GENERAL ROUGH INTEGRATION, LEVY ROUGH PATHS AND A LEVY-KINTCHINE-TYPE FORMULA

• 成果类型: Article
• 署名作者: Friz, Peter K.; Shekhar, Atul
• 署名单位: Technical University of Berlin; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Indian Statistical Institute; Indian Statistical Institute Bangalore
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1123
• 发表日期: 2017
• 页码: 2707-2765
• 关键词: differential-equations driven; brownian-motion; p-variation; signature; INEQUALITY

摘要

We consider rough paths with jumps. In particular, the analogue of Lyons' extension theorem and rough integration are established in a jump setting, offering a pathwise view on stochastic integration against cadlag processes. A class of Levy rough paths is introduced and characterized by a sub-ellipticity condition on the left-invariant diffusion vector fields and a certain integrability property of the Carnot-Caratheodory norm with respect to the Levy measure on the group, using Hunt's framework of Lie group valued Levy processes. Examples of Levy rough paths include a standard multidimensional Levy process enhanced with a stochastic area as constructed by D. Williams, the pure area Poisson process and Brownian motion in a magnetic field. An explicit formula for the expected signature is given.