Note on Viscosity Solution of Path-Dependent PDE and G-Martingales
Note on Viscosity Solution of Path-Dependent PDE and G-Martingales
复制标题
DOI：
--
复制
发表时间：
2011-06
期刊：
影响因子：
--
通讯作者：
S. Peng
中科院分区：
文献类型：
--
作者：
S. Peng
在本文的第二版中，我们引入了一类完全非线性抛物型路径依赖偏微分方程（P-PDE）粘性解的概念。然后，我们证明了这种新类型的方程的比较定理（或最大值原理），这是这个框架的关键属性。为了克服众所周知的困难的非紧性的路径空间的最大化，我们引入了一种新的方法，称为左冻结最大化的方法，使我们能够获得的比较原则，以及光滑的粘度解决方案的路径依赖PDE。一个倒向随机微分方程的解和一个G-期望下的G-鞅是这类P-PDE解的典型例子。经典偏微分方程粘性解的最大值原理是其特例，称为状态相关偏微分方程。
In the 2nd version of this note we introduce the notion of viscosity solution for a type of fully nonlinear parabolic path-dependent partial differential equations (P-PDE). We then prove the comparison theorem (or maximum principle) of this new type of equation which is the key property of this framework. To overcome the well-known difficulty of non-compactness of the space of paths for the maximization, we have introduced a new approach, called left frozen maximization approach which permits us to obtain the comparison principle for smooth as well as viscosity solutions of path-dependent PDE. A solution of a backward stochastic differential equation and a G-martingale under a G-expectation are typical examples of such type of solutions of P-PDE. The maximum principle for viscosity solutions of classical PDE, called state dependent PDE, is a special case.