A dynamical system pair with identical first two moments but different probability densities

Often in the literature, stochastic dynamical systems are approximated by moment closure techniques, closure in second moment being common practice. This refers to truncating the statistics generated by time varying probability density functions which evolve under the action of the trajectory-level dynamics. Although it is known that such moment closure approximations may lead to incorrect inferences, explicit examples at the dynamical systems level, are rare in the literature. In this paper, using optimal transport theory, we construct two dynamical systems such that starting from the same initial condition ensemble, their first two moments match at all times, but the underlying probability densities do not. This example serves as a motivation to consider the entire joint probability density function, as opposed to first few moments, for approximating stochastic systems in general, and stochastic jump linear systems in particular.