Distributional Replication

A function which transforms a continuous random variable such that it has specified distribution is called replicating function. We suppose functions may be assigned price, and study an optimization problem in the cheapest approximation to sought. Under suitable regularity conditions, including bound on entropy of set candidate approximations, we show optimal comes close achieving distributional replication, minimum cost among functions. discuss relevance our results financial literature hedge fund replication; this case, corresponds portfolio market index options delivers return distribution.