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Nonzero-Sum Stochastic Differential Games for Controlled Convection-Diffusion SPDEs

Nacira Agram, Eya Zougar

Abstract

This paper studies a two-player nonzero-sum stochastic differential game governed by a controlled convection-diffusion stochastic partial differential equation (SPDE) with spatially heterogeneous coefficients. The diffusion and transport operators depend on the players' controls, allowing each agent to influence the system dynamics. We prove the existence and uniqueness of solutions to both the forward uncontrolled SPDE and the associated adjoint backward SPDE (BSPDE) in a Hilbert space framework. Using a Hamiltonian approach, we derive sufficient and necessary maximum principles characterizing Nash equilibria. Special attention is given to operators with piecewise constant coefficients, where interface transmission conditions arise naturally. As an illustration, we provide two examples from composite materials where the game structure models the interaction between different material phases in a diffusion process.

Nonzero-Sum Stochastic Differential Games for Controlled Convection-Diffusion SPDEs

Abstract

This paper studies a two-player nonzero-sum stochastic differential game governed by a controlled convection-diffusion stochastic partial differential equation (SPDE) with spatially heterogeneous coefficients. The diffusion and transport operators depend on the players' controls, allowing each agent to influence the system dynamics. We prove the existence and uniqueness of solutions to both the forward uncontrolled SPDE and the associated adjoint backward SPDE (BSPDE) in a Hilbert space framework. Using a Hamiltonian approach, we derive sufficient and necessary maximum principles characterizing Nash equilibria. Special attention is given to operators with piecewise constant coefficients, where interface transmission conditions arise naturally. As an illustration, we provide two examples from composite materials where the game structure models the interaction between different material phases in a diffusion process.

Paper Structure

This paper contains 22 sections, 12 theorems, 202 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stochastic Convection--Diffusion SPDE
  3. Problem Formulation
  4. Hypothesis.
  5. Notation.
  6. Some properties of the operator
  7. Smooth coefficient case.
  8. Piecewise constant coefficient case.
  9. Forward SPDE existence and unicity
  10. Existence.
  11. Uniqueness.
  12. The Game Problem
  13. Objective Function and Hamiltonian System
  14. Sufficient Optimality Condition for Games
  15. Necessary Optimality Condition for Games
  16. ...and 7 more sections

Key Result

Lemma 4

Let $\rho,a,b:\mathbb R\to\mathbb R$ be measurable functions satisfying Hypothesis hypo1. Then, for any $\varphi_i\in C_c^\infty(\mathbb R)$, $i=1,2$, and any $\alpha_i\in\mathbb R$, $i=1,2$, the operator $\mathcal{A}_{\rho,a,b}$ satisfies $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Illustration of a two-player control game in a composite system with piecewise constant coefficients
  • Figure 2: Homogeneous thermal medium with two distributed actuators

Theorems & Definitions (29)

  • Lemma 4: Linearity
  • proof
  • Lemma 5: Coercivity in $\mathbb{H}$
  • proof
  • Lemma 6
  • proof
  • Lemma 7: Adjoint operator in the smooth case
  • proof
  • Remark 8
  • Lemma 9: Adjoint operator in the piecewise constant case
  • ...and 19 more