论文标题
在Hörmander的状况下，密度依赖路径依赖的SDE的光滑度
Smoothness of densities for path-dependent SDEs under Hörmander's condition
论文作者
论文摘要
我们确定了在Hörmander型条件下的溶液中的平滑密度的存在。在路径依赖性上下文中，基于简化的Malliavin矩阵的经典方案被证明是不可用的。我们通过将给定的$ n $维路径依赖性SDE提升为合适的$ L_P $ -L_P $ -TYPE BANACH空间来解决问题。然后，我们以$ \ mathbb r^n $为hörmander的括号条件，用于非预期的SDE系数，从功能上的ITôColculus的意义上，根据垂直衍生物定义了Lie括号。我们通往主要结果的途径与Banach空间，Malliavin微积分和粗糙路径技术的SDE分析之间的相互作用。
We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a Hörmander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given $n$-dimensional path-dependent SDE into a suitable $L_p$-type Banach space in such a way that the lifted Banach-space-valued equation becomes a state-dependent reformulation of the original SDE. We then formulate Hörmander's bracket condition in $\mathbb R^n$ for non-anticipative SDE coefficients defining the Lie brackets in terms of vertical derivatives in the sense of the functional Itô calculus. Our pathway to the main result engages an interplay between the analysis of SDEs in Banach spaces, Malliavin calculus, and rough path techniques.
