Direct-comparison approach to continuous time Linear Quadratic Gaussian control problem

Recently, a direct-comparison approach has been developed to control and optimize the performance of a stochastic Markov system, see for discrete time case, and for continuous time case. Compared with dynamic programming, the standard approach for stochastic control problem, this alternative approach is simple and intuitive. It is based on the direct comparison of the system performance under two policies, and discounting is not needed when dealing with long-run average criterion. By directly applying this approach, we studied the continuous time linear quadratic Gaussian (LQG) control problem, obtained the optimal policy for the long run average criterion without introducing discounting. The well known algebraic Riccati equation for the optimal policy can be easily obtained by this direct-comparison approach. This paper servers as an example to show the effectiveness of direct-comparison approach for the continuous time case.