A quadratic minimization problem for a linear stochastic system is solved in this paper. Both the finite and infinite terminal time cases are considered. Also two precise representations of the controlled stochastic process are considered. In one representation we include the correction term [6] for the state and control-dependent noise and in the other we do not. For the case with no correction terms, the optimal control is shown to be a linear feedback of the system state variables. Uniqueness and stability conditions are presented for this problem. The case with correction term is much harder to solve and we only determine the linear optimal control. An example is included which illustrates many results of the paper.

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