The Stochastic TR-BDF2 Scheme of Order 2

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.

💡 Research Summary

The paper introduces a novel stochastic numerical integrator that extends the well‑known deterministic TR‑BDF2 method to stochastic differential equations (SDEs) while preserving second‑order strong convergence and A‑stability. After a concise introduction that motivates the need for high‑order stochastic solvers in applications such as population dynamics, neuroscience, and chemical kinetics, the authors formulate the problem in a standard d‑dimensional SDE framework with globally Lipschitz drift and diffusion coefficients, ensuring existence and uniqueness of the strong solution.

Section 2 reviews the Itô–Taylor expansion, defining hierarchical index sets, the operators L₀ and L_j, and the construction of numerical schemes by truncating the expansion at a chosen order p. The familiar Euler–Maruyama (p = 0.5) and Milstein (p = 1) methods are recovered as special cases, illustrating how higher‑order schemes can be systematically derived.

Section 3 recalls the deterministic TR‑BDF2 scheme, a two‑stage method that combines a trapezoidal predictor with a backward differentiation formula of order two. The method depends on a parameter γ∈(0,1); for any γ it is A‑stable, and for the specific choice γ = 2 − √2 it also enjoys L‑stability.

The core contribution appears in Section 4, where the stochastic TR‑BDF2 method is constructed. The first sub‑step mirrors the deterministic trapezoidal predictor, while the second sub‑step incorporates the implicit BDF2 correction together with Milstein‑type stochastic integrals. Concretely, the scheme reads

 y_{n+γ}=y_n+½ h γ