Delay equations driven by rough paths

Andreas Neuenkirch (Johann Wolfgang Goethe-Universität Frankfurt)
Ivan Nourdin (Université Paris 6)
Samy Tindel (Université Nancy 1)

Abstract


In this article, we illustrate the flexibility of the algebraic integration formalism introduced in M. Gubinelli, J. Funct. Anal. 216, 86-140, 2004, Math. Review 2005k:60169, by establishing an existence and uniqueness result for delay equations driven by rough paths. We then apply our results to the case where the driving path is a fractional Brownian motion with Hurst parameter $H>1/3$.

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Pages: 2031-2068

Publication Date: November 11, 2008

DOI: 10.1214/EJP.v13-575

References

  1. E. Alòs, J.A. Léon and D. Nualart. Stochastic Stratonovich calculus for fractional Brownian motion with Hurst parameter lesser than 1/2. Taiwanese J. Math. 5 (2001), 609-632. Math. Review 2000m:60059
  2. E. Alòs and D. Nualart. Stochastic integration with respect to the fractional Brownian motion. Stochastics Stochastics Rep. 75 (2002), 129-152. Math. Review 2004b:60138
  3. P. Carmona, L. Coutin and G. Montseny. Stochastic integration with respect to fractional Brownian motion. Ann. Inst. Henri Poincaré, Probab. Stat. 39 (2003), 27-68. Math. Review 2003m:60095
  4. P. Cheridito and D. Nualart. Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H in (0,1/2). Ann. Inst. H. Poincaré, Probab. Stat. 41 (2005), 1049-1081. Math. Review 2006m:60051
  5. L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002), 108-140. Math. Review 2003c:60066
  6. L. Decreusefond. Stochastic integration with respect to fractional Brownian motion. In: P. Doukhan (ed.) et al. Theory and applications of long-range dependence. Birkhäuser Boston 2003, 203-226. Math. Review MR1956051
  7. M. Ferrante and C. Rovira. Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H>1/2. Bernoulli 12 (2006), 85-100. Math. Review 2007b:60144
  8. D. Feyel and A. de La Pradelle. Curvilinear integrals along enriched paths. Electron. J. Probab. 11 (2006), 860-892. Math. Review 2007k:60112
  9. M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004), 86-140. Math. Review 2005k:60169
  10. M. Gubinelli and S. Tindel. Rough evolution equations. Ann. Probab., to appear. Math. Review number not available.
  11. A. Lejay. An Introduction to Rough Paths. In: Azéma, J. (ed.) et al. Séminaire de probabilités 37, Lecture Notes in Mathematics 1832. Springer Berlin 2003, 1-59. Math. Review 2005e:60120
  12. T. Lyons and Z. Qian. System control and rough paths. Clarendon Press Oxford 2002. Math. Review 2005f:93001
  13. T. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), 215-310. Math. Review 2000c:60089
  14. S.-E. A. Mohammed. Stochastic functional differential equations. Research Notes in Mathematics 99, Pitman Advanced Publishing Program Boston-London-Melbourne 1984. Math. Review 86j:60151
  15. S.-E. A. Mohammed. Stochastic differential systems with memory: theory, examples and applications. In: L. Decreusefond (ed.) et al. Stochastic Analysis and Related Topics VI. Birkhäuser Boston 1998, 1-77. Math. Review 99k:60155
  16. A. Neuenkirch, I. Nourdin, A. Rößler and S. Tindel. Trees and asymptotic expansions for fractional stochastic differential equations. Ann. Inst. H. Poincaré, Probab. Stat., to appear. Math. Review number not available.
  17. I. Nourdin and T. Simon. Correcting Newton-Cotes integrals by Lévy areas. Bernoulli 13 (2007), 695-711. Math. Review MR2348747
  18. D. Nualart. Stochastic calculus with respect to the fractional Brownian motion and applications. Contemp. Math. 336 (2003), 3-39. Math. Review 2004m:60119
  19. D. Nualart and A. Răşcanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002), 55-81. Math. Review 2003f:60105
  20. V. Pérez-Abreu and C. Tudor. Multiple stochastic fractional integrals: A transfer principle for multiple stochastic fractional integrals. Bol. Soc. Mat. Mex., III. 8 (2002), 187-203. Math. Review 2003k:60122
  21. F. Russo and P. Vallois. Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields. 97 (1993), 403-421. Math. Review 94j:60113
  22. S. Tindel and I. Torrecilla. Fractional differential systems with H>1/4. In preparation. Math. Review number not available.
  23. M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields. 111 (1998), 333-374. Math. Review 99j:60073
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