Optimal control of LQ problem with anticipative partial observations

A stochastic linear-quadratic (LQ for short) problem with anticipative (i.e., not adapted) partial observations is studied. With the help of enlargement of filtration, we turn the anticipative signal observation system into a higher-dimensional adapted one, and obtain a linear filtering equation of the latter by the martingale representation theorem and a related equivalent control problem. By introducing a Riccati equation and an ordinary differential equation, we provide a unique optimal feedback control for another equivalent optimal control problem with the controlled state being the linear filtering equation. Finally, the optimal cost function for the original anticipative LQ problem is obtained, which is represented by the filtering of the extended adaptive system and some modified coefficients. Our result covers that of the classical stochastic LQ problem with adapted partial observations. As an application, the optimal control of an interception problem with anticipative radar tracking is explicitly given.