Coarse graining of stochastic differential equations: averaging and projection method

TL;DR

This paper compares averaging and projection methods for coarse-graining stochastic differential equations, providing theoretical insights, conditions for their equivalence, and illustrating their differences through examples.

Contribution

It offers a rigorous theoretical link between the projection method and Gy"ongy's approach, and clarifies when these methods coincide or differ.

Findings

Projection method relates to Gy"ongy's approach.

Averaging applies with time scale separation.

Methods may not always coincide, contrary to common assumptions.

Abstract

We study coarse-graining methods for stochastic differential equations. In particular we consider averaging and a type of projection operator method, sometimes referred to as effective dynamic via conditional expectations. The projection method (PM) we consider is related to the ``mimicking marginals'' coarse graining approach proposed by Gy\"ongy. The first contribution of this paper is to provide further theoretical background for the PM and a rigorous link to the Gy\"ongy method. Moreover, we compare PM and averaging. While averaging applies to systems with time scale separation, the PM can in principle be applied irrespective of this. However it is often assumed that the two methods coincide in presence of scale separation. The second contribution of this paper is to make this statement precise, provide sufficient conditions under which these two methods coincide and then show -- via examples and counterexamples -- that this needs not be the case.