Near Optimal Controls for Partially Observed Stochastic Linear Quadratic Problems

We consider a stochastic linear quadratic control problem with partial observation. A near optimal control in the weak formulation is characterized. The main features of this paper are the presence of the control in the diffusion term of the state equation; the circular dependence between the control process and the filtration generated by the observation; and the observation process contains an unbounded drift term. We address these difficulties by first restricting the control to a smaller domain, which enables us to apply the Girsanov theorem using a conditional argument and thereby break the circular dependence. Subsequently, we study the restricted problem using a non‐standard variation method. The desired near optimal control is then obtained by taking the limit of an approximating sequence.