Stochastic Fokker–Planck Equations for Conditional McKean–Vlasov Jump Diffusions and Applications to Optimal Control

• Published in: SIAM journal on control and optimization 2023-01, Vol.61 (3), p.1472-1493
• Main Authors: Agram, Nacira, Øksendal, Bernt
• Format: Article
• Language: English
• Subjects: common noise; conditional McKean-Vlasov differential equation; HJB equation; jump diffusion; optimal control; stochastic Fokker-Planck equation
• ISSN: 0363-0129; 1095-7138; 1095-7138
• DOI: 10.1137/21M1461034

The purpose of this paper is to study optimal control of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffu-sions, for short). To this end, we first prove a stochastic Fokker-Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate a Hamilton-Jacobi-Bellman equation for the optimal control of conditional McKean-Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker-Planck equation reduces to a stochastic par-tial differential equation for the Radon-Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the linear-quadratic optimal control problem of conditional stochastic McKean-Vlasov jump diffusions, and optimal consumption from a cash flow.