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SPDE Games Driven by a Brownian Sheet with Applications to Pollution Minimization

Nacira Agram, Bernt Øksendal, Frank Proske, Olena Tymoshenko

Abstract

This paper studies a nonzero-sum stochastic differential game in the context of shared spatial-domain pollution control. The pollution dynamics are governed by a stochastic partial differential equation (SPDE) driven by a Brownian sheet, capturing the stochastic nature of environmental fluctuations. Two players, representing different regions, aim to minimize their respective cost functionals, which balance pollution penalties with the cost of implementing control strategies. The nonzero-sum framework reflects the interdependent yet conflicting objectives of the players, where both cooperation and competition influence the outcomes. We derive necessary and sufficient conditions for Nash equilibrium strategies, using a maximum principle approach. This approach involves the introduction of a new pair of adjoint variables, (L_1, L_2), which do not appear in a corresponding formulation with the classical (1-parameter) Brownian motion. Finally, we apply our results to two case studies in pollution control, demonstrating how spatial and stochastic dynamics shape the equilibrium strategies.

SPDE Games Driven by a Brownian Sheet with Applications to Pollution Minimization

Abstract

This paper studies a nonzero-sum stochastic differential game in the context of shared spatial-domain pollution control. The pollution dynamics are governed by a stochastic partial differential equation (SPDE) driven by a Brownian sheet, capturing the stochastic nature of environmental fluctuations. Two players, representing different regions, aim to minimize their respective cost functionals, which balance pollution penalties with the cost of implementing control strategies. The nonzero-sum framework reflects the interdependent yet conflicting objectives of the players, where both cooperation and competition influence the outcomes. We derive necessary and sufficient conditions for Nash equilibrium strategies, using a maximum principle approach. This approach involves the introduction of a new pair of adjoint variables, (L_1, L_2), which do not appear in a corresponding formulation with the classical (1-parameter) Brownian motion. Finally, we apply our results to two case studies in pollution control, demonstrating how spatial and stochastic dynamics shape the equilibrium strategies.

Paper Structure

This paper contains 12 sections, 7 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Itô Formula
  4. Integration by Parts Formula
  5. Nonzero-sum stochastic differential games
  6. Sufficient Optimality Conditions for Two Players
  7. Necessary Optimality Conditions for Two Players
  8. Derivative Process for Two Players
  9. Applications: Pollution Control with Competing Regions
  10. Example 1
  11. Example 2
  12. Symmetry in Parameters

Key Result

Theorem 2.1

Let $Y(t,x)$ be a process as defined in (eq), and $\alpha$ be continuous function, $\beta \in \mathcal{L}^2_{a,1}$, $\psi \in \mathcal{L}^2_{a,2}$. Then the Itô formula for $f(Y(z))$ is given by: where

Figures (2)

  • Figure 1: Decision process leading to Nash equilibrium in a two-player game model.
  • Figure :

Theorems & Definitions (10)

  • Theorem 2.1: Itô Formula for $f(Y(z))$
  • Lemma 2.2
  • Theorem 2.3: Existence and Uniqueness
  • Definition 3.1: Convolution-like operator
  • Remark 3.2
  • Theorem 3.3: Sufficient optimality condition
  • Remark 3.4
  • Theorem 3.5
  • Theorem 4.1
  • Theorem 4.2