Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients II: finite moments and higher-order schemes

Yuying Zhao, Xiaojie Wang, Zhongqiang Zhang

Abstract

This paper is the second in a series of works on weak convergence of one-step schemes for solving stochastic differential equations (SDEs) with one-sided Lipschitz conditions. It is known that the super-linear coefficients may lead to a blowup of moments of solutions and numerical solutions and thus affect the convergence of numerical methods. Wang et al. (2023, IMA J. Numer. Anal.) have analyzed weak convergence of one-step numerical schemes when solutions to SDEs have all finite moments. Therein some modified Euler schemes have been discussed about their weak convergence orders. In this work, we explore the effects of limited orders of moments on the weak convergence of a family of explicit schemes. The schemes are based on approximations/modifications of terms in the Ito-Talyor expansion. We provide a systematic but simple way to establish weak convergence orders for these schemes. We present several numerical examples of these schemes and show their weak convergence orders.

Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients II: finite moments and higher-order schemes

Abstract

This paper is the second in a series of works on weak convergence of one-step schemes for solving stochastic differential equations (SDEs) with one-sided Lipschitz conditions. It is known that the super-linear coefficients may lead to a blowup of moments of solutions and numerical solutions and thus affect the convergence of numerical methods. Wang et al. (2023, IMA J. Numer. Anal.) have analyzed weak convergence of one-step numerical schemes when solutions to SDEs have all finite moments. Therein some modified Euler schemes have been discussed about their weak convergence orders. In this work, we explore the effects of limited orders of moments on the weak convergence of a family of explicit schemes. The schemes are based on approximations/modifications of terms in the Ito-Talyor expansion. We provide a systematic but simple way to establish weak convergence orders for these schemes. We present several numerical examples of these schemes and show their weak convergence orders.
Paper Structure (19 sections, 15 theorems, 122 equations, 4 figures, 7 tables)

This paper contains 19 sections, 15 theorems, 122 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Conditions on coefficients of SDEs
  4. The approximation theorem
  5. Schemes of weak convergence based on the Itô-Taylor expansion
  6. Assumptions on the maps for bounded moments
  7. Modified Euler schemes and the maps $\mathcal{T}_i$
  8. Weak convergence order for the scheme \ref{['eq:second-order-new-scheme']}
  9. Examples of the maps $\mathcal{T}_i$'s
  10. Examples of weak convergence order depending on moment bounds
  11. Numerical Experiments
  12. Weak convergence of the scheme \ref{['eq:second-order-new-scheme']}
  13. Moment bounds
  14. Weak convergence of the scheme \ref{['eq:second-order-new-scheme']}
  15. Conclusion and discussion
  16. ...and 4 more sections

Key Result

Lemma 2.4

(c.f.Cerrai2001second) Let Assumption ass:coefficient_function_assumption and inequality assu:a3'-mono-condi be fulfilled. Then, the solution to SDE eq:Problem_SDE$X(t, x; s), 0 \leq t \leq s \leq T$ is $2q+2$ times differentiable with respect to the initial data $x \in \mathbb{R}^d$ and for any $q where $C(T,\mathds{P},j)$ is a constant that depends on $T,\mathds{P},j$. Here $D^j$'s refer to der

Figures (4)

  • Figure 1: Illustration of approximations of truncation functions $x \mathds{1}_{\left\vert x\right\vert\leq h^{-1}}+\text{sign}(x)h^{-1}\mathds{1}_{\left\vert x\right\vert\geq h^{-1}}$, $\frac{x}{1+h\left\vert x\right\vert}$, and $h^{-1}\tanh(h x)$.
  • Figure 2: Example \ref{['ex:numerical_ex_linear_diff']}: weak convergence orders with $\varphi(x) = x^2$ (Left) and $\varphi(x) =\cos(x)$ (Right).
  • Figure 3: Case II in Example \ref{['ex:numerical_non_dri_diff_critical_case']}: weak convergence orders with $\varphi(x) = x^2$ (Left) and $\varphi(x) =\cos(x)$ (Right).
  • Figure 4: Example \ref{['exm:sde-twod']}: weak convergence orders of the scheme \ref{['eq:second-order-new-scheme']} with $\varphi(x) = x$ (Left) and $\varphi(x) = \cos(x)$ (Right).

Theorems & Definitions (39)

  • Remark 2.2
  • Lemma 2.4
  • Theorem 2.5: The approximation theorem, c.f. wang2021weak
  • Lemma 3.2
  • Theorem 3.3: Weak convergence order for the modified Euler scheme \ref{['eq:Euler-scheme']}
  • Remark 3.4
  • Remark 3.5
  • Lemma 3.6: Moment bounds of numerical solutions \ref{['eq:second-order-new-scheme']}
  • Theorem 3.8: Weak convergence order for the scheme \ref{['eq:second-order-new-scheme']}
  • Remark 3.9
  • ...and 29 more