Before revision:  Inspired by the work of Benamou and Brenier, we formulate a martingale optimal transport problem  which seeks a  diffusion process that matches the known densities of an asset price at two different dates and minimizes a chosen cost function.
After revision:  We formulate a time continuous martingale optimal transport problem which seeks a diffusion process that matches the known densities of an asset price at two different dates and minimizes a chosen cost function.