Abstract: In this talk, we will discuss how optimal transport can be used to compare stochastic processes in discrete time. While it is possible to use classical optimal transport, it turns out that this approach is ill-suited to accommodate the temporal structure that is crucial for stochastic processes. Hence the appropriate way to adapt optimal transport is to restrict the set of couplings considered to the subset of so-called bicausal couplings, which respect the temporal structure.
For classical optimal transport on the real line, the monotone coupling stands out as particularly important and solves the optimal transport problem in many cases. There exists a natural counterpart to the monotone coupling for the setting of stochastic processes and bicausal couplings, namely the Knothe-Rosenblatt coupling. We will give conditions under which this coupling is optimal. When the necessary conditions are only approximately satisfied, this coupling is likewise approximately optimal, and quantifiably so.