We propose a novel approach to numerically approximate McKean-Vlasov stochastic differential equations (MV-SDE) using stochastic gradient descent (SGD) while avoiding the use of interacting particle systems. 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 subspace. 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 to 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.

翻译：本文提出了一种通过随机梯度下降（SGD）数值逼近McKean-Vlasov随机微分方程（MV-SDE）的新方法，该方法避免了使用交互粒子系统。SGD技术被用于求解一个欧几里得最小化问题，该问题首先通过将MV-SDE表示为时间连续函数集合上的最小化问题获得，随后通过有限维子空间逼近定义域。通过证明相关随机过程（包括切向过程）的若干中间稳定性与矩估计，我们建立了算法的收敛性。数值实验表明，与交互粒子系统基准方法相比，我们基于SGD的方法具有竞争性性能。本工作为在MV-SDE数值逼近中使用SGD方法提供了理论基础，并为研究其稳定性和收敛性提供了分析工具。