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 socalled 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 KurtzProtter's uniformlycontrolledvariations (UCV) condition. A number of examples illustrate the scope of our results.
Original language  English 

Pages (fromto)  420463 
Number of pages  44 
Journal  Annals of Probability 
Volume  47 
Issue number  1 
DOIs  
Publication status  Published  13 Dec 2018 
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Profiles

Ilya Chevyrev
 School of Mathematics  Reader in Probability and Stochastic Analysis
Person: Academic: Research Active (Teaching)