Deep Learning for Mean-Field Systems with Common Noise

Abstract

The current paper focuses on using deep learning methods to optimize the control of conditional McKean-Vlasov jump diffusions. We begin by exploring the dynamics of multi-particle jump-diffusion and presenting the propagation of chaos. The optimal control problem in the context of conditional McKean-Vlasov jump-diffusion is introduced along with the verification theorem (HJB equation). A linear quadratic conditional mean-field (LQ CMF) is discussed to illustrate these theoretical concepts. Then, we introduce a deep-learning algorithm that combines neural networks for optimization with path signatures for conditional expectation estimation. The algorithm is applied to practical examples, including LQ CMF and interbank systemic risk, and we share the resulting numerical outcomes.

Keywords: McKean-Vlasov jump diffusion, signatures, common noise, deep learning