Non-linear expectations in spaces of Colombeau generalized functions

Abstract

We introduce the notion of a non-linear expectation in spaces of Colombeau generalized functions and provide its characterization in terms of the upper expectation over a set of probability measures. We then study a fully non-linear backward stochastic differential equation in the Colombeau setting via its connection with the corresponding fully non-linear partial differential equation.

Keywords:

• Generalized solutions
• Colombeau algebras
• non-linear heat equation
• G-expectation
• backward stochastic differential equations

Mathematics Subject Classifications:

• 60G44; 60H30; 35D99

Additional information

Funding

This work was supported by Australian Research Council grant 130103517, Social Science and Humanities Research Council of Canada 1002262.