Quadratic Backward Stochastic Diﬀerential Equations with Unbounded Coeﬃcients and their Application
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.