Pathwise Taylor Expansions for It^o Random Fields
Pathwise Taylor Expansions for It^o Random Fields
复制标题
It^o 随机场的路径泰勒展开式
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Jin Ma
中科院分区：
文献类型：
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作者：
R. Buckdahn;I. Bulla;Jin Ma
本文研究了一类It^o型随机场的{it-路径随机Taylor展开式}，在我们以前的工作引用{Buckdahn_Ma_02}的意义下，其中扩散部分允许包含随机场本身及其空间导数.这种“自激”类型的随机场特别包含曲率驱动扩散的完全非线性随机偏微分方程，以及某些随机Hamilton-Jacobi-Bellman方程。我们引入了一个新的概念“$n$倍”的衍生物的随机场，作为一个基本的设备，以科普特殊的自激性质。与我们以前的工作引用{Buckdahn_Ma_02}不同，我们的新扩展可以围绕任何随机时空点$（ ，xi）$，其中时间分量$ $甚至不一定是停止时间。此外，例外空集与随机点$（ ，xi）$。作为一个应用程序，我们展示了这种新形式的路径泰勒展开可能会导致一个不同的治疗一类完全非线性的SPDE的扩散项涉及的解决方案和它的梯度的随机特性，从而导致的定义{它的随机粘性解决方案}这样的SPDE，这是新的文献。
In this paper we study the {it pathwise stochastic Taylor expansion}, in the sense of our previous work cite{Buckdahn_Ma_02}, for a class of It^o-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an "self-exciting" type particularly contains the fully nonlinear stochastic PDEs of curvature driven diffusion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of "$n$-fold" derivatives of a random field, as a fundamental device to cope with the special self-exciting nature. Unlike our previous work cite{Buckdahn_Ma_02}, our new expansion can be defined around any random time-space point $( ,xi)$, where the temporal component $ $ does not even have to be a stopping time. Moreover, the exceptional null set is independent of the choice of the random point $( ,xi)$. As an application, we show how this new form of pathwise Taylor expansion could lead to a different treatment of the stochastic characteristics for a class of fully nonlinear SPDEs whose diffusion term involves both the solution and its gradient, and hence lead to a definition of the {it stochastic viscosity solution} for such SPDEs, which is new in the literature.