On the parameter dependence of a class of rational matrix equations occurring in stochastic optimal control

This paper is concerned with rational matrix equations occuring in stochastic control that play an analogous role as the algebraic Riccati equation does in deterministic control. We will therefore sometimes refer to these equations as stochastic (algebraic) Riccati equations. A first rigorous treatment of a stochastic Riccati equation from LQ-control theory seems to have been undertaken by Wonham in [21]. Since then different versions of stochastic Riccati equations have been obtained in various control and stabilization problems (e.g. [15], [18], [6], [11], [19], [23]) and it is widely agreed that their analysis constitutes a challenging problem. Algebraic tools such as canonical transformations, Hamiltonians or factorization methods used for the deterministic Riccati equation, appear to be not available for the stochastic Riccati equation. Our approach is of an analytical nature and based on a nonlocal convergence result for Newton’s method applied to a certain class of nonlinear matrix equations. Resolvent positive and concave operators play a crucial role. We specify solvability conditions for stochastic Riccati equations of different kinds and analyze the dependence of their largest solution on parameters. The paper is organized as follows. In Sec. 2 we introduce the stochastic Riccati operator and specify some notations. Then in Sec. 3 we discuss several problems of stochastic control that lead to stochastic Riccati inequalities with structurally different parameter matrices. These inequalities and the corresponding equations are discussed in Sec. 4, which contains our main results.