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

Lejay, Antoine

ESAIM: Probability and Statistics, Tome 10 (2006), pp. 356-379.

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.

[1] D.G. Aronson, Non-negative solutions of linear parabolic equation. Ann. Scuola Norm. Sup. Pisa 22 (1968) 607-693. | Numdam | Zbl

[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. | Zbl

[3] K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. 65 (1957) 163-178. | Zbl

[4] L. Coutin and A. Lejay, Semi-martingales and rough paths theory. Electron. J. Probab. 10 (2005) 761-785. | Zbl

[5] L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108-140. | Zbl

[6] F. Coquet and L. Słomiński, On the convergence of Dirichlet processes. Bernoulli 5 (1999) 615-639. | Zbl

[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. | Zbl

[8] H. Föllmer, Calcul d'Itô sans probabilités, in Séminaire de Probabilités, XV. Lect. Notes Math. 850 (1981) 143-150. Springer, Berlin. | Numdam | Zbl

[9] H. Föllmer, Dirichlet processes, in Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lect. Notes Math. 851 (1981) 476-478. Springer, Berlin. | Zbl

[10] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Process. De Gruyter (1994). | MR | Zbl

[11] F. Flandoli and F. Russo, Generalized integration and stochastic ODEs. Ann. Probab. 30 (2002) 270-292. | Zbl

[12] P. Friz and N. Victoir, A note on the notion of geometric rough paths. To appear in Probab. Theory Related Fields (2006). | MR | Zbl

[13] B.M. Hambly and T.J. Lyons, Stochastic area for Brownian motion on the Sierpinski gasket. Ann. Probab. 26 (1998) 132-148. | Zbl

[14] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes1989). | MR | Zbl

[15] H. Kunita, Stochastic flows and stochastic differential equations. Cambridge University Press (1990). | MR | Zbl

[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/$\sim$lejay/.

[17] A. Lejay, An introduction to rough paths, in Séminaire de probabilités, XXXVII. Lect. Notes Math. 1832 (2003) 1-59, Springer, Berlin. | Zbl

[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. | Zbl

[19] A. Lejay, Stochastic Differential Equations driven by processes generated by divergence form operators II: Convergence results. Institut Élie Cartan de Nancy (preprint), 2003.

[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. | Zbl

[22] T. Lyons and Z. Qian, System Control and Rough Paths. Oxford Mathematical Monographs. Oxford University Press (2002). | MR | Zbl

[23] T.J. Lyons and L. Stoica, The limits of stochastic integrals of differential forms. Ann. Probab. 27 (1999) 1-49. | Zbl

[24] T.J. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215-310. | Zbl

[25] A. Lejay and N. Victoir, On $(p,q)$-rough paths. J. Differential Equations 225 (2006) 103-133. | Zbl

[26] Z. Ma and M. Röckner, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext. Springer-Verlag (1991). | Zbl

[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). | Zbl

[28] A. Rozkosz, Stochastic Representation of Diffusions Corresponding to Divergence Form Operators. Stochastic Process. Appl. 63 (1996) 11-33. | Zbl

[29] A. Rozkosz, On Dirichlet processes associated with second order divergence form operators. Potential Anal. 14 (2001) 123-148. | Zbl

[30] A. Rozkosz and L. Slomiński, Extended Convergence of Dirichlet Processes. Stochastics Stochastics Rep. 65 (1998) 79-109. | Zbl

[31] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag (1990). | Zbl

[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. | Numdam | Zbl

[34] D.R.E. Williams, Path-wise solutions of SDE's driven by Lévy processes. Rev. Mat. Iberoamericana 17 (2002) 295-330. arXiv:math.PR/0001018. | Zbl

[35] E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist. 36 (1965) 1560-1564. | Zbl