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Optimal control of SPDEs driven by time-space Brownian motion

Nacira Agram, Bernt Øksendal, Frank Proske, Olena Tymoshenko

Abstract

In this paper we study a Pontryagin type stochastic maximum principle for the optimal control of a system, where the state dynamics satisfy a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion (also called Brownian sheet). We first discuss some properties of a Brownian sheet driven linear SPDE which models the growth of an ecosystem. Further, applying time-space white noise calculus we derive sufficient conditions and necessary conditions of optimality of the control. Finally, we illustrate our results by solving a linear quadratic control problem and an optimal harvesting problem in the plane. We also study possible applications to machine learning.

Optimal control of SPDEs driven by time-space Brownian motion

Abstract

In this paper we study a Pontryagin type stochastic maximum principle for the optimal control of a system, where the state dynamics satisfy a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion (also called Brownian sheet). We first discuss some properties of a Brownian sheet driven linear SPDE which models the growth of an ecosystem. Further, applying time-space white noise calculus we derive sufficient conditions and necessary conditions of optimality of the control. Finally, we illustrate our results by solving a linear quadratic control problem and an optimal harvesting problem in the plane. We also study possible applications to machine learning.
Paper Structure (13 sections, 14 theorems, 149 equations)

This paper contains 13 sections, 14 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. A general formulation
  3. A discussion of the solution of the SPDE
  4. Background
  5. The Itô formula
  6. BSPDEs in the plane
  7. Maximum principle approaches
  8. The sufficient maximum principle
  9. The necessary maximum principle
  10. Applications
  11. Return to the optimal harvesting problem in the plane
  12. Return to the linear-quadratic (LQ) problem in the plane
  13. Example related to machine learning

Key Result

Theorem 3.1

The solution $Y(t,x)$ of the equation Y5 is not always positive

Theorems & Definitions (22)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Remark 2.1
  • Theorem 3.1
  • Remark 3.2
  • Theorem 4.1: Itô formula, Wang & Zakai WZ
  • Theorem 4.2
  • Lemma 4.3: Integration by parts
  • Theorem 5.1: Wong & Zakai WZ
  • ...and 12 more