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Error Distribution for One-Dimensional Stochastic Differential Equation Driven By Fractional Brownian Motion

Kento Ueda

TL;DR

The paper addresses the problem of quantifying asymptotic errors for one-dimensional SDEs driven by fractional Brownian motion, introducing a unified pathwise evaluation based on Hölder dynamics and controlled paths. It develops a general remainder estimate (maintheorem) and uses a novel Y^{(m,\rho)} interpolation to decompose and bound the error, enabling precise asymptotic characterizations of numerical schemes. The main contributions are the complete determination of the Milstein scheme's asymptotic error for arbitrary order and the new determination of Crank-Nicolson's asymptotic error in the regime $1/4<H\leq 1/3$, under suitable smoothness and drift conditions, along with a rigorous framework that handles the non-Markovian, rough-path nature of fBm. These results advance rigorous error analysis for non-Markovian SDE solvers, guiding scheme choice and enabling accurate error predictions in applications with fBm dynamics.

Abstract

This paper deals with asymptotic errors, limit theorems for errors between numerical and exact solutions of stochastic differential equation (SDE) driven by one-dimensional fractional Brownian motion (fBm). The Euler-Maruyama, higher-order Milstein, and Crank-Nicolson schemes are among the most studied numerical schemes for SDE (fSDE) driven by fBm. Most previous studies of asymptotic errors have derived specific asymptotic errors for these schemes as main theorems or their corollary. Even in the one-dimensional case, the asymptotic error was not determined for the Milstein or the Crank-Nicolson method when the Hurst exponent is less than or equal to $1/3$ with a drift term. We obtained a new evaluation method for convergence and asymptotic errors. This evaluation method improves the conditions under which we can prove convergence of the numerical scheme and obtain the asymptotic error under the same conditions. We have completely determined the asymptotic error of the Milstein method for arbitrary orders. In addition, we have newly determined the asymptotic error of the Crank-Nicolson method for $1/4<H\leq 1/3$.

Error Distribution for One-Dimensional Stochastic Differential Equation Driven By Fractional Brownian Motion

TL;DR

The paper addresses the problem of quantifying asymptotic errors for one-dimensional SDEs driven by fractional Brownian motion, introducing a unified pathwise evaluation based on Hölder dynamics and controlled paths. It develops a general remainder estimate (maintheorem) and uses a novel Y^{(m,\rho)} interpolation to decompose and bound the error, enabling precise asymptotic characterizations of numerical schemes. The main contributions are the complete determination of the Milstein scheme's asymptotic error for arbitrary order and the new determination of Crank-Nicolson's asymptotic error in the regime , under suitable smoothness and drift conditions, along with a rigorous framework that handles the non-Markovian, rough-path nature of fBm. These results advance rigorous error analysis for non-Markovian SDE solvers, guiding scheme choice and enabling accurate error predictions in applications with fBm dynamics.

Abstract

This paper deals with asymptotic errors, limit theorems for errors between numerical and exact solutions of stochastic differential equation (SDE) driven by one-dimensional fractional Brownian motion (fBm). The Euler-Maruyama, higher-order Milstein, and Crank-Nicolson schemes are among the most studied numerical schemes for SDE (fSDE) driven by fBm. Most previous studies of asymptotic errors have derived specific asymptotic errors for these schemes as main theorems or their corollary. Even in the one-dimensional case, the asymptotic error was not determined for the Milstein or the Crank-Nicolson method when the Hurst exponent is less than or equal to with a drift term. We obtained a new evaluation method for convergence and asymptotic errors. This evaluation method improves the conditions under which we can prove convergence of the numerical scheme and obtain the asymptotic error under the same conditions. We have completely determined the asymptotic error of the Milstein method for arbitrary orders. In addition, we have newly determined the asymptotic error of the Crank-Nicolson method for .
Paper Structure (17 sections, 50 theorems, 198 equations, 7 tables)

This paper contains 17 sections, 50 theorems, 198 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Definition and Notation
  3. Pathwise integral driven by Hölder path
  4. Discrete Hölder norm
  5. Main theorem
  6. Asymptotic error of the ($k$)-Milstein scheme and the Crank-Nicolson scheme
  7. Interpolation between exact and numerical solution
  8. Representation of the error term
  9. Decomposition of the error term
  10. Estimate of each term and whole remainder term.
  11. Estimate of $Y,M,Z,N$
  12. Estimate of the decomposed terms
  13. Proof of Theorem \ref{['theorem:maintheorem']}
  14. Calculation of asymptotic error
  15. Decomposition of $\epsilon^{1,(m,A)}$
  16. ...and 2 more sections

Key Result

Theorem 1

(Theorem theorem:Theorem_of_distribution_M) Let $\sigma\in C_b^{k+1}, b\in C_b^2,0<H<1,k\in \mathbb{Z}_{\geq 2}, l\in \mathbb{Z}_{\geq 1}$.

Theorems & Definitions (101)

  • Theorem 1
  • Theorem 2: Theorem \ref{['theorem:Theorem_of_distribution_CN']}
  • Remark
  • Remark
  • Proposition 3: Integral driven by $x$
  • Proposition 4
  • Proposition 5: Operation for a controlled path
  • Proposition 6: Stochastic differential equation
  • Proposition 7: Itô's formula
  • Proposition 8: Garsia-Rodemich-Ramsey inequality
  • ...and 91 more