A new probabilistic approach to non local and fully non linear second order partial differential equations

TL;DR

This paper develops a probabilistic framework for solving fully nonlinear second order PDEs, including non local cases, using martingale problems and dynamic procedures with Feller properties.

Contribution

It introduces a new probabilistic approach for non local and fully nonlinear second order PDEs via martingale problems and dynamic procedures, extending existing methods.

Findings

Constructed time consistent convex dynamic procedures in non dominated settings.

Established a link between these procedures and viscosity solutions of PDEs.

Provided explicit constructions for diffusions with Levy generators.

Abstract

We prove that weakly continuous solutions to martingale problems admit a canonical regular conditional probability distribution. This allows for the construction of time consistent convex dynamic procedures in a non dominated setting. Making use of the martingale problem approach for continuous diffusions and diffusions with Levy generator, we give an explicit construction of such procedures having furthermore a Feller property. These procedures lead to viscosity solution of fully non linear second order partial differential equations in case of continuous diffusions. In case of diffusions with Levy generator this provides a probabilistic approach for the resolution of non local fully non linear second order PDE.