Stationary solution for a fractional stochastic delay differential equations with multiple delays

Abstract

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.