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Perturbation estimates for order-one strong approximations of SDEs without globally monotone coefficients

Lei Dai, Xiaojie Wang

TL;DR

This paper addresses the difficulty of obtaining strong convergence rates for SDEs with non-globally monotone coefficients by developing new perturbation estimates that bypass the limitations of global monotonicity. Using an Itô expansion of the drift and BDG-type inequalities, the authors establish order-one pathwise uniform convergence for the stopped increment-tamed Euler–Maruyama scheme (SITEM) in additive-noise and multiplicative-noise second-order SDEs, and they introduce a positivity-preserving explicit Milstein-type scheme for stochastic Lotka–Volterra models that also achieves order-one convergence. The theoretical results are supported by numerical experiments illustrating the superior convergence and stability properties compared to prior $1/2$-order results. Collectively, the work extends strong convergence theory to a broader class of SDEs and provides practical, structure-preserving numerical schemes for applications in physics, biology, and economics.

Abstract

To obtain strong convergence rates of numerical schemes, an overwhelming majority of existing works impose a global monotonicity condition on coefficients of SDEs. Nevertheless, there are still many SDEs from applications that do not have globally monotone coefficients. As a recent breakthrough, the authors of [Hutzenthaler, Jentzen, Ann. Probab., 2020] originally presented a perturbation theory for stochastic differential equations (SDEs), which is crucial to recovering strong convergence rates of numerical schemes in a non-globally monotone setting. However, only a convergence rate of order 1/2 was obtained there for time-stepping schemes such as a stopped increment-tamed Euler-Maruyama (SITEM) method. An interesting question arises, also raised by the aforementioned work, as to whether a higher convergence rate than 1/2 can be obtained when higher order schemes are used. The present work attempts to give a positive answer to this question. To this end, we develop some new perturbation estimates that are able to reveal the order-one strong convergence of numerical methods. As the first application of the newly developed estimates, we identify the expected order-one pathwise uniformly strong convergence of the SITEM method for additive noise driven SDEs and multiplicative noise driven second order SDEs with non-globally monotone coefficients. As the other application, we propose and analyze a positivity preserving explicit Milstein-type method for Lotka-Volterra competition model driven by multi-dimensional noise, with a pathwise uniformly strong convergence rate of order one recovered under mild assumptions. These obtained results are completely new and significantly improve the existing theory. Numerical experiments are also provided to confirm the theoretical findings.

Perturbation estimates for order-one strong approximations of SDEs without globally monotone coefficients

TL;DR

This paper addresses the difficulty of obtaining strong convergence rates for SDEs with non-globally monotone coefficients by developing new perturbation estimates that bypass the limitations of global monotonicity. Using an Itô expansion of the drift and BDG-type inequalities, the authors establish order-one pathwise uniform convergence for the stopped increment-tamed Euler–Maruyama scheme (SITEM) in additive-noise and multiplicative-noise second-order SDEs, and they introduce a positivity-preserving explicit Milstein-type scheme for stochastic Lotka–Volterra models that also achieves order-one convergence. The theoretical results are supported by numerical experiments illustrating the superior convergence and stability properties compared to prior -order results. Collectively, the work extends strong convergence theory to a broader class of SDEs and provides practical, structure-preserving numerical schemes for applications in physics, biology, and economics.

Abstract

To obtain strong convergence rates of numerical schemes, an overwhelming majority of existing works impose a global monotonicity condition on coefficients of SDEs. Nevertheless, there are still many SDEs from applications that do not have globally monotone coefficients. As a recent breakthrough, the authors of [Hutzenthaler, Jentzen, Ann. Probab., 2020] originally presented a perturbation theory for stochastic differential equations (SDEs), which is crucial to recovering strong convergence rates of numerical schemes in a non-globally monotone setting. However, only a convergence rate of order 1/2 was obtained there for time-stepping schemes such as a stopped increment-tamed Euler-Maruyama (SITEM) method. An interesting question arises, also raised by the aforementioned work, as to whether a higher convergence rate than 1/2 can be obtained when higher order schemes are used. The present work attempts to give a positive answer to this question. To this end, we develop some new perturbation estimates that are able to reveal the order-one strong convergence of numerical methods. As the first application of the newly developed estimates, we identify the expected order-one pathwise uniformly strong convergence of the SITEM method for additive noise driven SDEs and multiplicative noise driven second order SDEs with non-globally monotone coefficients. As the other application, we propose and analyze a positivity preserving explicit Milstein-type method for Lotka-Volterra competition model driven by multi-dimensional noise, with a pathwise uniformly strong convergence rate of order one recovered under mild assumptions. These obtained results are completely new and significantly improve the existing theory. Numerical experiments are also provided to confirm the theoretical findings.
Paper Structure (8 sections, 11 theorems, 131 equations, 11 figures)

This paper contains 8 sections, 11 theorems, 131 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Burkholder-Davis-Gundy type inequalities
  5. New perturbation estimates for SDEs
  6. Order-one pathwise uniformly strong convergence of the SITEM scheme
  7. A positive preserving Milstein type scheme for the stochastic LV competition model with order-one pathwise uniformly strong convergence
  8. Conclusion

Key Result

Lemma 2.1

Let $S:[0,T] \times \Omega \rightarrow \mathbb{R}^{d \times m}$ be a predictable stochastic process satisfying $\mathbb{P}(\int_0^T\|S_t\|^2$${\rm d}t<\infty)=1$ and let $\{W_t\}_{t\geq0}$ be a $m$-dimensional standard Brownian motion. Then for any $p\geq2$,

Figures (11)

  • Figure 1: A comparison of strong convergence rates for Langevin dynamics
  • Figure 2: Invariant density approximation: (a) $h=2^{-7}$; (b) $h=2^{-4}$
  • Figure 3: Strong convergence rate for stochastic van der Pol oscillator
  • Figure 4: Sample average trajectory and phase plane with $h=2^{-7}$
  • Figure 5: Sample average trajectory and phase plane with $h=2^{-4}$
  • ...and 6 more figures

Theorems & Definitions (11)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Theorem 4.2
  • Lemma 5.2
  • Proposition 5.3: Positivity preserving
  • Proposition 5.4
  • Lemma 5.5: Bounded moments
  • ...and 1 more