A generalized coupling approach for the weak approximation of stochastic functional differential equations

In this paper, we study functional type weak approximation of weak solutions of stochastic functional differential equations by means of the Euler--Maruyama scheme. Under mild assumptions on the coefficients, we provide a quantitative error estimate for the weak approximation in terms of the L\'evy--Prokhorov metric of probability laws on the path space. The weak error estimate obtained in this paper is sharp in the topological and quantitative senses in some special cases. We apply our main result to ten concrete examples appearing in a wide range of science and obtain a weak error estimate for each model. The proof of the main result is based on the so-called generalized coupling of probability measures.