Mimicking an Itô process by a solution of a stochastic differential equation

Given a multi-dimensional Itô process whose drift and diffusion terms are adapted processes, we construct a weak solution to a stochastic differential equation that matches the distribution of the Itô process at each fixed time. Moreover, we show how to match the distributions at each fixed time of functionals of the Itô process, including the running maximum and running average of one of the components of the process. A consequence of this result is that a wide variety of exotic derivative securities have the same prices when the underlying asset price is modeled by the original Itô process or the mimicking process that solves the stochastic differential equation.