This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

该手稿涉及具有随机系数和/或随机项的泛函微分方程（FDE）解的稳定性的研究。我们专注于研究随机/随机功能系统的不同类型的稳定性，特别是随机延迟微分方程（SDDE）。引入适当的Lyapunov泛函使我们能够研究随机稳定性，渐近随机稳定性，渐近均方稳定性，均方指数稳定性，整体指数均方稳定性和实际均匀指数稳定性的必要条件。给出了一些带有数值模拟的例子，以加强理论结果。使用我们的理论研究，可以揭示流行病学和生态数学模型的重要方面。在生态学上 研究了尼科尔森s蝇方程的动力学。研究了蝇类灭绝的平衡点的随机稳定性和随机全局指数稳定性的条件。在金融领域，还研究了由具有随机变量系数和时间延迟的布朗运动驱动的布莱克－斯科尔斯市场模型的动力学。