Stationary properties of the Gauss-Galerkin QMoM truncation of MV-SDEs

TL;DR

This paper investigates stationary solutions of the Gauss-Galerkin QMoM truncation of MV-SDEs, revealing multiple solutions, bifurcation structures, and stability properties, with applications to polynomial MV-SDEs and the Dawson-Shiino model.

Contribution

It provides a detailed analysis of stationary solutions for the GG-QMoM closure, including bifurcation and stability analysis, independent of the number of moments retained.

Findings

Multiple stationary solutions exist, corresponding to potential extrema.

Bifurcation diagrams are preserved in the deterministic limit.

Scaling properties allow direct probing of stability changes.

Abstract

The Gauss Galerkin Method/Quadrature method of moments (GG-QMoM) closure scheme, introduced by Dawson, closes a truncated set of moment equations of an SDE by a Galerkin approximation of its law in the space of probability measures. Here, results are presented on stationary solutions of the closed equations, irrespective of the number of moments retained (hence not dependent on the convergence theorem). These are applied to polynomial MV-SDEs, with explicit dependence on its moments in the drift, which can possess multiple stationary solutions. Particularly, we show in the deterministic limit, there are as many stationary solutions as extrema of the potential (critically, not just the minima), preserving the bifurcation diagram of the full equations. Further, a scaling property of solutions is proven, allowing changes of stability to be directly probed. Finally, this is applied to the GG-QMoM truncation of Dawson-Shiino model, whose stationary measures and change in stability has been previously studied.