Stratonovich SDEs in High Dimensions: Weak Convergence Approach to Generalized Trivial Coupling

This paper uses a generalized trivial coupling method to study the weak convergence of Stratonovich stochastic differential equations (SDEs) in high-dimensional environments. In contrast to standard methods that depend on the invertibility of the diffusion matrix or are restricted to low dimensions, our methodology deals with degenerate and non-invertible systems in arbitrary dimensions. We develop a theoretical framework for weak approximation, establishing the weak order of convergence under standard regularity assumptions. The method is integrated with multi-level Monte Carlo (MLMC) techniques to achieve optimal computational complexity O(ϵ−2) for mean square error ϵ2. We provide a detailed analysis of the variance decay, proving that Vℓ=O(δℓ2) for the coupled estimator, which leads to optimal complexity. Representative numerical tests on a 4-dimensional non-invertible system validate the theoretical results, proving the usefulness of the suggested approach. The numerical results demonstrate the computing benefits of the multi-level technique in high dimensions and validate the theoretical convergence rates. By offering both theoretical insights and useful computational methods for high-dimensional stochastic systems, this work advances the expanding field of computational stochastic analysis.