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Exact Moment Estimation of Stochastic Differential Dynamics

Shenghua Feng, Jie An, Naijun Zhan, Fanjiang Xu

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.

Exact Moment Estimation of Stochastic Differential Dynamics

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.
Paper Structure (3 sections, 2 theorems, 11 equations, 1 algorithm)

This paper contains 3 sections, 2 theorems, 11 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Reduction of EME Problem to ODE Solving

Key Result

theorem 1

Let $\{X_t\}_{t\ge 0}$ be the solution of eq:sde. If $f \in C^{2}(\mathbb{R}^n)$ has compact support, then for all $t\ge 0$, where $\mathcal{A}$ is the infinitesimal generator of eq:sde given by

Theorems & Definitions (6)

  • definition 1: Moment-solvable SDE
  • theorem 1: Dynkin’s formula oksendal2013stochastic
  • remark 1
  • theorem 2: Explicit moment computation
  • proof
  • remark 2