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Path-dependent Hamilton-Jacobi equations with u-dependence and time-measurable Hamiltonians

Elena Bandini, Christian Keller

TL;DR

This work develops a rigorous well-posedness theory for path-dependent Hamilton–Jacobi equations with time-measurable Hamiltonians on the history space $D([0,T],\mathbb{R}^d)$. By formulating minimax solutions and leveraging Perron’s method, the authors address existence and uniqueness without requiring continuity in time, and they establish a robust comparison principle via a doubled- equation approach. They apply the framework to delay-type optimal control problems with discount factors, proving that the value function is the unique minimax solution of the associated path-dependent HJB and proving regularity along Lipschitz paths. The paper also clarifies the relationship between path-dependent and classical, non-path-dependent HJB theory, including a non-path-dependent counterpart and a consistency result that unifies the two settings. These results advance the analysis of non-local, history-driven control problems under minimal time-regularity assumptions and underpin the companion work on non-Markovian stochastic control.

Abstract

We establish existence and uniqueness of minimax solutions for a fairly general class of path-dependent Hamilton-Jacobi equations. In particular, the relevant Hamiltonians can contain the solution and they only need to be measurable with respect to time. We apply our results to optimal control problems of (delay) functional differential equations with cost functionals that have discount factors and with time-measurable data. Our main results are also crucial for our companion paper Bandini and Keller [arXiv preprint arXiv:2408.02147 (2024)], where non-local path-dependent Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control of non-Markovian piecewise deterministic processes are studied.

Path-dependent Hamilton-Jacobi equations with u-dependence and time-measurable Hamiltonians

TL;DR

This work develops a rigorous well-posedness theory for path-dependent Hamilton–Jacobi equations with time-measurable Hamiltonians on the history space . By formulating minimax solutions and leveraging Perron’s method, the authors address existence and uniqueness without requiring continuity in time, and they establish a robust comparison principle via a doubled- equation approach. They apply the framework to delay-type optimal control problems with discount factors, proving that the value function is the unique minimax solution of the associated path-dependent HJB and proving regularity along Lipschitz paths. The paper also clarifies the relationship between path-dependent and classical, non-path-dependent HJB theory, including a non-path-dependent counterpart and a consistency result that unifies the two settings. These results advance the analysis of non-local, history-driven control problems under minimal time-regularity assumptions and underpin the companion work on non-Markovian stochastic control.

Abstract

We establish existence and uniqueness of minimax solutions for a fairly general class of path-dependent Hamilton-Jacobi equations. In particular, the relevant Hamiltonians can contain the solution and they only need to be measurable with respect to time. We apply our results to optimal control problems of (delay) functional differential equations with cost functionals that have discount factors and with time-measurable data. Our main results are also crucial for our companion paper Bandini and Keller [arXiv preprint arXiv:2408.02147 (2024)], where non-local path-dependent Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control of non-Markovian piecewise deterministic processes are studied.
Paper Structure (16 sections, 17 theorems, 139 equations)

This paper contains 16 sections, 17 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Further related literature
  3. Some difficulties and their resolutions
  4. Organization of the rest of the paper
  5. Setting and notation
  6. Path derivatives
  7. Minimax solutions
  8. Comparison principle
  9. Existence
  10. Proof of Proposition \ref{['P:NonContSol']}
  11. Proof of Theorem \ref{['T:Perron']}
  12. Optimal control with discount factors
  13. The non-path-dependent case
  14. General theory
  15. Optimal control
  16. ...and 1 more sections

Key Result

Lemma 4.7

A function $u:[0,T]\times\Omega\to{\mathbb{R}}$ is a minimax $L$-supersolution of E:PPDE:H if and only if $u\in\mathrm{LSC}([0,T]\times\Omega)$, $u(T,\cdot)\ge h$, and, for each $(s_0,x_0,z)\in [0,T)\times\Omega\times{\mathbb{R}}^d$, $y_0\ge u(s_0,x_0)$, and $t\in (s_0,T]$, there is an $(x,y)\in\mat

Theorems & Definitions (43)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Remark 3.2
  • Remark 4.3
  • Remark 4.4
  • Definition 4.5
  • Remark 4.6
  • Lemma 4.7
  • ...and 33 more