 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.
<meaning-changed>  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.
 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.