On Information Controls

Abstract

In this paper we study an optimization problem in which the control is information, more precisely, the control is a $\ddot{I}ƒ$-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 ${\mathcal{P}}_2({\mathcal{P}}_2({\mathbb{R}}^d))$, the space of laws of random probability measures. By using a new Itô's formula for smooth functions on ${\mathcal{P}}_2({\mathcal{P}}_2({\mathbb{R}}^d))$, we characterize the value function of the information control problem by an Hamilton-Jacobi-Bellman equation on this space.