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Differential Equations Driven by Π-Rough Paths

Part of: Control systems Stochastic analysis

Published online by Cambridge University Press:  21 January 2016

Lajos Gergely Gyurkó
Lajos Gergely Gyurkó*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK Oxford-Man Institute of Quantitative Finance, University of Oxford, Eagle House, Walton Well Road, Oxford OX6 2ED, UK (gyurko@maths.ox.ac.uk)
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Abstract

This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric Π-rough paths in our terminology) sketched by Lyons in 1998. Although geometric Π-rough paths can be treated as p-rough paths for a sufficiently large p, and the theory of integration of Lipγ one-forms (γ > p–1) along geometric p-rough paths applies, we prove the existence of integrals of one-forms under weaker conditions. Moreover, we consider differential equations driven by geometric Π-rough paths and give sufficient conditions for existence and uniqueness of solution.

Keywords

rough pathsdifferential equationsstochastic integralsstochastic differential equations

MSC classification

Secondary: 60H05: Stochastic integrals 60H10: Stochastic ordinary differential equations 93C15: Systems governed by ordinary differential equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 741 - 758
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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