Generalized Riccati Equations Arising in Stochastic Games

We study a class of rational matrix differential equations that generalize the Riccati differential equations. The generalization involves replacing positive definite “weighting” matrices in the usual Riccati equations with either semidefinite or indefinite matrices that arise in linear quadratic control problems and differential games−both stochastic and deterministic. The purpose of this paper is to prove some fundamental properties such as comparison, monotonicity and existence theorems. These properties are well-known for classical Riccati differential equations when certain matrices are assumed definite. As applications, we obtain conditions for the existence of solutions to the algebraic Riccati equation and to equations with periodic coefficients.