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

ESAIM: Probability and Statistics (2006)

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

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topLejay, 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- D.G. Aronson, Non-negative solutions of linear parabolic equation. Ann. Scuola Norm. Sup. Pisa22 (1968) 607–693.
- 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.
- K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math.65 (1957) 163–178.
- L. Coutin and A. Lejay, Semi-martingales and rough paths theory. Electron. J. Probab.10 (2005) 761–785.
- L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields122 (2002) 108–140.
- F. Coquet and L. Słomiński, On the convergence of Dirichlet processes. Bernoulli5 (1999) 615–639.
- 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.
- H. Föllmer, Calcul d'Itô sans probabilités, in Séminaire de Probabilités, XV. Lect. Notes Math.850 (1981) 143–150. Springer, Berlin.
- H. Föllmer, Dirichlet processes, in Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lect. Notes Math.851 (1981) 476–478. Springer, Berlin.
- M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Process. De Gruyter (1994).
- F. Flandoli and F. Russo, 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).
- B.M. Hambly and T.J. Lyons, 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/.
- A. Lejay, An introduction to rough paths, in Séminaire de probabilités, XXXVII. Lect. Notes Math.1832 (2003) 1–59, Springer, Berlin.
- 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.
- 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).
- M. Ledoux, T. Lyons and Z. Qian, 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).
- T.J. Lyons and L. Stoica, The limits of stochastic integrals of differential forms. Ann. Probab.27 (1999) 1–49.
- T.J. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoamericana14 (1998) 215–310.
- A. Lejay and N. Victoir, On (p,q)-rough paths. J. Differential Equations225 (2006) 103–133.
- 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).
- A. Rozkosz, Stochastic Representation of Diffusions Corresponding to Divergence Form Operators. Stochastic Process. Appl.63 (1996) 11–33.
- A. Rozkosz, On Dirichlet processes associated with second order divergence form operators. Potential Anal.14 (2001) 123–148.
- A. Rozkosz and L. Slomiński, Extended Convergence of Dirichlet Processes. Stochastics Stochastics Rep.65 (1998) 79–109.
- 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).
- 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.
- D.R.E. Williams, Path-wise solutions of SDE's driven by Lévy processes. Rev. Mat. Iberoamericana17 (2002) 295–330. arXiv:math.. URIPR/0001018
- E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist.36 (1965) 1560–1564.

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