian motion defining the equation. h being the step size, it is known that the order of convergence of such approximations is h in the general case, and of h in some particular cases (one-dimensional Brownian for example). Among the approximation methods with optimal order of convergence, some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of the quadratic error. We prove that the proposed method, which is based on the representation of diffusions as flows of an ordinary differential equation, is asymptotically efficient.