Uniqueness for nonlinear parabolic systems in stochastic game theory with application to financial economics

Abstract

The objective of this paper is to study nonlinear partial differential systems like $$ \partial_t {\bf u}- \Delta {\bf u} +{\bf H}(x,t,{\bf u}, \nabla {\bf u})={\bf G}(x,t), $$ with applications to the solution of stochastic differential games with $N$ players, where $N$ is arbitrarily large. It is assumed that the Hamiltonian ${\bf H}$ of the nonlinear system satisfies a quadratic growth condition in $\nabla {\bf u}$ and has a positive definite Jacobian ${\bf H_u}$. An energy estimate and the uniqueness property for bounded weak solutions are proved. Moreover, applications to stochastic games and financial economics such as modern portfolio theory are discussed.