Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T12:12:42.846Z Has data issue: false hasContentIssue false
  • English
  • Français

Differential Equations Driven by Π-Rough Paths

Part of: Stochastic analysis Control systems

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)
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Coutin, L. and Qian, Z., Stochastic analysis, rough paths analysis and fractional Brownian motion, Prob. Theory Relat. Fields 122 (2002), 108–140.Google Scholar
2. Friz, P. and Victoir, N., Approximations of the Brownian rough path with applications to stochastic analysis, Annales Inst. H. Poincaré Prob. Statist. 41 (2005), 703–724.CrossRefGoogle Scholar
3. Friz, P. and Victoir, N., Differential equations driven by Gaussian signals, Annales Inst. H. Poincaré Prob. Statist. 46 (2010), 369–413.CrossRefGoogle Scholar
4. Friz, P. and Victoir, N., Multidimensional stochastic processes as rough paths (Cambridge University Press, 2010).Google Scholar
5. Gyurkó, L. G., Numerical methods for approximating solutions to rough differential equations, PhD thesis, University of Oxford (2009) (available at http://ora.ox.ac.uk/objects/uuid:d977be17–76c6–46d6–8691–6d3b7bd51f7a).Google Scholar
6. Hairer, M. and Kelly, D., Geometric versus non-geometric rough paths, Annales Inst. H. Poincaré Prob. Statist. 51(1) (2015), 207–251.CrossRefGoogle Scholar
7. Lejay, A. and Victoir, N., On (p, q)-rough paths, J. Diff. Eqns 225(1) (2006), 103–133.CrossRefGoogle Scholar
8. Lyons, T. J., Differential equations driven by rough signals, Rev. Mat. Iber. 14(2) (1998), 215–310.CrossRefGoogle Scholar
9. Lyons, T. J. and Qian, Z., System control and rough paths, Oxford Mathematical Monographs (Clarendon Press, Oxford, 2002).Google Scholar
10. Lyons, T. J., Caruana, M. and Lévy, T., Differential equations driven by rough paths: École d’Eté de Probabilités de Saint-Flour XXXIV – 2004, Lecture Notes in Mathematics, Volume 1908 (Springer, 2007).CrossRefGoogle Scholar
11. Mille, A. and Sanz-Solé, M., Approximation of rough paths of fractional Brownian motion, in Seminar on stochastic analysis, random fields and applications V, Progress in Probability, Volume 59, pp. 275–303 (Birkhäuser, 2008).Google Scholar