Quadratic Backward Stochastic Differential Equations with Unbounded Coefficients and their Application

Abstract

We focus on solving problems on quadratic backward stochastic differential equations (BSDEs). We improve fundamental results on quadratic BSDEs by using more general conditions on the coefficients. Firstly, we consider quadratic BSDEs with possibly unbounded coefficients. We prove a mono- tonicity theorem, which gives the main argument of the existence result. Then, we give sufficient conditions for the existence of a solution pair. Sec- ondly, we consider the general one-dimensional case of Riccati BSDEs. We prove that some integrability conditions, which are weaker than the existing ones, are sufficient for the existence of a solution to the Riccati BSDE. As an application, we obtain the existence of a solution to linear-quadratic optimal control problems with possibly unbounded coefficients. Thirdly, we study a certain class of Riccati BSDEs with possibly unbounded coefficients. Integra- bility conditions on the coefficients are derived that ensure the existence and uniqueness of the solution pair.