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Strong convergence rates for long-time approximations of SDEs with non-globally Lipschitz continuous coefficients

Xiaoming Wu, Xiaojie Wang

TL;DR

The paper addresses the challenge of obtaining reliable long-time strong approximations for SDEs with non-globally Lipschitz coefficients. It develops a long-time fundamental strong convergence theorem for general one-step schemes under a contractive monotone framework, and applies it to the backward Euler and projected Euler methods to establish strong convergence rates of order $1/2$ over infinite time. It also provides moment bounds for the projected Euler method and extends the analysis to invariant-measure MLMC settings, supported by numerical experiments that corroborate the theory. The results advance secure long-horizon simulations and statistical inference for nonlinear SDEs in applied sciences, including MLMC contexts for invariant distributions.

Abstract

This paper is concerned with long-time strong approximations of SDEs with non-globally Lipschitz coefficients.Under certain non-globally Lipschitz conditions, a long-time version of fundamental strong convergence theorem is established for general one-step time discretization schemes. With the aid of the fundamental strong convergence theorem, we prove the expected strong convergence rate over infinite time for two types of schemes such as the backward Euler method and the projected Euler method in non-globally Lipschitz settings. Numerical examples are finally reported to confirm our findings.

Strong convergence rates for long-time approximations of SDEs with non-globally Lipschitz continuous coefficients

TL;DR

The paper addresses the challenge of obtaining reliable long-time strong approximations for SDEs with non-globally Lipschitz coefficients. It develops a long-time fundamental strong convergence theorem for general one-step schemes under a contractive monotone framework, and applies it to the backward Euler and projected Euler methods to establish strong convergence rates of order over infinite time. It also provides moment bounds for the projected Euler method and extends the analysis to invariant-measure MLMC settings, supported by numerical experiments that corroborate the theory. The results advance secure long-horizon simulations and statistical inference for nonlinear SDEs in applied sciences, including MLMC contexts for invariant distributions.

Abstract

This paper is concerned with long-time strong approximations of SDEs with non-globally Lipschitz coefficients.Under certain non-globally Lipschitz conditions, a long-time version of fundamental strong convergence theorem is established for general one-step time discretization schemes. With the aid of the fundamental strong convergence theorem, we prove the expected strong convergence rate over infinite time for two types of schemes such as the backward Euler method and the projected Euler method in non-globally Lipschitz settings. Numerical examples are finally reported to confirm our findings.
Paper Structure (8 sections, 18 theorems, 125 equations, 4 figures)

This paper contains 8 sections, 18 theorems, 125 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The long-time fundamental strong convergence theorem
  3. Applications: Strong Convergence Rate of the Backward Euler Method over Infinite Time
  4. Applications: Strong Convergence Rate of the Projected Euler Method over Infinite Time
  5. Bounded moments of the projected Euler method
  6. Strong convergence rate of the projected Euler method
  7. Numerical Experiments
  8. Appendix

Key Result

Lemma 2.2

Let $r : [0,\infty) \rightarrow [0,\infty)$ and $\xi : [0,\infty) \rightarrow [0,\infty)$ be nonnegative continuous functions. If there exists a positive constant $\beta$ such that for any $0 \leq s < t < \infty$, then

Figures (4)

  • Figure 1: Strong convergence rate of the backward Euler method for \ref{['eq:Ginzburg_Landau_equation']}.
  • Figure 2: Strong convergence rate of the projected Euler method for \ref{['eq:Ginzburg_Landau_equation']}.
  • Figure 3: Strong convergence rate of the backward Euler method for \ref{['eq:SLSDE5']}.
  • Figure 4: Strong convergence rate of the projected Euler method for \ref{['eq:SLSDE5']}.

Theorems & Definitions (20)

  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 10 more