Estimating stationary characteristic functions of stochastic systems via semidefinite programming

TL;DR

This paper introduces a semidefinite programming approach to estimate the stationary characteristic functions of polynomial-driven stochastic differential equations with Lévy noise, leveraging positive definiteness to bound moments and functions.

Contribution

It presents a novel method to estimate stationary characteristic functions of complex stochastic systems using semidefinite programming and moment bounds.

Findings

Successfully estimates stationary characteristic functions in examples.

Provides bounds on stationary moments using positive definiteness.

Demonstrates applicability to systems driven by Lévy noise.

Abstract

This paper proposes a methodology to estimate characteristic functions of stochastic differential equations that are defined over polynomials and driven by L\'evy noise. For such systems, the time evolution of the characteristic function is governed by a partial differential equation; consequently, the stationary characteristic function can be obtained by solving an ordinary differential equation (ODE). However, except for a few special cases such as linear systems, the solution to the ODE consists of unknown coefficients. These coefficients are closely related with the stationary moments of the process, and bounds on these can be obtained by utilizing the fact that the characteristic function is positive definite. These bounds can be further used to find bounds on other higher order stationary moments and also estimate the stationary characteristic function itself. The method is finally illustrated via examples.