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Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients

Chenxu Pang, Xiaojie Wang, Yue Wu

TL;DR

This work addresses weak ergodic approximation for semi-linear SDEs with non-globally Lipschitz coefficients by introducing the linear-theta-projected Euler (LTPE) scheme, which combines implicit handling of linear stiffness with a projection to control nonlinearity. It proves exponential convergence to invariant measures for both the SDE and the LTPE method, and establishes a time-independent weak error of order one between the two invariant measures for test functions in $C_b^3$. The analysis rests on dissipativity and polynomial-growth assumptions, uniform moment bounds, and time-independent regularity for the Kolmogorov equation, with a continuous-time LTPE surrogate aiding the weak-error decomposition. Numerical experiments corroborate the theory, demonstrating the scheme’s ergodicity-preserving behavior and convergence rate in stiff and non-globally Lipschitz settings, highlighting computational advantages over fully implicit schemes.

Abstract

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.

Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients

TL;DR

This work addresses weak ergodic approximation for semi-linear SDEs with non-globally Lipschitz coefficients by introducing the linear-theta-projected Euler (LTPE) scheme, which combines implicit handling of linear stiffness with a projection to control nonlinearity. It proves exponential convergence to invariant measures for both the SDE and the LTPE method, and establishes a time-independent weak error of order one between the two invariant measures for test functions in . The analysis rests on dissipativity and polynomial-growth assumptions, uniform moment bounds, and time-independent regularity for the Kolmogorov equation, with a continuous-time LTPE surrogate aiding the weak-error decomposition. Numerical experiments corroborate the theory, demonstrating the scheme’s ergodicity-preserving behavior and convergence rate in stiff and non-globally Lipschitz settings, highlighting computational advantages over fully implicit schemes.

Abstract

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.
Paper Structure (18 sections, 16 theorems, 237 equations, 4 figures)

This paper contains 18 sections, 16 theorems, 237 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Settings and main result
  3. Invariant measure of semi-linear SDE
  4. Invariant measure of the LTPE scheme
  5. Time-independent weak error analysis
  6. Numerical experiments
  7. Proof of Lemmas in Section \ref{['subsection: Invariant measure of semi-linear SDE']}
  8. Proof of Lemma \ref{['lemma:uniform-moments-bound-of-SDEs']}
  9. Proof of Lemma \ref{['lemma:contractivity-of-sde']}
  10. Proof of Lemma \ref{['lemma:cauchy-sequence-of-sode']}
  11. Proof of Lemmas in Section \ref{['subsection:Invariant-measure-of-the-numerical-scheme']}
  12. Proof of Lemma \ref{['lemma:necessary-estimates']}
  13. Proof of Lemma \ref{['lemma:contractivity-of-the-theta-linear-projected-Euler-method']}
  14. Proof of Lemmas in Section \ref{['subsection: Time-independent weak error analysis']}
  15. Proof of Lemma \ref{['lemma:differentiability-of-solutions']}
  16. ...and 3 more sections

Key Result

Theorem 2.5

(Main result) Let Assumptions assumption:one-side-Lipschitz-condition-for-linear-operator-assumption:growth-condition-of-frechet-derivatives-of-drift-and-diffusion hold with $p_{0} \geq \max\{4\gamma+1, 5\gamma-4\}$ and $2\lambda_{1} > \max\{L_{1}, L_{2}\}$ and consider SDE eq:semi-linear-SODE. Give where $\lambda_{f}:=C_{1}(1+2h^{\frac{1}{2}})$, $C_{1}$ is a constant depending only on the drift $

Figures (4)

  • Figure 1: Weak convergence rates of the explicit projected Euler method for stochastic Ginzburg-Landau model \ref{['equation:ginzburg-landau-model']}
  • Figure 2: Probability density of LTPE scheme method for discretizing the mean reverting model \ref{['eq:mean-reverting-model']} with different $\theta$.
  • Figure 3: Weak convergence rates of semi-linear-implicit projected Euler method for the mean reverting model \ref{['eq:mean-reverting-model']}
  • Figure 4: Weak convergence rates of linear-implicit projected Euler method for model \ref{['eq:SPDE-SODE-system']} (K=4).

Theorems & Definitions (30)

  • Theorem 2.5
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • proof : Proof of Lemma \ref{['lemma:uniform-moments-bound-of-the-LTPE-method']}
  • ...and 20 more