Stochastic LQ Control with Extra Measurability Restriction

Different from the standard linear quadratic (LQ) problem for stochastic systems, the LQ problem considered in the paper has extra measurability restrictions. The problem also appears in the LQ control problem for stochastic systems with delays, rational expectations problems, asymmetric information control, and so on. The essential difficulty lies in that one has to optimize the input and its conditional expectations simultaneously. The stochastic maximum principle (SMP) and orthogonal decomposition technique are the key tools. Firstly, the authors establish the SMP and convert the original problem into forward and backward stochastic difference equations (FBSDEs) with extra measurability restrictions. Secondly, the authors resolve the FBSDEs by using the orthogonal decomposition technique and obtain the analytical solution to the underlying problem. Thirdly, the authors explore the essential distinction between the problem and the standard stochastic LQ control problem. Finally, numerical examples are given to illustrate the obtained results.