A Probabilistic Characterization of g-Harmonic Functions

Abstract

Associated with a quasi-linear generator function g, we give a definition of g-harmonic functions. The relation between the g-harmonic functions and g-martingales will be delineated. It is direct to construct such relation for smooth case, but for continuous case we need the theory of viscosity solution. Under the nonlinear expectation mechanism, we can also get the similar relation between harmonic functions and martingales. The strict converse problem of mean value property of g-harmonic functions are discussed finally.

Keywords

• BSDE| g-martingale| g-harmonic function| nonlinear Feynman-Kac formula| viscosity solution