On the Solutions of the Matrix Equations in Optimal Stochastic Control

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