论文标题
分配依赖性SDE的狮子衍生物的Bismut公式
Bismut Formula for Lions Derivative of Distribution-Path Dependent SDEs
论文作者
论文摘要
为了表征初始分布中分布路径依赖性SDE的规律性，该分布在路径空间上的概率度量等级方面有所不同，我们引入了Banach空间上的概率测量的内在和狮子衍生物，并证明了Banach值随机变量的分布的狮子衍生物的链条规则。通过使用Malliavin演算，我们建立了具有分布依赖性漂移的SDE的狮子衍生物的Bismut类型公式。当噪声项也依赖于路径，因此bismut公式无效时，我们建立了渐近的bismut公式。都考虑了非脱位和退化的噪声。本文的主要结果概括并改善了最近在文献中衍生出的相应的结果，该文献带有内存和McKean-Vlasov SDE的经典SDE，而没有内存。
To characterize the regularity of distribution-path dependent SDEs in the initial distribution which varies in the class of probability measures on the path space, we introduce the intrinsic and Lions derivatives for probability measures on Banach spaces, and prove the chain rule of the Lions derivative for the distribution of Banach-valued random variables. By using Malliavin calculus, we establish the Bismut type formula for the Lions derivatives of functional solutions to SDEs with distribution-path dependent drifts. When the noise term is also path dependent so that the Bismut formula is invalid, we establish the asymptotic Bismut formula. Both non-degenerate and degenerate noises are considered. The main results of this paper generalize and improve the corresponding ones derived recently in the literature for the classical SDEs with memory and McKean-Vlasov SDEs without memory.
