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The Stochastic TR-BDF2 Scheme of Order 2

Tomás Caraballo, Macarena Gómez-Mármol, Ignacio Roldán

TL;DR

This work extends the deterministic TR-BDF2 scheme to stochastic differential equations, achieving second-order strong convergence and preserving A-stability. By leveraging a refined partition and Itô--Taylor corrections up to order two, the authors derive a two-stage stochastic method with explicit update formulas and stochastic correction terms, plus several reduced variants. They prove that the full stochastic TR-BDF2 method attains order $2$ under standard regularity and growth conditions, and establish $A$-stability and mean-square stability in relevant regimes, including comparisons to Itô--Taylor order-2 schemes. Numerical experiments on stiff and non-stiff problems validate the theoretical results and illustrate the method’s advantages for slow-fast dynamics and stiff stochastic systems.

Abstract

Our main objective in this paper is to develop a second-order stochastic numerical method which generalizes the well-known deterministic TR-BDF2 scheme. Since most stochastic techniques used for approximating the solution of a stochastic differential equation may have lower order compared to the deterministic case, we have elaborated a scheme which not only preserves the second-order accuracy of the original scheme in the stochastic framework, but also its $A$-stability. Once we obtain the scheme and prove its second-order accuracy and $A$-stability, which is not a trivial task, we also state a result concerning its $MS$-stability. This concept is also analyzed for different parameter ranges in our scheme and the It{ô}--Taylor approximation of order 2, revealing scenarios where, for certain time step sizes, the developed method is $MS$-stable while the It{ô}--Taylor one is not. This concept is really useful to tackle slow-fast problems such as stiff ones, which we aim to explore further in future work. Finally, we validate the theoretical results with some academic test cases.

The Stochastic TR-BDF2 Scheme of Order 2

TL;DR

This work extends the deterministic TR-BDF2 scheme to stochastic differential equations, achieving second-order strong convergence and preserving A-stability. By leveraging a refined partition and Itô--Taylor corrections up to order two, the authors derive a two-stage stochastic method with explicit update formulas and stochastic correction terms, plus several reduced variants. They prove that the full stochastic TR-BDF2 method attains order under standard regularity and growth conditions, and establish -stability and mean-square stability in relevant regimes, including comparisons to Itô--Taylor order-2 schemes. Numerical experiments on stiff and non-stiff problems validate the theoretical results and illustrate the method’s advantages for slow-fast dynamics and stiff stochastic systems.

Abstract

Our main objective in this paper is to develop a second-order stochastic numerical method which generalizes the well-known deterministic TR-BDF2 scheme. Since most stochastic techniques used for approximating the solution of a stochastic differential equation may have lower order compared to the deterministic case, we have elaborated a scheme which not only preserves the second-order accuracy of the original scheme in the stochastic framework, but also its -stability. Once we obtain the scheme and prove its second-order accuracy and -stability, which is not a trivial task, we also state a result concerning its -stability. This concept is also analyzed for different parameter ranges in our scheme and the It{ô}--Taylor approximation of order 2, revealing scenarios where, for certain time step sizes, the developed method is -stable while the It{ô}--Taylor one is not. This concept is really useful to tackle slow-fast problems such as stiff ones, which we aim to explore further in future work. Finally, we validate the theoretical results with some academic test cases.
Paper Structure (7 sections, 1 theorem, 18 equations)

This paper contains 7 sections, 1 theorem, 18 equations.

Table of Contents

  1. Subjclass:
  2. Keywords:
  3. Problem Formulation
  4. The Itô--Taylor analysis
  5. The Itô--Taylor expansion
  6. Numerical method based on Itô--Taylor expansion
  7. The stochastic TR-BDF2 method

Key Result

Theorem 1.1

Suppose that Then, the stochastic differential equation est has a path-wise unique strong solution $X_{t}$ on $\mathopen{}\mathclose{\left[0, T\right]$ with $E\mathopen{}\mathclose{\left[\sup_{0 \leq t \leq T}}\mathopen{}\mathclose{\left\|X_{t}\right\|^{2}}\right] < \infty$.

Theorems & Definitions (6)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3