Canonical RDEs and general semi-martingales as rough paths

Ilya Chevyrev, Peter K. Friz

Research output: Contribution to journalArticlepeer-review

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 via Kurtz-Protter's uniformly-controlled-variations (UCV) condition. A number of examples illustrate the scope of our results.
Original languageEnglish
Pages (from-to)420-463
Number of pages44
JournalAnnals of Probability
Volume47
Issue number1
DOIs
Publication statusPublished - 13 Dec 2018

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