  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.
