Quadratic BSDE with $\mathbb{L}^{2}$-terminal data: Krylov’s estimate, Itô–Krylov’s formula and existence results
Quadratic BSDE with $\mathbb{L}^{2}$-terminal data: Krylov’s estimate, Itô–Krylov’s formula and existence results
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DOI：
10.1214/16-aop1115
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发表时间：
2017-07
影响因子：
2.3
通讯作者：
K. Bahlali;M. Eddahbi;Y. Ouknine
中科院分区：
文献类型：
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作者：
K. Bahlali;M. Eddahbi;Y. Ouknine
本文建立了一维二次倒向随机微分方程解的Krylov型估计和Ito-Krylov变元公式，该方程具有一个可测生成元和一个任意终端数据.这使我们能够证明各种存在性和唯一性的结果与平方可积终端条件，有时仅仅是一个可测量的发电机的某些类QBSDES。结果表明，终端数据的指数矩的存在性和生成元的连续性对于解的存在性和唯一性都不是必要的。我们还建立了一类特殊的具有可测生成元的QBSDE解的比较定理。作为一个副产品，我们得到了一个特殊的二次偏微分方程（QPDE）的粘性解的存在性平方可积终端数据。
We establish a Krylov-type estimate and an Ito–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.