Bounding stationary averages of polynomial diffusions via semidefinite programming

TL;DR

This paper presents a semidefinite programming approach to compute bounds on polynomial stationary averages of diffusions, with applications in Bayesian inference, Lyapunov exponents, and structural reliability.

Contribution

The paper introduces a novel algorithm that provides converging bounds on stationary averages of polynomial diffusions using semidefinite programming.

Findings

Bounds converge to true stationary averages in certain cases

Numerical experiments demonstrate the effectiveness of the bounds

Applications include Bayesian inference, Lyapunov exponents, and structural mechanics

Abstract

We introduce an algorithm based on semidefinite programming that yields increasing (resp. decreasing) sequences of lower (resp. upper) bounds on polynomial stationary averages of diffusions with polynomial drift vector and diffusion coefficients. The bounds are obtained by optimising an objective, determined by the stationary average of interest, over the set of real vectors defined by certain linear equalities and semidefinite inequalities which are satisfied by the moments of any stationary measure of the diffusion. We exemplify the use of the approach through several applications: a Bayesian inference problem; the computation of Lyapunov exponents of linear ordinary differential equations perturbed by multiplicative white noise; and a reliability problem from structural mechanics. Additionally, we prove that the bounds converge to the infimum and supremum of the set of stationary averages for certain SDEs associated with the computation of the Lyapunov exponents, and we provide numerical evidence of convergence in more general settings.