Model reduction of stochastic processes using Wasserstein pseudometrics

We consider the problem of finding reduced models of stochastic processes. We use Wasserstein pseudometrics to quantify the difference between processes. The method proposed in this paper is applicable to any continuous-time stochastic process with output, and pseudometrics between processes are defined only in terms of the available outputs. We demonstrate how to approximate a wide class of behavioral pseudometrics and how to optimize parameter values to minimize Wasserstein pseudometrics between processes. In particular, we introduce an algorithm that allows for the approximation of Wasserstein pseudometrics from sampled data, even in the absence of models for the processes. We illustrate the approach with an example from systems biology.