The maximum principle, semicontinuity and nonlinear PDE´s

Scalar, fully nonlinear, second-order partial differential equations arise in stochastic control and the theory of stochastic differential games. The proper notion of solution of these equations and associated questions of uniqueness of these solutions (subject perhaps to boundary conditions) have evolved in a striking way and have many applications in control and differential games. One approach to the central uniqueness question relies on a maximum principle for semicontinuous functions. The notion of maximum principle and the role it plays in the uniqueness theory are explained