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On Information Controls

Zihao Gu, Jianfeng Zhang

TL;DR

This work studies stochastic control where the controller directly selects information structures (filtrations or σ-algebras) under a slightly strengthened (H*)-hypothesis, deriving a dynamic programming principle and law-invariance to recast the problem on the space ${\cal P}_2({\cal P}_2(\mathbb{R}^d))$. A novel Itô formula on this lifted space is developed and used to formulate a Hamilton–Jacobi–Bellman equation on measure-valued spaces, along with a verification theorem that ties smooth solutions to optimal filtrations. The authors also provide a continuous-time insider-trading example and a dynamic insider-trading problem to illustrate the framework, including explicit solutions in Gaussian settings and reductions to one-dimensional HJBs. Finally, a generalized Itô formula accommodating common noise and higher-order derivatives is established, ensuring consistency with mean-field type calculus and offering a pathway to viscosity solutions. Overall, the paper advances a rigorous, information-centric viewpoint in stochastic control with potential implications for persuasion games, insider trading, and information design in dynamic settings.

Abstract

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, assuming a condition slightly stronger than the (H)-hypothesis for the admissible filtration, we establish the dynamic programming principle and the law invariance of the value function. The latter 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 through an Hamilton-Jacobi-Bellman equation on this space.

On Information Controls

TL;DR

This work studies stochastic control where the controller directly selects information structures (filtrations or σ-algebras) under a slightly strengthened (H*)-hypothesis, deriving a dynamic programming principle and law-invariance to recast the problem on the space ${\cal P}_2({\cal P}_2(\mathbb{R}^d))$. A novel Itô formula on this lifted space is developed and used to formulate a Hamilton–Jacobi–Bellman equation on measure-valued spaces, along with a verification theorem that ties smooth solutions to optimal filtrations. The authors also provide a continuous-time insider-trading example and a dynamic insider-trading problem to illustrate the framework, including explicit solutions in Gaussian settings and reductions to one-dimensional HJBs. Finally, a generalized Itô formula accommodating common noise and higher-order derivatives is established, ensuring consistency with mean-field type calculus and offering a pathway to viscosity solutions. Overall, the paper advances a rigorous, information-centric viewpoint in stochastic control with potential implications for persuasion games, insider trading, and information design in dynamic settings.

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, assuming a condition slightly stronger than the (H)-hypothesis for the admissible filtration, we establish the dynamic programming principle and the law invariance of the value function. The latter enables us to define the value function on , the space of laws of random probability measures. By using a new Itô's formula for smooth functions on , we characterize the value function of the information control problem through an Hamilton-Jacobi-Bellman equation on this space.
Paper Structure (14 sections, 13 theorems, 187 equations, 1 figure)

This paper contains 14 sections, 13 theorems, 187 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some related literature
  3. A toy model of insider trading
  4. A primal for the space ${\cal P}({\cal P}(\mathbb{R}^d))$
  5. A continuous time information control problem
  6. Stochastic calculus on $\mathcal{P}_2(\mathcal{P}_2(\mathbb R^d))$
  7. Hamilton–Jacobi–Bellman equations on ${\cal P}_2({\cal P}_2(\mathbb{R}^d))$
  8. A dynamic insider trading problem
  9. A generalized Itô formula
  10. Some examples
  11. The Braess's Paradox
  12. An example of nonzero sum Nash game
  13. Persuasion Games
  14. Some technical proofs

Key Result

Proposition 2.1

Let $X$ be a standard normal. Then the problem insider-V2 has an explicit solution: where $R_0:= \mathbb{E}[|X|] = \sqrt{\tfrac{2}{\pi}}$, and $a_R = I^{-1}(R)$ for the following funtion $I$:

Figures (1)

  • Figure 1: Braess’s Paradox. Adding an apparently beneficial link to the network can unexpectedly lead to worse outcomes for all drivers.

Theorems & Definitions (24)

  • Proposition 2.1
  • Remark 2.2
  • Proposition 3.1
  • Example 3.2
  • Theorem 3.3
  • proof
  • Example 4.1
  • Lemma 4.2
  • Example 4.3
  • Proposition 4.4: DPP
  • ...and 14 more