
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 -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 , the space of laws of random probability measures. By using a new It\^o's formula for smooth functions on , we characterize the value function of the information control problem by an Hamilton-Jacobi-Bellman equation on this space.
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