分数布朗运动驱动的含多重时滞的分数阶随机延迟微分方程的平稳解
Stationary solution for a fractional stochastic delay differential equations with multiple delays
浏览论文内容
中文总结 AI 辅助
本文研究分数布朗运动驱动的线性随机延迟微分方程，通过基本解和路径wise积分获得显式解，证明在稳定性条件下解收敛于平稳高斯过程，并构造了唯一平稳解，为含延迟和分数记忆的平稳系统建模提供了框架。
中文摘要 AI 辅助
本文研究了一类由分数布朗运动驱动且包含有限个离散时滞的线性随机延迟微分方程。该模型结合了两种记忆来源：漂移项中的延迟反馈以及由分数布朗运动的Hurst参数引起的时间依赖性。我们首先引入了相应确定性延迟方程的基本解，并利用它获得了该解的显式表示。利用基本解的有限变差性质，关于分数布朗运动的随机卷积被逐路径地定义为Riemann-Stieltjes积分。这一表示使我们能够计算该过程的均值和自协方差函数。在关于漂移系数的与延迟无关的稳定性条件下，我们证明了基本解指数衰减，并推导了随机延迟方程的长时间行为。特别地，该解在有限维分布上收敛于一个平稳高斯过程，其极限均值和协方差被显式给出。当Hurst参数满足\(H>1/2\)时，我们证明了极限协方差保留了驱动分数布朗运动的长记忆行为，并渐近地按\(h^{2H-2}\)衰减。最后，通过将噪声扩展为双边分数布朗运动，我们构造了该方程的平稳解并证明了其唯一性。这些结果为建模同时具有延迟效应和分数记忆的平稳系统提供了一个易于处理的框架，并可能应用于随机波动率、能源建模以及其他表现出持续性的时间序列。
英文摘要
This paper studies a linear stochastic delay differential equation driven by fractional Brownian motion and involving a finite number of discrete delays. The model combines two sources of memory: delayed feedback in the drift and temporal dependence induced by the Hurst parameter of the fractional Brownian motion. We first introduce the fundamental solution associated with the corresponding deterministic delay equation and use it to obtain an explicit representation of the solution. The stochastic convolution with respect to fractional Brownian motion is defined pathwise as a Riemann--Stieltjes integral, using the finite-variation properties of the fundamental solution. This representation allows us to compute the mean and auto-covariance function of the process. Under a delay-independent stability condition on the drift coefficients, we prove that the fundamental solution decays exponentially and derive the long-time behaviour of the stochastic delay equation. In particular, the solution converges in finite-dimensional distributions to a stationary Gaussian process, whose limiting mean and covariance are given explicitly. When the Hurst parameter satisfies \(H>1/2\), we show that the limiting covariance preserves the long-memory behaviour of the driving fractional Brownian motion and decays asymptotically as \(h^{2H-2}\). Finally, by extending the noise to a double-sided fractional Brownian motion, we construct a stationary solution of the equation and prove its uniqueness. The results provide a tractable framework for modelling stationary systems with both delay effects and fractional memory, with potential applications to stochastic volatility, energy modelling, and other time series exhibiting persistence.
