The Annals of Probability

Canonical RDEs and general semimartingales as rough paths

Abstract

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.

Article information

Source
Ann. Probab., Volume 47, Number 1 (2019), 420-463.

Dates
Received: April 2017
Revised: December 2017
First available in Project Euclid: 13 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1544691625

Digital Object Identifier
doi:10.1214/18-AOP1264

Mathematical Reviews number (MathSciNet)
MR3909973

Zentralblatt MATH identifier
07036341

Citation

Chevyrev, Ilya; Friz, Peter K. Canonical RDEs and general semimartingales as rough paths. Ann. Probab. 47 (2019), no. 1, 420--463. doi:10.1214/18-AOP1264. https://projecteuclid.org/euclid.aop/1544691625

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