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

Journal Article

Eddahbi, M. . 2017

• Publication Work Type: Research paper
• Publication Online URL: The Annals of Probability
• Issue Number: 4
• Volume Number: 45
• Pages: 2377 to 2397

Publication Abstract

We establish a Krylov's type estimate and an Itô-Krylov's 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.