Deep learning for conditional McKean-Vlasov Jump diffusions

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 and the verification theorem (HJB equation) are introduced. A linear quadratic conditional mean-field control 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 linear quadratic conditional mean-field control and interbank systemic risk, and we share the resulting numerical outcomes.