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Convergence in Density of Splitting AVF Scheme for Stochastic Langevin Equation

Jianbo Cui, Jialin Hong, Derui Sheng

TL;DR

The paper investigates convergence in density for numerical approximations of a stochastic Langevin equation with non-globally monotone drift using a splitting AVF scheme. It develops a Malliavin-calculus framework to prove exponential integrability and the existence and smoothness of densities for both the exact and numerical solutions, and establishes uniform non-degeneracy of the Malliavin covariance. The authors obtain an optimal strong convergence rate of order $h^{1/2}$ in the Malliavin–Sobolev sense and, leveraging Donsker’s delta, prove a density-convergence rate of order $O(h)$ for the numerical scheme. These results bridge strong-path convergence with probabilistic density convergence under Hörmander-type nondegeneracy, enabling reliable density approximations for degenerate SDEs with super-linear drift in practical simulations.

Abstract

In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. To deal with the non-globally monotone coefficient in the considered equation, we first present the exponential integrability properties of the exact and numerical solutions. Then we show the existence and smoothness of the density function of the numerical solution by proving its uniform non-degeneracy in Malliavin sense. In order to analyze the approximate error between the density function of the exact solution and that of the numerical solution, we derive the optimal strong convergence rate in every Malliavin--Sobolev norm of the numerical scheme via Malliavin calculus. Combining the approximation result of Donsker's delta function and the smoothness of the density functions, we prove that the convergence rate in density coincides with the optimal strong convergence rate of the numerical scheme.

Convergence in Density of Splitting AVF Scheme for Stochastic Langevin Equation

TL;DR

The paper investigates convergence in density for numerical approximations of a stochastic Langevin equation with non-globally monotone drift using a splitting AVF scheme. It develops a Malliavin-calculus framework to prove exponential integrability and the existence and smoothness of densities for both the exact and numerical solutions, and establishes uniform non-degeneracy of the Malliavin covariance. The authors obtain an optimal strong convergence rate of order in the Malliavin–Sobolev sense and, leveraging Donsker’s delta, prove a density-convergence rate of order for the numerical scheme. These results bridge strong-path convergence with probabilistic density convergence under Hörmander-type nondegeneracy, enabling reliable density approximations for degenerate SDEs with super-linear drift in practical simulations.

Abstract

In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. To deal with the non-globally monotone coefficient in the considered equation, we first present the exponential integrability properties of the exact and numerical solutions. Then we show the existence and smoothness of the density function of the numerical solution by proving its uniform non-degeneracy in Malliavin sense. In order to analyze the approximate error between the density function of the exact solution and that of the numerical solution, we derive the optimal strong convergence rate in every Malliavin--Sobolev norm of the numerical scheme via Malliavin calculus. Combining the approximation result of Donsker's delta function and the smoothness of the density functions, we prove that the convergence rate in density coincides with the optimal strong convergence rate of the numerical scheme.

Paper Structure

This paper contains 16 sections, 20 theorems, 173 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Malliavin calculus on the Wiener space
  4. Regularity of probability laws
  5. Main assumptions
  6. Stochastic Langevin equation
  7. Exponential integrability property of the exact solution
  8. Probability density function of the exact solution
  9. Splitting AVF scheme
  10. Exponential integrability property of the numerical approximation
  11. Probability Density Function
  12. Strong convergence
  13. Convergence in probability density function
  14. Convergence in $\mathbb{D}^{\alpha,p}(\mathbb{R}^{2m})$
  15. Convergence in probability density function
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let Assumption F2 hold, $h_0$ be a sufficiently small positive constant and $p\ge1$. There exists some positive constant $C=C(p,T,\sigma,X(0))$ such that for any $h\in(0,h_0]$,

Figures (1)

  • Figure 1: Mean square convergence rate of splitting AVF method for stochastic Langevin equations.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Remark 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • ...and 27 more