Quadratic FBSDE with generalized Burgers' type nonlinearities, perturbations and large deviations

TL;DR

This paper studies backward stochastic differential equations with polynomial nonlinearities in the driver, establishing existence, uniqueness, comparison results, and connections to PDE perturbations and large deviations.

Contribution

It introduces new conditions for existence and uniqueness of solutions for BSDEs with polynomial nonlinearities and links these to PDE perturbation theory and large deviations principles.

Findings

Established existence and uniqueness conditions for BSDEs with polynomial nonlinearities.

Connected BSDE solutions to PDE perturbation theory.

Proved a large deviations principle for the first component of the BSDE solution.

Abstract

We discuss BSDE with drivers containing nonlinearities of the type $p(y)|z|$ and $p(y)|z|^2$ with $p$ a polynomial of any degree. Sufficient conditions are given for existence and uniqueness of solutions as well as comparison results. We then connect the results to the Markovian FBSDE setting, discussing applications in the theory of PDE perturbation and stating a result concerning a large deviations principle for the first component of the solution to the BSDE.