Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-27T23:19:22.475Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  08 September 2006

Antoine Lejay
Antoine Lejay*
Affiliation:
Projet OMEGA (INRIA Lorraine), IECN, Campus scientifique, BP 239, 54506 Vandœuvre-lès-Nancy Cedex, France; [email protected]
Get access

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.

Keywords

Rough pathsstochastic differential equationsstochastic process generated by divergence-form operatorsDirichlet processapproximation of trajectories.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 356 - 379
Copyright
© EDP Sciences, SMAI, 2006

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

Aronson, D.G., Non-negative solutions of linear parabolic equation. Ann. Scuola Norm. Sup. Pisa 22 (1968) 607–693.
Bass, R.F., Hambly, B. and Lyons, T.J., Extending the Wong-Zakai theorem to reversible Markov processes. J. Eur. Math. Soc. 4 (2002) 237–269. CrossRef
Chen, K.-T., Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. 65 (1957) 163–178. CrossRef
Coutin, L. and Lejay, A., Semi-martingales and rough paths theory. Electron. J. Probab. 10 (2005) 761–785. CrossRef
Coutin, L. and Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108–140. CrossRef
Coquet, F. and Słomiński, L., On the convergence of Dirichlet processes. Bernoulli 5 (1999) 615–639. CrossRef
Dupoiron, K., Mathieu, P. and San, J. martin, Formule d'Itô pour des diffusions uniformément elliptiques et processus de Dirichlet. Potential Anal. 21 (2004) 7–3. CrossRef
Föllmer, H., Calcul d'Itô sans probabilités, in Séminaire de Probabilités, XV. Lect. Notes Math. 850 (1981) 143–150. Springer, Berlin. CrossRef
Föllmer, H., Dirichlet processes, in Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lect. Notes Math. 851 (1981) 476–478. Springer, Berlin. CrossRef
M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Process. De Gruyter (1994).
Flandoli, F. and Russo, F., Generalized integration and stochastic ODEs. Ann. Probab. 30 (2002) 270–292.
P. Friz and N. Victoir, A note on the notion of geometric rough paths. To appear in Probab. Theory Related Fields (2006).
Hambly, B.M. and Lyons, T.J., Stochastic area for Brownian motion on the Sierpinski gasket. Ann. Probab. 26 (1998) 132–148.
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes. North Holland, 2nd edition (1989).
H. Kunita, Stochastic flows and stochastic differential equations. Cambridge University Press (1990).
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/.
Lejay, A., An introduction to rough paths, in Séminaire de probabilités, XXXVII. Lect. Notes Math. 1832 (2003) 1–59, Springer, Berlin. CrossRef
Lejay, A., Probabilistic Representation, A 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. CrossRef
A. Lejay, Stochastic Differential Equations driven by processes generated by divergence form operators II: Convergence results. Institut Élie Cartan de Nancy (preprint), 2003.
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).
Ledoux, M., Lyons, T. and Qian, Z., Lévy area of Wiener processes in Banach spaces. Ann. Probab. 30 (2002) 546–578.
T. Lyons and Z. Qian, System Control and Rough Paths. Oxford Mathematical Monographs. Oxford University Press (2002).
Lyons, T.J. and Stoica, L., The limits of stochastic integrals of differential forms. Ann. Probab. 27 (1999) 1–49.
Lyons, T.J., Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215–310. CrossRef
Lejay, A. and Victoir, N., On (p,q)-rough paths. J. Differential Equations 225 (2006) 103–133. CrossRef
Z. Ma and M. Röckner, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext. Springer-Verlag (1991).
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).
Rozkosz, A., Stochastic Representation of Diffusions Corresponding to Divergence Form Operators. Stochastic Process. Appl. 63 (1996) 11–33. CrossRef
Rozkosz, A., Dirichlet, On processes associated with second order divergence form operators. Potential Anal. 14 (2001) 123–148. CrossRef
Rozkosz, A. and Slomiński, L., Extended Convergence of Dirichlet Processes. Stochastics Stochastics Rep. 65 (1998) 79–109. CrossRef
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag (1990).
E.-M. Sipiläinen, A pathwise view of solutions of stochastic differential equations. Ph.D. thesis, University of Edinburgh (1993).
Stroock, D.W., Diffusion Semigroups Corresponding to Uniformly Elliptic Divergence Form Operator, in Séminaire de Probabilités XXII. Lect. Notes Math. 1321 (1988) 316–347. Springer-Verlag. CrossRef
Williams, D.R.E., Path-wise solutions of SDE's driven by Lévy processes. Rev. Mat. Iberoamericana 17 (2002) 295–330. arXiv:math.PR/0001018.
Wong, E. and Zakai, M., On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist. 36 (1965) 1560–1564. CrossRef