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On the convergence of adaptive approximations for stochastic differential equations

James Foster, Andraž Jelinčič

TL;DR

This work addresses the convergence of adaptive, potentially non-previsible, step-size schemes for stochastic differential equations. By leveraging rough path theory, it proves pathwise convergence of standard solvers (Milstein, Heun) and the SPaRK scheme under no-skip, dyadic partitions and unbiased Lévy-area terms, without requiring fine discretization of second iterated integrals. A counterexample shows the necessity of the no-skip condition, while numerical experiments on the SABR model illustrate practical gains from adaptive controls. The results pave the way for robust adaptive SDE solvers and introduce SPaRK as a promising, higher-accuracy alternative in adaptive contexts.

Abstract

In this paper, we study numerical approximations for stochastic differential equations (SDEs) that use adaptive step sizes. In particular, we consider a general setting where decisions to reduce step sizes are allowed to depend on the future trajectory of the underlying Brownian motion. Since these adaptive step sizes may not be previsible, the standard mean squared error analysis cannot be directly applied to show that the numerical method converges to the solution of the SDE. Building upon the pioneering work of Gaines and Lyons, we instead use rough path theory to establish pathwise convergence for a wide class of adaptive numerical methods on general Stratonovich SDEs (with sufficiently smooth vector fields). To our knowledge, this is the first convergence guarantee that applies to standard solvers, such as the Milstein and Heun methods, with non-previsible step sizes. In our analysis, we require adaptive step sizes to have a "no skip" property and to take values at only dyadic times. Secondly, in contrast to the Euler-Maruyama method, we require the SDE solver to have unbiased "Lévy area" terms in its Taylor expansion. We conjecture that for adaptive SDE solvers more generally, convergence is still possible provided the method does not introduce "Lévy area bias". We present a simple example where the step size control can skip over previously considered times, resulting in the numerical method converging to an incorrect limit (i.e. not the Stratonovich SDE). Finally, we conclude with an experiment demonstrating the accuracy of Heun's method and a newly introduced Splitting Path-based Runge-Kutta scheme (SPaRK) when used with adaptive step sizes.

On the convergence of adaptive approximations for stochastic differential equations

TL;DR

This work addresses the convergence of adaptive, potentially non-previsible, step-size schemes for stochastic differential equations. By leveraging rough path theory, it proves pathwise convergence of standard solvers (Milstein, Heun) and the SPaRK scheme under no-skip, dyadic partitions and unbiased Lévy-area terms, without requiring fine discretization of second iterated integrals. A counterexample shows the necessity of the no-skip condition, while numerical experiments on the SABR model illustrate practical gains from adaptive controls. The results pave the way for robust adaptive SDE solvers and introduce SPaRK as a promising, higher-accuracy alternative in adaptive contexts.

Abstract

In this paper, we study numerical approximations for stochastic differential equations (SDEs) that use adaptive step sizes. In particular, we consider a general setting where decisions to reduce step sizes are allowed to depend on the future trajectory of the underlying Brownian motion. Since these adaptive step sizes may not be previsible, the standard mean squared error analysis cannot be directly applied to show that the numerical method converges to the solution of the SDE. Building upon the pioneering work of Gaines and Lyons, we instead use rough path theory to establish pathwise convergence for a wide class of adaptive numerical methods on general Stratonovich SDEs (with sufficiently smooth vector fields). To our knowledge, this is the first convergence guarantee that applies to standard solvers, such as the Milstein and Heun methods, with non-previsible step sizes. In our analysis, we require adaptive step sizes to have a "no skip" property and to take values at only dyadic times. Secondly, in contrast to the Euler-Maruyama method, we require the SDE solver to have unbiased "Lévy area" terms in its Taylor expansion. We conjecture that for adaptive SDE solvers more generally, convergence is still possible provided the method does not introduce "Lévy area bias". We present a simple example where the step size control can skip over previously considered times, resulting in the numerical method converging to an incorrect limit (i.e. not the Stratonovich SDE). Finally, we conclude with an experiment demonstrating the accuracy of Heun's method and a newly introduced Splitting Path-based Runge-Kutta scheme (SPaRK) when used with adaptive step sizes.
Paper Structure (15 sections, 17 theorems, 112 equations, 3 figures, 1 table)

This paper contains 15 sections, 17 theorems, 112 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Examples of adaptive numerical methods
  3. Organisation of the paper
  4. Notation
  5. Main result
  6. Rough path convergence of piecewise polynomial approximations
  7. Pathwise convergence of "polynomial-driven" CDEs with small local errors
  8. Numerical methods for SDEs viewed as approximations of CDEs
  9. Counterexample involving non-nested step sizes
  10. Numerical example
  11. Conclusion
  12. Improved local accuracy of the SPaRK method (\ref{['eq:srk']})
  13. Generating increments and integrals of Brownian motion
  14. Norm of the determinant for a Gaussian random matrix
  15. A lower bound for the SABR-specific previsible step size

Key Result

Theorem 1.5

Suppose that the vector fields of the SDE (eq:strat_SDE) are bounded and differentiable with bounded derivatives. Moreover, suppose $g$ is twice differentiable with $g^{\prime\prime}$ bounded and $\alpha$-Hölder continuous where $\alpha\in(0,1)$. We assume the following about the SDE solver: \newlab Then, letting $Y^n$ denote the numerical solution computed over $\mathcal{D}_n = \{t_k^n\}_{k\geq 0

Figures (3)

  • Figure 1: In the Monte Carlo paradigm, Brownian motion is discretized and then mapped to a numerical solution of the SDE. Despite the random fluctuations of underlying Brownian motion, which can occasionally be very large, numerical methods for SDEs typically only use fixed step sizes.
  • Figure 1: Estimated convergence rates for the different numerical methods and step size controls. Errors for the constant and variable step size SDE solvers were estimated with $N = 100,000$ samples. As the Euler-Maruyama, Heun and SPaRK methods use $1,2$ and $3$ vector field evaluations per step, the x-axis is obtained as the average number of steps multiplied by these numbers.
  • Figure 1: Illustration of the Brownian increments and areas in Theorem \ref{['thm:levy_with_areas']} (diagram from foster2020thesis)

Theorems & Definitions (59)

  • Definition 1.1
  • Definition 1.2: No-skip partition
  • Definition 1.3
  • Example 1.4
  • Theorem 1.5: Convergence of adaptive SDE approximations, informal version
  • Remark 1.6
  • Definition 1.7: No$\space$-area Milstein method with embedded Euler for Itô SDEs
  • Remark 1.8
  • Definition 1.9: Heun's method with embedded Euler-Maruyama
  • Definition 1.10: Splitting Path Runge-Kutta (SPaRK) with embedded Heun
  • ...and 49 more