ON THE APPROXIMATION OF STOCHASTIC DELAY EQUATIONS IN INFINITE-DIMENSIONAL SPACES

Анотація

The article presents a detailed scheme for the mean square approximation of evolutionary stochastic delay equations in infinite-dimensional spaces. The primary focus lies in substituting the original system with delay by a system of evolutionary stochastic equations without delay. The proposed approach involves partitioning the delay interval into subintervals and constructing a corresponding system of equations that approximates the original system's behavior. Notably, the number of equations in the approximating system grows as the number of partition subintervals increases. A significant result of this study demonstrates that, as the partitioning becomes finer (i.e., the number of subintervals approaches infinity), the mean square distance between the solutions of the delay equation and the solutions of the delay-free approximating system converges to zero. The theoretical framework of the approximation method leverages key concepts and results from infinite-dimensional stochastic analysis, incorporating tools to address the challenges posed by the functional nature of the delay term and the unboundedness of the state space. The study not only generalizes earlier finite-dimensional results to the infinite-dimensional setting but also extends the methods used for deterministic delay systems to stochastic systems. The methodology builds on the classical idea of decomposing the solution of the delay equation using a Taylor expansion in terms of the delay length $h > 0$. This decomposition allows the construction of an approximating system that replaces the original delay equation with a system of Cauchy problems for ordinary differential equations (ODEs). The results have significant implications for practical applications, particularly for systems where delays naturally arise, such as in stochastic control, population dynamics, and infinite-dimensional systems described by stochastic PDEs. The ability to replace complex delay systems with delay-free approximations not only simplifies numerical computations but also provides insight into the underlying dynamics of these systems. By rigorously establishing the conditions under which the approximation is valid, this work contributes to the theoretical foundation of stochastic delay equations in infinite-dimensional spaces and offers a robust tool for analyzing and simulating such systems.

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