Stochastic differential equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem

Antoine Lejay

ESAIM: Probability and Statistics (2006)

  • Volume: 10, page 356-379
  • ISSN: 1292-8100

Abstract

top
We show in this article how the theory of “rough paths” allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.

How to cite

top

Lejay, Antoine. "Stochastic differential equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem." ESAIM: Probability and Statistics 10 (2006): 356-379. <http://eudml.org/doc/249761>.

@article{Lejay2006,
abstract = { We show in this article how the theory of “rough paths” allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately. },
author = {Lejay, Antoine},
journal = {ESAIM: Probability and Statistics},
keywords = {Rough paths; stochastic differential equations; stochastic process generated by divergence-form operators; Dirichlet process; approximation of trajectories.; rough paths; stochastic process generated by divergence-form operators; approximation of trajectories},
language = {eng},
month = {9},
pages = {356-379},
publisher = {EDP Sciences},
title = {Stochastic differential equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem},
url = {http://eudml.org/doc/249761},
volume = {10},
year = {2006},
}

TY - JOUR
AU - Lejay, Antoine
TI - Stochastic differential equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem
JO - ESAIM: Probability and Statistics
DA - 2006/9//
PB - EDP Sciences
VL - 10
SP - 356
EP - 379
AB - We show in this article how the theory of “rough paths” allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.
LA - eng
KW - Rough paths; stochastic differential equations; stochastic process generated by divergence-form operators; Dirichlet process; approximation of trajectories.; rough paths; stochastic process generated by divergence-form operators; approximation of trajectories
UR - http://eudml.org/doc/249761
ER -

References

top
  1. D.G. Aronson, Non-negative solutions of linear parabolic equation. Ann. Scuola Norm. Sup. Pisa22 (1968) 607–693.  Zbl0182.13802
  2. R.F. Bass, B. Hambly and T.J. Lyons, Extending the Wong-Zakai theorem to reversible Markov processes. J. Eur. Math. Soc.4 (2002) 237–269.  Zbl1010.60070
  3. K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math.65 (1957) 163–178.  Zbl0077.25301
  4. L. Coutin and A. Lejay, Semi-martingales and rough paths theory. Electron. J. Probab.10 (2005) 761–785.  Zbl1109.60035
  5. L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields122 (2002) 108–140.  Zbl1047.60029
  6. F. Coquet and L. Słomiński, On the convergence of Dirichlet processes. Bernoulli5 (1999) 615–639.  Zbl0953.60001
  7. K. Dupoiron, P. Mathieu and J. San martin, Formule d'Itô pour des diffusions uniformément elliptiques et processus de Dirichlet. Potential Anal.21 (2004) 7–3.  
  8. H. Föllmer, Calcul d'Itô sans probabilités, in Séminaire de Probabilités, XV. Lect. Notes Math.850 (1981) 143–150. Springer, Berlin.  Zbl0461.60074
  9. H. Föllmer, Dirichlet processes, in Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lect. Notes Math.851 (1981) 476–478. Springer, Berlin.  
  10. M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Process. De Gruyter (1994).  Zbl0838.31001
  11. F. Flandoli and F. Russo, Generalized integration and stochastic ODEs. Ann. Probab.30 (2002) 270–292.  Zbl1022.60054
  12. P. Friz and N. Victoir, A note on the notion of geometric rough paths. To appear in Probab. Theory Related Fields (2006).  Zbl1108.34052
  13. B.M. Hambly and T.J. Lyons, Stochastic area for Brownian motion on the Sierpinski gasket. Ann. Probab.26 (1998) 132–148.  Zbl0936.60073
  14. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes. North Holland, 2nd edition (1989).  Zbl0684.60040
  15. H. Kunita, Stochastic flows and stochastic differential equations. Cambridge University Press (1990).  Zbl0743.60052
  16. A. Lejay, Méthodes probabilistes pour l'homogénéisation des opérateurs sous forme-divergence : cas linéaires et semi-linéaires. Ph.D. thesis, Université de Provence, Marseille, France (2000). www.iecn.u-nancy.fr/~lejay/.  
  17. A. Lejay, An introduction to rough paths, in Séminaire de probabilités, XXXVII. Lect. Notes Math.1832 (2003) 1–59, Springer, Berlin.  Zbl1041.60051
  18. A. Lejay, A Probabilistic Representation of the Solution of some Quasi-Linear PDE with a Divergence-Form Operator. Application to Existence of Weak Solutions of FBSDE. Stochastic Process. Appl.110 (2004) 145–176.  Zbl1075.60070
  19. A. Lejay, Stochastic Differential Equations driven by processes generated by divergence form operators II: Convergence results. Institut Élie Cartan de Nancy (preprint), 2003.  Zbl1185.60061
  20. A. Lejay and T.J. Lyons, On the Importance of the Lévy Area for Systems Controlled by Converging Stochastic Processes. Application to Homogenization, in New Trend in Potential Theory, D. Bakry, L. Beznea, Gh. Bucur and M. Röckner Eds., The Theta Foundation (2006).  
  21. M. Ledoux, T. Lyons and Z. Qian, Lévy area of Wiener processes in Banach spaces. Ann. Probab.30 (2002) 546–578.  Zbl1016.60071
  22. T. Lyons and Z. Qian, System Control and Rough Paths. Oxford Mathematical Monographs. Oxford University Press (2002).  Zbl1029.93001
  23. T.J. Lyons and L. Stoica, The limits of stochastic integrals of differential forms. Ann. Probab.27 (1999) 1–49.  Zbl0969.60078
  24. T.J. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoamericana14 (1998) 215–310.  Zbl0923.34056
  25. A. Lejay and N. Victoir, On (p,q)-rough paths. J. Differential Equations225 (2006) 103–133.  Zbl1097.60048
  26. Z. Ma and M. Röckner, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext. Springer-Verlag (1991).  Zbl0826.31001
  27. E.J. McShane. Stochastic differential equations and models of random processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pp. 263–294. Univ. California Press (1972).  
  28. A. Rozkosz, Stochastic Representation of Diffusions Corresponding to Divergence Form Operators. Stochastic Process. Appl.63 (1996) 11–33.  Zbl0870.60073
  29. A. Rozkosz, On Dirichlet processes associated with second order divergence form operators. Potential Anal.14 (2001) 123–148.  Zbl0974.60064
  30. A. Rozkosz and L. Slomiński, Extended Convergence of Dirichlet Processes. Stochastics Stochastics Rep.65 (1998) 79–109.  Zbl0917.60076
  31. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag (1990).  Zbl1087.60040
  32. E.-M. Sipiläinen, A pathwise view of solutions of stochastic differential equations. Ph.D. thesis, University of Edinburgh (1993).  
  33. D.W. Stroock, Diffusion Semigroups Corresponding to Uniformly Elliptic Divergence Form Operator, in Séminaire de Probabilités XXII. Lect. Notes Math.1321 (1988) 316–347. Springer-Verlag.  
  34. D.R.E. Williams, Path-wise solutions of SDE's driven by Lévy processes. Rev. Mat. Iberoamericana17 (2002) 295–330. arXiv:math..  URIPR/0001018
  35. E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist.36 (1965) 1560–1564.  Zbl0138.11201

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.