Smoothness of densities for path-dependent SDEs under Hörmander’s condition

november, 2021
Publication type:
Paper in peer-reviewed journals
Journal:
Journal of Functional Analysis, vol. 281, pp. 109-225
Publisher:
Elsevier
arXiv:
assets/images/icons/icon_arxiv.png 2011.04089
Keywords :
Hörmander’s theorem; Path-dependent SDEs; SDEs in Banach spaces; Rough paths
Abstract:
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 Lp-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 Rnfor 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.
BibTeX:
@article{Oha-Rus-Sha-2021,
    author={Alberto Ohashi and Francesco Russo and Evelina Shamarova },
    title={Smoothness of densities for path-dependent SDEs under 
           Hörmander’s condition },
    doi={10.1016/j.jfa.2021.1092250022-1236 },
    journal={Journal of Functional Analysis },
    year={2021 },
    month={11},
    volume={281 },
    pages={109--225},
}