Exact Moment Estimation of Stochastic Differential Dynamics

TL;DR

This paper introduces a symbolic method to compute exact moments of certain stochastic differential equations with polynomial coefficients, enabling precise analysis of nonlinear stochastic systems.

Contribution

It formalizes the class of moment-solvable SDEs, introduces a symbolic procedure for exact moment computation, and characterizes pro-solvable SDEs with a block-triangular structure.

Findings

Exact moments can be computed for a broad class of polynomial SDEs.

The proposed method is effective on nonlinear models.

Pro-solvable SDEs include many nonlinear stochastic systems.

Abstract

Moment estimation for stochastic differential equations (SDEs) is fundamental to the formal reasoning and verification of stochastic dynamical systems, yet remains challenging and is rarely available in closed form. In this paper, we study time-homogeneous SDEs with polynomial drift and diffusion, and investigate when their moments can be computed exactly. We formalize the notion of moment-solvable SDEs and propose a generic symbolic procedure that, for a given monomial, attempts to construct a finite linear ordinary differential equation (ODE) system governing its moment, thereby enabling exact computation. We introduce a syntactic class of pro-solvable SDEs, characterized by a block-triangular structure, and prove that all polynomial moments of any pro-solvable SDE admit such finite ODE representations. This class strictly generalizes linear SDEs and includes many nonlinear models. Experimental results demonstrate the effectiveness of our approach.