Asymptotics of the Solution to a Multidimensional Renewal Equation in Matrix Form

• Дата: 2025
• Автор: Ярова, Оксана
• Формат: Стаття
• Мова: Українська
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2025
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/342050
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The paper investigates multidimensional renewal equations, which represent an important class of integral equations associated with stochastic processes possessing renewal moments. Such equations naturally arise in the theory of random evolutions, Markov and semi-Markov processes, as well as in the modeling of systems that periodically return to their initial state. Particular attention is given to the case where the equation is represented in matrix form, which makes it possible to generalize classical scalar relations to systems of equations suitable for describing multicomponent processes. A renewal equation with a nonlinear normalizing factor is considered. This factor complicates analytical analysis but makes the model more flexible and closer to real-world applications. To obtain the solution, the Laplace transform method is applied, allowing the transition from the integral form of the renewal equation to an algebraic matrix representation suitable for further analysis. An explicit expression for the Laplace transform of the solution to the renewal equation is derived, which is a key step toward recovering the time-dependent behavior of the solution. In addition to the theoretical analysis, the paper presents an example for a specific renewal function illustrating the effectiveness of the proposed approach. The main characteristics of the solution are calculated, and the influence of the renewal function parameters on the system’s behavior is analyzed. The obtained results can be applied to the study of stochastic systems with a renewal structure, as well as in problems of applied probability, reliability theory, and modeling of complex processes with multidimensional dynamics