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Causal Hamilton-Jacobi-Bellman Equations for Anticipative Stochastic Optimal Control

Peter Bank, Franziska Bielert

TL;DR

The paper develops a causal Hamilton-Jacobi-Bellman framework for anticipative stochastic control with rough-path dynamics, where the controller can see the future Brownian motion over a moving horizon. By merging the martingale optimality principle with a pathwise, Dupire-based functional Itô calculus, it derives a deterministic HJB in terms of causal derivatives and a transport term reflecting anticipativity, linking the value function $v$ to a path-dependent map $u$. A verification theorem and a regular-value analysis are provided, leveraging conditional distributions and pathwise quadratic variation to establish both sufficiency and necessity of the HJB under suitable regularity. The approach is illustrated through insider-investment and front-running-style examples, demonstrating how anticipative information can be incorporated into a robust rough-path control framework with concrete HJB characterizations.

Abstract

We consider a stochastic optimal control problem where the controller can anticipate the evolution of the driving noise over some dynamically changing time window. The controlled state dynamics are understood as a rough differential equation. We combine the martingale optimality principle with a functional form of Itô's formula to derive a Hamilton-Jacobi-Bellman (HJB) equation for this problem. This HJB equation is formulated in terms of Dupire's functional derivatives and involves a transport equation arising from the anticipativity of the problem.

Causal Hamilton-Jacobi-Bellman Equations for Anticipative Stochastic Optimal Control

TL;DR

The paper develops a causal Hamilton-Jacobi-Bellman framework for anticipative stochastic control with rough-path dynamics, where the controller can see the future Brownian motion over a moving horizon. By merging the martingale optimality principle with a pathwise, Dupire-based functional Itô calculus, it derives a deterministic HJB in terms of causal derivatives and a transport term reflecting anticipativity, linking the value function to a path-dependent map . A verification theorem and a regular-value analysis are provided, leveraging conditional distributions and pathwise quadratic variation to establish both sufficiency and necessity of the HJB under suitable regularity. The approach is illustrated through insider-investment and front-running-style examples, demonstrating how anticipative information can be incorporated into a robust rough-path control framework with concrete HJB characterizations.

Abstract

We consider a stochastic optimal control problem where the controller can anticipate the evolution of the driving noise over some dynamically changing time window. The controlled state dynamics are understood as a rough differential equation. We combine the martingale optimality principle with a functional form of Itô's formula to derive a Hamilton-Jacobi-Bellman (HJB) equation for this problem. This HJB equation is formulated in terms of Dupire's functional derivatives and involves a transport equation arising from the anticipativity of the problem.

Paper Structure

This paper contains 19 sections, 16 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Optimal Control Problem With Anticipative Controls
  4. State Equation
  5. Rough Differential Equations
  6. Stochastic Dynamics with Anticipative Controls
  7. Value Process
  8. Dynamic Programming Principle
  9. Hamilton-Jacobi-Bellmann Equation
  10. Quadratic Variation
  11. Functional Calculus
  12. Causal Derivatives
  13. Rough Funtional Itô Formula
  14. Verification Theorem
  15. Regular Value Functions Solve HJB
  16. ...and 4 more sections

Key Result

Theorem 3.4

Let $\mathrm{\mathbf{X}}=(X,\mathbb{X})$ be an $\alpha$-Hölder rough path over $\mathop{\mathrm{\mathbb{R}}}\nolimits^d$ and $(Y,Y')$ be an $(\alpha,\beta)$-Hölder $X$-controlled rough path over $\mathop{\mathrm{\mathbb{R}}}\nolimits^{n\times d}$ such that $\beta\in(0,\alpha]$ and $2\alpha+\beta>1$. is a well-defined limit along partitions $\mathcal{P}$ of $[0,T]$ that we call rough integral. More

Theorems & Definitions (49)

  • Definition 3.1: $\alpha$-Hölder Rough Path
  • Example 3.2: Itô Lift
  • Definition 3.3: $(\alpha, \beta)$-Hölder $X$-controlled Rough Path
  • Theorem 3.4: Rough Integration
  • Theorem 3.5: Existence and Properties of Solutions
  • proof
  • Example 3.6
  • Remark 3.7: Measurability and Random Initial Data
  • Lemma 4.1
  • proof
  • ...and 39 more