Weak existence and uniqueness for forward–backward SDEs
Weak existence and uniqueness for forward–backward SDEs
复制标题
前向-后向 SDE 的弱存在性和唯一性
DOI：
10.1016/j.spa.2006.05.002
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发表时间：
2006
影响因子：
1.4
通讯作者：
G. Guatteri
中科院分区：
文献类型：
--
作者：
F. Delarue;G. Guatteri
我们的目标是建立一类具有一维后向分量的非简并确定性 FBSDE 弱解的存在性和唯一性。经典的 Lipschitz 框架被部分削弱：假设扩散矩阵和最终条件是空间 Hölder 连续的，而漂移和后向驱动器在 x 上可能是不连续的。后向驱动器的增长最多允许与梯度项成二次方。该策略分为三个不同的步骤。我们首先为相关的偏微分方程建立一个良好控制的解决方案，并作为副产品为前向-后向系统建立一个弱解决方案。然后我们采用 Ma、Protter 和 Yong 的四步方案中引入的“解耦策略”[J. Ma，P. Protter，J. Yong，显式求解前向-后向随机微分方程 - 四步方案，Probab。理论相关领域 98 (1994) 339–359] 证明唯一性。
We aim to establish the existence and uniqueness of weak solutions to a suitable class of non-degenerate deterministic FBSDEs with a one-dimensional backward component. The classical Lipschitz framework is partially weakened: the diffusion matrix and the final condition are assumed to be space Hölder continuous whereas the drift and the backward driver may be discontinuous in x. The growth of the backward driver is allowed to be at most quadratic with respect to the gradient term. The strategy holds in three different steps. We first build a well controlled solution to the associated PDE and as a by-product a weak solution to the forward–backward system. We then adapt the “decoupling strategy” introduced in the four-step scheme of Ma, Protter and Yong [J. Ma, P. Protter, J. Yong, Solving forward–backward stochastic differential equations explicitly — a four step scheme, Probab. Theory Related Fields 98 (1994) 339–359] to prove uniqueness.