Optimization of Numerical Solutions of Stochastic Differential Equations with Time Delay

The research focuses on the optimization of numerical solutions of neutral stochastic differential equations with time delay. Analyzing approaches such as Euler-Maruyama, backward Euler, and θ-Euler-Maruyama methods, the goal is to investigate the characteristics of approximate solutions, especially stability and boundedness. This study contributes to the understanding of the complexity of stochastic processes, offering a perspective for further mathematical modeling and optimization. The study of the characteristics of approximate solutions includes a detailed analysis of their stability and limitations, providing insight into the system’s behavior in dynamic conditions. This analysis lays the foundations for the improvement of numerical methods and more precise modeling of stochastic processes with time delay. The aforementioned approaches, such as the Euler-Maruyama, backward Euler, and θ-Euler-Maruyama methods, provide tools for understanding and solving complex mathematical challenges. Through an interdisciplinary approach, this study sheds light on the field of optimization of numerical solutions, encouraging further development of theoretical and practical aspects of stochastic differential equations.