论文标题
设定值向后随机微分方程
Set-Valued Backward Stochastic Differential Equations
论文作者
论文摘要
在本文中，我们建立了一个分析框架，用于研究设置值向后的随机微分方程（Set值BSDE），这在很大程度上是由当前对多资产或基于网络财务模型的动态设置价值风险度量的研究进行的。我们的框架将利用集合之间的Hukuhara差异概念，以补偿传统Minkowski添加的“反相反”操作，从而在Set值分析中矢量空间结构。在证明一类设定值的BSDE的适合性的同时，我们还将解决有关广义Aumann-Itô积分的一些基本问题，尤其是当它与Martingale代表定理连接时。特别是，我们提出了一些积分的必要扩展，这些扩展可用于代表具有非单明子初始值的设置值。事实证明，这一扩展对于研究值BSDE是必不可少的。
In this paper, we establish an analytic framework for studying set-valued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will make use of the notion of Hukuhara difference between sets, in order to compensate the lack of "inverse" operation of the traditional Minkowski addition, whence the vector space structure in set-valued analysis. While proving the well-posedness of a class of set-valued BSDEs, we shall also address some fundamental issues regarding generalized Aumann-Itô integrals, especially when it is connected to the martingale representation theorem. In particular, we propose some necessary extensions of the integral that can be used to represent set-valued martingales with non-singleton initial values. This extension turns out to be essential for the study of set-valued BSDEs.
