Canonical RDEs and general semimartingales as rough paths

Journal article

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 Lépingle’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–Mémin–Pagès). A number of examples illustrate the scope of our results.

Authors: Chevyrev, I; Friz, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Institute of Mathematical Statistics
• Journal: Annals of Probability
• Volume: 47
• Issue: 1
• Pages: 420-463
• Publication date: 2018-12-13
• Acceptance date: 2018-02-23
• DOI: 10.1214/18-AOP1264
• ISSN: 0091-1798
• Keywords: stochastic and rough differential equations with jumps; general semimartingales; Marcus canonical equations; càdlàg rough paths; limit theorems