On the solutions of the matrix equations in optimal stochastic control

In this paper, the matrix algebraic equations involved in the optimal control problem of time-invariant linear Itô stochastic systems, named Riccati — Itô equations in the paper, are investigated. The necessary and sufficient condition for the existence of positive definite solutions of the Riccati-Itô equations is obtained and an iterative solution to the Riccati-Itô equations is also given in the paper thus a complete solution to the basic problem of optimal control of time-invariant linear Itô stochastic systems is then obtained. An example is given at the end of the paper to illustrate the application of the result of the paper.