Numerical approximation of McKean–Vlasov SDEs via stochastic gradient descent

We propose a novel approach to numerically approximate McKean-Vlasov stochastic differential equations (MV-SDEs) using stochastic gradient descent (SGD) while avoiding the use of interacting particle systems (IPSs) and the associated simulation costs required to achieve the 'propagation of chaos' limit. The SGD technique is deployed to solve a Euclidean minimization problem, obtained by first representing the MV-SDE as a minimization problem over the set of continuous functions of time, and then approximating the domain with a finite-dimensional sub-space. Convergence is established by proving certain intermediate stability and moment estimates of the relevant stochastic processes, including the tangent processes. Numerical experiments illustrate the competitive performance of our SGD-based method compared with the IPS benchmarks. This work offers a theoretical foundation for using the SGD method in the context of numerical approximation of MV-SDEs, and provides analytical tools to study its stability and convergence.