Mimicking an It\^(o) process by a solution of a stochastic differential equation

Given a multi-dimensional It\^{o} 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 It\^{o} process at each fixed time. Moreover, we show how to match the distributions at each fixed time of functionals of the It\^{o} 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 It\^{o} process or the mimicking process that solves the stochastic differential equation.