On Quadratic BSDEs with Final Condition in L2

TL;DR

This thesis investigates quadratic backward stochastic differential equations (BSDEs) with L2 terminal conditions, establishing existence, uniqueness, and stability results using novel regularization techniques and providing an alternative proof for stability of quadratic semimartingale BSDEs.

Contribution

Introduces Lipschitz-quadratic regularization to prove existence and uniqueness of quadratic BSDEs with minimal assumptions, and offers an alternative proof for the stability of quadratic semimartingale BSDEs.

Findings

Proved existence and uniqueness of solutions for a broad class of BSDEs.

Established stability results for quadratic semimartingale BSDEs.

Provided an alternative proof for the monotone stability of quadratic semimartingales.

Abstract

This thesis consists of three parts. In the first part, we study $\mathbb{L}^p$ solutions of a large class of BSDEs. Existence, comparison theorem, uniqueness and a stability result are proved. In the second part, we establish the solvability of quadratic semimartingale BSDEs. In contrast to current literature, we use Lipschitz-quadratic regularization and obtain the existence and uniqueness results with minimal assumptions. The third part is a brief summary of quadratic semimartingales and the monotone stability result. This provides an alternative proof of monotone stability result for quadratic semimartingales BSDEs.