The Euler Scheme for Fractional Stochastic Delay Differential Equations with Additive Noise

ArXiv ID: 2402.08513 “View on arXiv”

Authors: Unknown

Abstract

In this paper we consider the Euler-Maruyama scheme for a class ofstochastic delay differential equations driven by a fractional Brownian motion with index $H\in(0,1)$. We establish the consistency of the scheme and study the rate of convergence of the normalized error process. This is done by checking that the generic rate of convergence of the error process with stepsize $Δ_{“n”}$ is $Δ_{“n”}^{"\min{H+\frac{1"}{“2”},3H,1}}$. It turned out that such a rate is suboptimal when the delay is smooth and $H>1/2$. In this context, and in contrast to the non-delayed framework, we show that a convergence of order $H+1/2$ is achievable.

Keywords: Stochastic Delay Differential Equations, Fractional Brownian Motion, Euler-Maruyama Scheme, Numerical Methods, Convergence Rate, Quantitative Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 2.1/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, featuring advanced concepts like fractional Brownian motion, Malliavin calculus, and detailed convergence proofs, but it is purely theoretical with no empirical backtests, datasets, or implementation details.

  flowchart TD
    A["Research Goal: Analyze Euler-Maruyama Scheme<br>for Fractional Stochastic Delay DEs"] --> B["Methodology: Consistency & Convergence Analysis"]
    B --> C["Input: Fractional Brownian Motion<br>Index H ∈ 0,1"]
    C --> D["Computational Process:<br>Normalized Error Process<br>with Step Size Δn"]
    D --> E{"Rate Calculation:<br>Δn^min{H+0.5, 3H, 1"}}
    E --> F["Suboptimal Rate<br>identified for H > 0.5"]
    F --> G["Key Finding: Improved Rate H + 0.5<br>achieved with smooth delay"]