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

TL;DR

This paper develops a framework for analyzing nonzero-sum stochastic differential games governed by controlled convection-diffusion SPDEs, establishing existence, uniqueness, and equilibrium conditions in complex heterogeneous systems.

Contribution

It introduces a novel Hamiltonian approach for such games, including cases with piecewise constant coefficients and interface conditions, with applications to composite materials.

Findings

Proved existence and uniqueness of solutions for the forward and backward SPDEs.

Derived maximum principles characterizing Nash equilibria.

Applied the theory to models of material phase interactions in diffusion processes.

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.