Canonical RDEs and general semimartingales as rough paths

Author: 

Chevyrev, I
Friz, P

Publication Date: 

13 December 2018

Journal: 

Annals of Probability

Last Updated: 

2020-06-01T22:04:03.647+01:00

Issue: 

1

Volume: 

47

DOI: 

10.1214/18-AOP1264

page: 

420-463

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.

Symplectic id: 

826142

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article