Matching Statistics of an Itô Process by a Process of Diffusion Type

Suppose we are given a multi-dimensional Itô process, which can be regarded as a model for an underlying asset price together with related stochastic processes, e.g., volatility. The drift and diffusion terms for this Itô process are permitted to be arbitrary adapted processes. We construct a weak solution to a diffusion-type equation that matches the distribution of the Itô process at each fixed time. Moreover, we show how to also match the distribution at each fixed time of statistics 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 written on the original Itô process as when written on the mimicking process. Partially supported by the National Science Foundation under Grants No. DMS-0404682 and DMS-0903475.