A Chen-Fliess approximation for diffusion functionals

We show that an interesting class of functionals of stochastic differential equations can be approximated by a Chen-Fliess series of iterated stochastic integrals and give a L^{2} error estimate, thus generalizing the standard stochastic Taylor expansion. The coefficients in this series are given a very intuitive meaning by using functional derivatives, recently introduced by B. Dupire.