CANONICAL RDES AND GENERAL SEMIMARTINGALES AS ROUGH PATHS

• 成果类型: Article
• 署名作者: Chevyrev, Ilya; Friz, Peter K.
• 署名单位: University of Oxford; Technical University of Berlin; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1264
• 发表日期: 2019
• 页码: 420-463
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; Levy processes; brownian-motion; random-walks; p-variation; jump type; driven; MANIFOLDS; SEQUENCES; FLOWS

摘要

In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of Lepingle's BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased a uniformly-controlled-variations (UCV) condition (Kurtz-Protter, Jakubowski-Memin-Pages). A number of examples illustrate the scope of our results.