On Information Controls

TL;DR

This paper investigates an optimization problem where the control variable is an information structure, specifically a sigma-algebra or filtration, and characterizes its value function through a Hamilton-Jacobi-Bellman equation in a complex probability space.

Contribution

It establishes the dynamic programming principle and law invariance for information control problems, introducing a new Itô's formula on a space of probability measures.

Findings

Proves the dynamic programming principle for information control.

Shows law invariance under a strengthened (H)-hypothesis.

Characterizes the value function via a Hamilton-Jacobi-Bellman equation.

Abstract

In this paper we study an optimization problem in which the control is information, more precisely, the control is a $\sigma$-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 It\^o'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.