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January 2019 Canonical RDEs and general semimartingales as rough paths
Ilya Chevyrev, Peter K. Friz
Ann. Probab. 47(1): 420-463 (January 2019). DOI: 10.1214/18-AOP1264

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.

Citation

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Ilya Chevyrev. Peter K. Friz. "Canonical RDEs and general semimartingales as rough paths." Ann. Probab. 47 (1) 420 - 463, January 2019. https://doi.org/10.1214/18-AOP1264

Information

Received: 1 April 2017; Revised: 1 December 2017; Published: January 2019
First available in Project Euclid: 13 December 2018

zbMATH: 07036341
MathSciNet: MR3909973
Digital Object Identifier: 10.1214/18-AOP1264

Subjects:
Primary: 60H99
Secondary: 60H10

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 1 • January 2019
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