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1/2 order convergence rate of Euler-type methods for time-changed stochastic differential equations with super-linearly growing drift and diffusion coefficients

Shuai Wang, Yuanling Niu, Ying Zhang

TL;DR

The paper addresses the numerical approximation of time-changed SDEs with super-linear coefficients by developing two Euler-type schemes: Backward Euler (implicit) and Projected Euler (explicit). Using a duality principle that links time-changed SDEs to classical SDEs, and a discretization of the inverse subordinator, the authors prove strong convergence in $L_2$ with the optimal rate of $1/2$. They establish precise convergence conditions under growth and Lipschitz-type assumptions and demonstrate that BEM offers broader applicability while PEM provides computational efficiency. Numerical experiments in 1D and 2D illustrate the theoretical rates and highlight the stability advantages of BEM in stiff regimes and the efficiency gains of PEM in non-stiff settings, validating the practical relevance of the proposed methods.

Abstract

This paper investigates the strong convergence properties of two Euler-type methods for a class of time-changed stochastic differential equations (TCSDEs) with super-linearly growing drift and diffusion coefficients. Building upon existing research, we propose a backward Euler method (BEM) and introduce its explicit counterpart -- the projected Euler method (PEM). We prove that both methods converge strongly in the $L_2$-sense at the optimal rate of 1/2. This result extends the applicability of both the BEM and the PEM to a broader class of TCSDEs. Moreover, the two methods offer complementary strengths: while BEM possesses wide applicability, PEM is computationally more efficient. Numerical simulations confirm our theoretical findings and illustrate practical performance of both schemes.

1/2 order convergence rate of Euler-type methods for time-changed stochastic differential equations with super-linearly growing drift and diffusion coefficients

TL;DR

The paper addresses the numerical approximation of time-changed SDEs with super-linear coefficients by developing two Euler-type schemes: Backward Euler (implicit) and Projected Euler (explicit). Using a duality principle that links time-changed SDEs to classical SDEs, and a discretization of the inverse subordinator, the authors prove strong convergence in with the optimal rate of . They establish precise convergence conditions under growth and Lipschitz-type assumptions and demonstrate that BEM offers broader applicability while PEM provides computational efficiency. Numerical experiments in 1D and 2D illustrate the theoretical rates and highlight the stability advantages of BEM in stiff regimes and the efficiency gains of PEM in non-stiff settings, validating the practical relevance of the proposed methods.

Abstract

This paper investigates the strong convergence properties of two Euler-type methods for a class of time-changed stochastic differential equations (TCSDEs) with super-linearly growing drift and diffusion coefficients. Building upon existing research, we propose a backward Euler method (BEM) and introduce its explicit counterpart -- the projected Euler method (PEM). We prove that both methods converge strongly in the -sense at the optimal rate of 1/2. This result extends the applicability of both the BEM and the PEM to a broader class of TCSDEs. Moreover, the two methods offer complementary strengths: while BEM possesses wide applicability, PEM is computationally more efficient. Numerical simulations confirm our theoretical findings and illustrate practical performance of both schemes.

Paper Structure

This paper contains 19 sections, 14 theorems, 102 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Setting and assumptions
  5. Numerical schemes and main results
  6. Approximation of time-changed process
  7. BEM with convergence result
  8. PEM with convergence result
  9. Numerical experiments
  10. One-dimensional non-linear TCSDE
  11. 2D stiff TCSDE
  12. Proof of main results in Section \ref{['sec3']}
  13. Proof overview
  14. Error estimates for time-changed process
  15. Error estimates for SDE
  16. ...and 4 more sections

Key Result

Lemma 2.1

(Duality principle) Under Assumptions A1-A4, if $\{Y (t)\}_{t\geq0}$ is the unique strong solution to SDE do, then the time-changed process $\{Y (E(t))\}_{t\geq0}$ is the unique strong solution to TCSDE fc. Conversely, if $\{X (t)\}_{t\geq0}$ is the unique strong solution to TCSDE fc, then the proce

Figures (4)

  • Figure 1: Simulated sample paths
  • Figure 2: $L_2$ errors between the exact solution and numerical solutions obtained using BEM and PEM with step sizes $h \in\{2^{-6},2^{-7},2^{-8},2^{-9}\}$
  • Figure 3: One path of reference solution $\{X(t)\}_{t\in[0,1]}=\{[X_1(t),X_2(t)]\}_{t\in[0,1]}$ obtained using BEM and PEM with $h_0=2^{-16}$
  • Figure 4: Convergence results for BEM and PEM with different step sizes

Theorems & Definitions (36)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.1
  • Remark 3.4
  • Theorem 3.2
  • Remark 3.5
  • ...and 26 more