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In classical optimal transport, the contributions of Benamou$-$Brenier andMcCann regarding the time-dependent version of the problem are cornerstones ofthe field and form the basis for a variety of applications in othermathematical areas.  Stretched Brownian motion provides an analogue for the martingale version ofthis problem. In this article we provide a characterization in terms ofgradients of convex functions, similar to the characterization of optimizers inthe classical transport problem for quadratic distance cost.

The structure of martingale Benamou$-$Brenier in $\mathbb{R}^{d}$