On quasilinear parabolic systems and FBSDEs of quadratic growth

TL;DR

This paper develops probabilistic techniques to derive a priori estimates for quasilinear parabolic PDEs with quadratic gradient growth, leading to existence results for classical solutions and related FBSDEs.

Contribution

It introduces new probabilistic methods to handle quadratic growth in PDE systems, establishing existence of classical solutions and solutions to associated FBSDEs.

Findings

Established a priori estimates for PDE systems with quadratic gradient growth

Proved existence of classical solutions to these PDE systems

Demonstrated existence of solutions to related FBSDEs

Abstract

Using probabilistic methods, we establish a-priori estimates for two classes of quasilinear parabolic systems of partial differential equations (PDEs). We treat in particular the case of a nonlinearity which has quadratic growth in the gradient of the unknown. As a result of our estimates, we obtain the existence of classical solutions of the PDE system. From this, we infer the existence of solutions to a corresponding class of forward-backward stochastic differential equations.