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Non-local Hamilton-Jacobi-Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes

Elena Bandini, Christian Keller

TL;DR

This work develops a rigorous framework for stochastic optimal control of path-dependent piecewise deterministic processes and proves that the value function is the unique minimax solution to a non-local path-dependent HJB equation. The analysis leverages a fixed-point approach inspired by Davis and Farid, adapted to time-measurable Hamiltonians on Skorokhod spaces, and establishes existence, uniqueness, and a comparison principle for nonsmooth solutions. By connecting the control problem to a non-local PPDE, the authors provide a precise bridge between path-dependent PDP dynamics and optimal control, with potential applications to memory effects such as delayed Hodgkin-Huxley models. The results constitute a first well-posedness theory for fully nonlinear non-local PPDEs and open avenues for path-dependent pricing, control with jumps, and delay-differential structures in stochastic systems.

Abstract

We study the optimal control of path-dependent piecewise deterministic processes. An appropriate dynamic programming principle is established. We prove that the associated value function is the unique minimax solution of the corresponding non-local path-dependent Hamilton-Jacobi-Bellman equation. This is the first well-posedness result for nonsmooth solutions of fully nonlinear non-local path-dependent partial differential equations.

Non-local Hamilton-Jacobi-Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes

TL;DR

This work develops a rigorous framework for stochastic optimal control of path-dependent piecewise deterministic processes and proves that the value function is the unique minimax solution to a non-local path-dependent HJB equation. The analysis leverages a fixed-point approach inspired by Davis and Farid, adapted to time-measurable Hamiltonians on Skorokhod spaces, and establishes existence, uniqueness, and a comparison principle for nonsmooth solutions. By connecting the control problem to a non-local PPDE, the authors provide a precise bridge between path-dependent PDP dynamics and optimal control, with potential applications to memory effects such as delayed Hodgkin-Huxley models. The results constitute a first well-posedness theory for fully nonlinear non-local PPDEs and open avenues for path-dependent pricing, control with jumps, and delay-differential structures in stochastic systems.

Abstract

We study the optimal control of path-dependent piecewise deterministic processes. An appropriate dynamic programming principle is established. We prove that the associated value function is the unique minimax solution of the corresponding non-local path-dependent Hamilton-Jacobi-Bellman equation. This is the first well-posedness result for nonsmooth solutions of fully nonlinear non-local path-dependent partial differential equations.
Paper Structure (30 sections, 20 theorems, 179 equations)

This paper contains 30 sections, 20 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Related research
  3. A simplified presentation of the control problem
  4. Our approach in dealing with specific difficulties concerning the non-local path-dependent HJB equation \ref{['E:Intro:PPDE']}
  5. The delayed Hodgkin--Huxley model as possible application
  6. Organization of the rest of the paper
  7. Notation and preliminaries
  8. The canonical setup
  9. The canonical path space
  10. The canonical space of controlled marked point processes
  11. Optimal control of path-dependent PDPs
  12. The data
  13. The flow and related notation
  14. The continuous-time optimal control problem
  15. A related discrete-time optimal control problem and the dynamic programming principle
  16. ...and 15 more sections

Key Result

Lemma 4.5

The map $(s,x,\alpha)\mapsto \phi^{s,x,\alpha}$, ${\mathbb{R}}_+\times\Omega\times\mathcal{A}\to\Omega$, is measurable from $\mathcal{B}({\mathbb{R}}_+)\otimes {\mathcal{F}}^0\otimes\mathcal{B}(\mathcal{A})$ to ${\mathcal{F}}^0$.

Theorems & Definitions (63)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 4.3
  • Example 4.4
  • Lemma 4.5
  • ...and 53 more