Asymptotic error distribution for stochastic Runge--Kutta methods of strong order one

2025/06/10 by Diancong Jin, Jin, Diancong · 2 citations
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Probability (math.PR) #Simulation Techniques and Applications #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2506.08937

openalex publication_date 2025/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work gives the asymptotic error distribution of the stochastic Runge--Kutta (SRK) method of strong order 1 applied to Stratonovich-type stochastic differential equations. For dealing with the implicitness introduced in the diffusion term, we propose a framework to derive the asymptotic error distribution of diffusion-implicit or fully implicit numerical methods, which enables us to construct a fully explicit numerical method sharing the same asymptotic error distribution as the SRK method. Further, we show that the limit distribution U(T) satisfies \mathbf E|U(T)|²≤ e^L₁T(1+η₁)T³ for some η₁≥0 only depending on the coefficients of the SRK method. Thus, we infer that η₁ is the key parameter reflecting the growth rate of the mean-square error of the SRK method. Especially, among the SRK methods of strong order 1, those of weak order 2 correspond to η₁=0, sharing the unified asymptotic error distribution, and have the smallest mean-square errors after a long time. This property is also found for the case of additive noise. It seems that we are the first to give the asymptotic error distribution of fully implicit numerical methods for stochastic differential equations.

Citations

• Asymptotic error distribution of accelerated exponential Euler method for parabolic SPDEs
• Superiority Of Symplectic Methods For Stochastic Hamiltonian System Via Asymptotic Error Distribution
• Backward Euler method for stochastic differential equations with non-Lipschitz coefficients
• Error Distribution Of The Euler Approximation Scheme For Stochastic Volterra Integral Equations
• Convergence analysis for minimum action methods coupled with a finite difference method
• Asymptotic error distribution for the Euler scheme with locally\n Lipschitz coefficients
• A Fundamental Mean-Square Convergence Theorem for SDEs with Locally Lipschitz Coefficients and Its Applications
• Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion

Cited by

• Symplectic methods for stochastic Hamiltonian systems: asymptotic error distributions and Hamiltonian-specific analysis
• Asymptotic error distribution of numerical methods for parabolic SPDEs with multiplicative noise

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