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July 2017 Quadratic BSDE with $\mathbb{L}^{2}$-terminal data: Krylov’s estimate, Itô–Krylov’s formula and existence results
Khaled Bahlali, M’hamed Eddahbi, Youssef Ouknine
Ann. Probab. 45(4): 2377-2397 (July 2017). DOI: 10.1214/16-AOP1115


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.


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Khaled Bahlali. M’hamed Eddahbi. Youssef Ouknine. "Quadratic BSDE with $\mathbb{L}^{2}$-terminal data: Krylov’s estimate, Itô–Krylov’s formula and existence results." Ann. Probab. 45 (4) 2377 - 2397, July 2017.


Received: 1 December 2013; Revised: 1 March 2016; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 1379.60057
MathSciNet: MR3693965
Digital Object Identifier: 10.1214/16-AOP1115

Primary: 60H10, 60H20

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 4 • July 2017
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