A transformation method to study the solvability of fully coupled FBSDEs
A transformation method to study the solvability of fully coupled FBSDEs
复制标题
研究全耦合 FBSDE 可解性的变换方法
DOI：
10.1080/17442508.2021.1903466
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发表时间：
2022
期刊：
影响因子：
0.9
通讯作者：
Julian Wendt
中科院分区：
文献类型：
--
作者：
Stefan Ankirchner;Alexander Fromm;Julian Wendt
考虑完全耦合的正倒向随机微分方程，其中所有函数参数是Lipschitz连续的，终端条件是单调的，前向部分的扩散系数单调依赖于后向部分的控制过程分量z.我们证明了存在一类线性变换，将FB_n变换为辅助FB_n，使得前向扩散系数的Lipschitz常数w.r.t. z小于前向分量x的终端条件的Lipschitz常数的倒数。后一个条件允许通过分析解耦场的空间导数来验证全局解的存在性。这是有用的，因为通过将逆线性变换应用于辅助FBbn的解，我们获得原始FBbn的解。我们说明了几个例子如何线性变换，结合分析的解耦场的梯度，可以用来证明全局可解性FBSDES。
We consider fully coupled forward–backward stochastic differential equations (FBSDEs), where all function parameters are Lipschitz continuous, the terminal condition is monotone and the diffusion coefficient of the forward part depends monotonically onz, the control process component of the backward part. We show that there exists a class of linear transformations turning the FBSDE into an auxiliary FBSDE for which the Lipschitz constant of the forward diffusion coefficient w.r.t.zis smaller than the inverse of the Lipschitz constant of the terminal condition w.r.t. the forward componentx. The latter condition allows to verify existence of a global solution by analysing the spatial derivative of the decoupling field. This is useful since by applying the inverse linear transformation to a solution of the auxiliary FBSDE we obtain a solution to the original one. We illustrate with several examples how linear transformations, combined with an analysis of the decoupling field's gradient, can be used for proving global solvability of FBSDEs.
DOI：
10.1137/17m1152401
发表时间：
2018
期刊：
SIAM J. Control. Optim.
影响因子：
--
作者：
Stefan Ankirchner;Alexander Fromm
通讯作者：
Alexander Fromm
DOI：
10.1515/9781400833115.77
发表时间：
2008
期刊：
Brain research. Cognitive brain research
影响因子：
--
作者：
P. Barrieu;N. Karoui
通讯作者：
N. Karoui