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High order splitting methods for SDEs satisfying a commutativity condition

James Foster, Goncalo dos Reis, Calum Strange

TL;DR

This work introduces a practical, non-rough-path-based framework for constructing high-order strong splitting methods for Stratonovich SDEs with a commutativity condition, by replacing the Brownian/timelike driver with a carefully designed piecewise-linear path. Through Stratonovich and path-controlled Taylor expansions, the authors develop local-error and global-convergence results that, together with Milstein–Tretyakov style analysis, yield strong convergence rates of up to $O(h^{3/2})$ under commutativity, using path-moments and iterated-integral matching. They provide explicit path constructions (Lie–Trotter, Strang, high-order variants, and Shifted ODE schemes) and demonstrate substantial empirical gains (e.g., CIR, stochastic oscillator, FHN, Langevin dynamics) over existing methods. The paper also develops unbiased estimators for higher-order iterated integrals (Lévy areas) and details integral-cancellation mechanisms under commutativity, enabling practical high-order solvers. Overall, the approach offers a versatile, implementable framework for advancing strong convergence in SDE splitting methods with broad applicability in finance, physics, and data science.

Abstract

In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs $-$ which can be discretized to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox-Ingersoll-Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh-Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of $O(h^{3/2})$ and outperform schemes previously proposed in the literature.

High order splitting methods for SDEs satisfying a commutativity condition

TL;DR

This work introduces a practical, non-rough-path-based framework for constructing high-order strong splitting methods for Stratonovich SDEs with a commutativity condition, by replacing the Brownian/timelike driver with a carefully designed piecewise-linear path. Through Stratonovich and path-controlled Taylor expansions, the authors develop local-error and global-convergence results that, together with Milstein–Tretyakov style analysis, yield strong convergence rates of up to under commutativity, using path-moments and iterated-integral matching. They provide explicit path constructions (Lie–Trotter, Strang, high-order variants, and Shifted ODE schemes) and demonstrate substantial empirical gains (e.g., CIR, stochastic oscillator, FHN, Langevin dynamics) over existing methods. The paper also develops unbiased estimators for higher-order iterated integrals (Lévy areas) and details integral-cancellation mechanisms under commutativity, enabling practical high-order solvers. Overall, the approach offers a versatile, implementable framework for advancing strong convergence in SDE splitting methods with broad applicability in finance, physics, and data science.

Abstract

In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs which can be discretized to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox-Ingersoll-Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh-Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of and outperform schemes previously proposed in the literature.
Paper Structure (18 sections, 19 theorems, 171 equations, 9 figures, 6 tables)

This paper contains 18 sections, 19 theorems, 171 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Main assumption and preliminary results
  4. Taylor expansions and error analysis
  5. Stratonovich Taylor expansion
  6. Controlled Taylor expansion
  7. Main result
  8. Application to commutative SDEs
  9. Paths
  10. Examples
  11. Cox-Ingersoll-Ross model
  12. Additive noise SDEs, such as a stochastic anharmonic oscillator
  13. FitzHugh-Nagumo model
  14. Underdamped Langevin dynamics
  15. Conclusion and future work
  16. ...and 3 more sections

Key Result

Theorem 1.4

Given a fixed number of steps $N$, we will define a numerical solution $Y = \{Y_k\}_{0\space\leq\space k\space\leq N}$ for the SDE (eq:strat SDE) over the finite time horizon $[0,T]$ as follows, where each piecewise linear path $\gamma_k$ is constructed from $\{W_t : t\in[\frac{kT}{N},\frac{(k+1)T}{N}]\}$, is sufficiently regular (see Assumption assump:scaling), and for some fixed $p\in \{\frac{m

Figures (9)

  • Figure 1: In the Monte Carlo paradigm, information about the Brownian motion is generated and then mapped to a numerical solution of the SDE. Typically, only Brownian increments are sampled.
  • Figure 1: Illustration of piecewise linear paths associated with various splitting methods for SDEs. (these diagrams are not drawn accurately; the "vertical" pieces are only the same in distribution)
  • Figure 1: $S_N$ estimated for (\ref{['eq:cir']}) with 100,000 sample paths as a function of step size $h = \frac{T}{ N}$.
  • Figure 1: Space-time Lévy swing gives the side where the path has greater space-time Lévy area.
  • Figure 2: Space-time Lévy area gives the area between a Brownian path and its linear approximant.
  • ...and 4 more figures

Theorems & Definitions (66)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Theorem 1.4: Convergence of path-based splitting for SDEs (informal version)
  • Remark 2.1: Comment on Assumption \ref{['assump:scaling']}
  • Theorem 2.2: Fourth moment bound for CDEs
  • Proof 1
  • Proposition 3.1: Stochastic Taylor expansion of the Stratonovich SDE (\ref{['eq:strat SDE']}) bayer2006geometry, kloeden1992numerical
  • Proposition 3.2
  • Remark 3.3
  • ...and 56 more