Quadratic BSDE with $\mathbb{L}^{2}$-terminal data: Krylov’s estimate, Itô–Krylov’s formula and existence results

Abstract

We establish a Krylov-type estimate and an Itô–Krylov change of variable formula for the solutions of one-dimensional quadratic backward stochastic differential equations (QBSDEs) with a measurable generator and an arbitrary terminal datum. This allows us to prove various existence and uniqueness results for some classes of QBSDEs with a square integrable terminal condition and sometimes a merely measurable generator. It turns out that neither the existence of exponential moments of the terminal datum nor the continuity of the generator are necessary to the existence and/or uniqueness of solutions. We also establish a comparison theorem for solutions of a particular class of QBSDEs with measurable generator. As a byproduct, we obtain the existence of viscosity solutions for a particular class of quadratic partial differential equations (QPDEs) with a square integrable terminal datum.