On Information Controls

In this paper we study an optimization problem in which the control is information, more precisely, the control is a σ-algebra or a filtration. In a dynamic setting, we establish the dynamic programming principle and the law invariance of the value function. The latter requires a condition slightly stronger than the (H)-hypothesis for the admissible filtration, and enables us to define the value function on mathcal P2(mathcal P2(mathbb Rd)), the space of laws of random probability measures. By using a new Ito's formula for smooth functions on mathcal P2(mathcal P2(mathbb Rd)), we characterize the value function of the information control problem by an Hamilton-Jacobi-Bellman equation on this space.