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 technique of SGD is deployed to solve a Euclidean minimization problem, which is obtained by first representing the MV-SDE as a minimization problem over the set of continuous functions of time, and then by 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 ones). 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：本文提出了一种新颖的数值逼近McKean-Vlasov随机微分方程（MV-SDE）的方法，该方法采用随机梯度下降（SGD）技术，同时避免了交互粒子系统的使用。将SGD技术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，然后利用有限维子空间逼近定义域而得到。通过证明相关随机过程（包括切向过程）的某些中间稳定性与矩估计，建立了收敛性。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数值逼近背景下使用SGD方法提供了理论基础，并提供了分析其稳定性与收敛性的解析工具。