Linear–quadratic stochastic teams and zero-sum differential games for jump-diffusion systems with Markovian-switching coefficients under partial observations

We consider the two-player linear–quadratic (LQ) team and zero-sum differential game under partial observations for stochastic differential equations (SDEs) with Markovian-switching coefficients driven by (Brownian and Poisson) jump-diffusion processes. In both problems, each player has access to own partial observations of jump-diffusion processes, where the intersection of these two partial observations may not be empty. Our problems describe a practical situation that two players in a network require to optimize the same objective functional, where they have individual partial observations, while sharing a common observation (the intersection of the individual partial observations). In both team and zero-sum game, we obtain the explicit feedback representations of the open-loop type optimal solutions in terms of the filtering processes given partial observations, provided that the corresponding coupled Riccati differential equations (CRDEs) admit the strongly regular solutions. Several new techniques, including the variational approach, the decoupling method of forward-backward SDEs via the four-step scheme, the completion-of-squares method, and the Riccati approach, have to be developed to obtain the optimal solutions to both problems. Examples are presented for both problems to demonstrate the existence of the strongly regular solutions to the CRDEs with and without certain standard definiteness conditions of weighting matrices.