# Stochastic differential equations driven by processes generated by divergence form operators II: convergence results

ESAIM: Probability and Statistics (2008)

- Volume: 12, page 387-411
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topLejay, Antoine. "Stochastic differential equations driven by processes generated by divergence form operators II: convergence results." ESAIM: Probability and Statistics 12 (2008): 387-411. <http://eudml.org/doc/250413>.

@article{Lejay2008,

abstract = {
We have seen in a previous article how the theory of “rough paths”
allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one
can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary,
we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals already constructed
for stochastic processes generated by divergence form operators by using time-reversal techniques.
},

author = {Lejay, Antoine},

journal = {ESAIM: Probability and Statistics},

keywords = {Rough paths; stochastic differential equations;
stochastic process generated by divergence form
operators; Condition UTD; convergence of stochastic integrals; rough paths; stochastic process generated by divergence form operators; condition UTD},

language = {eng},

month = {7},

pages = {387-411},

publisher = {EDP Sciences},

title = {Stochastic differential equations driven by processes generated by divergence form operators II: convergence results},

url = {http://eudml.org/doc/250413},

volume = {12},

year = {2008},

}

TY - JOUR

AU - Lejay, Antoine

TI - Stochastic differential equations driven by processes generated by divergence form operators II: convergence results

JO - ESAIM: Probability and Statistics

DA - 2008/7//

PB - EDP Sciences

VL - 12

SP - 387

EP - 411

AB -
We have seen in a previous article how the theory of “rough paths”
allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one
can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary,
we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals already constructed
for stochastic processes generated by divergence form operators by using time-reversal techniques.

LA - eng

KW - Rough paths; stochastic differential equations;
stochastic process generated by divergence form
operators; Condition UTD; convergence of stochastic integrals; rough paths; stochastic process generated by divergence form operators; condition UTD

UR - http://eudml.org/doc/250413

ER -

## References

top- R. Adams, Sobolev spaces. Academic Press (1975). Zbl0314.46030
- F. Baudoin, An introduction to the geometry of stochastic flows. Imperial College Press, London (2004). Zbl1085.60002
- A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland (1978). Zbl0404.35001
- P. Billingsley, Convergence of Probability Measures. Wiley (1968).
- C.J.K. Batty, O. Bratteli, P.E.T. Jørgensen and D.W. Robinson, Asymptotics of periodic subelliptic operators. J. Geom. Anal.5 (1995) 427–443. Zbl0861.43005
- L. Capogna, D. Danielli, S.D. Pauls and J.T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Progress in Mathematics, Vol. 259. Birkhäuser (2007). Zbl1138.53003
- L. Coutin, P. Friz and N. Victoir, Good Rough Path Sequences and Applications to Anticipating Stochastic Calculus. Ann. Prob.35 (2007) 1172–1193. Zbl1132.60053
- L. Coutin and A. Lejay, Semi-martingales and rough paths theory. Electron. J. Probab.10 (2005) 761–785. Zbl1109.60035
- F. Coquet and L. Słomiński, On the convergence of Dirichlet processes. Bernoulli5 (1999) 615–639. Zbl0953.60001
- S.N. Ethier and T.G. Kurtz, Markov Processes, Characterization and Convergence. Wiley (1986).
- H. Föllmer, Dirichlet processes, in Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), Lecture Notes in Math.851 476–478. Springer, Berlin (1981).
- M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Process. De Gruyter (1994). Zbl0838.31001
- P. Friz and N. Victoir, A note on the notion of geometric rough path. Probab. Theory Related Fields136 (2006) 395–416. Zbl1108.34052
- P. Friz and N. Victoir, On Uniformly Subelliptic Operators and Stochastic Area. Preprint Cambridge University (2006). <arXiv:math.PR/0609007>. Zbl1151.31009
- V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer-Verlag (1994).
- T.G. Kurtz and P. Protter, Weak Convergence of Stochastic Integrals and Differential Equations, in Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995, Talay D. and Tubaro L. Eds., Lecture Notes in Math.1629 1–41. Springer-Verlag (1996). Zbl0862.60041
- I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag, 2nd edition (1991). Zbl0734.60060
- 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). <url: >. URIhttp://www.iecn.u-nancy.fr/~lejay/
- A. Lejay, A Probabilistic Approach of the Homogenization of Divergence-Form Operators in Periodic Media. Asymptot. Anal.28 (2001) 151–162. Zbl0999.60065
- A. Lejay, On the convergence of stochastic integrals driven by processes converging on account of a homogenization property. Electron. J. Probab.7 1–18 (2002). Zbl1007.60018
- A. Lejay, An introduction to rough paths, in Séminaire de probabilités, XXXVII, Lect. Notes Math.1832 1–59, Springer, Berlin (2003). Zbl1041.60051
- A. Lejay, Stochastic Differential Equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem. ESAIM: PS10 (2006) 356–379. Zbl1181.60085
- A. Lejay, Yet another introduction to rough paths. Preprint, Institut Élie Cartan, Nancy (2006). <>. Zbl1198.60002URIhttp://hal.inria.fr/inria-00107460
- 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).
- A. Lejay and N. Victoir, On (p,q)-rough paths. J. Diff. Equ.225 (2006) 103–133. Zbl1097.60048
- T. Lyons and Z. Qian, System Control and Rough Paths. Oxford Mathematical Monographs. Oxford University Press (2002). Zbl1029.93001
- T.J. Lyons and L. Stoica, The limits of stochastic integrals of differential forms. Ann. Probab.27 (1999) 1–49. Zbl0969.60078
- T.J. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoamericana14 (1998) 215–310. Zbl0923.34056
- P. Marcellini, Convergence of Second Order Linear Elliptic Operator. Boll. Un. Mat. Ital. B (5)16 (1979) 278–290. Zbl0408.35028
- R. Montgomery, A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs 91. American Mathematical Society, Providence, RI (2002). Zbl1044.53022
- A. Rozkosz, Stochastic Representation of Diffusions Corresponding to Divergence Form Operators. Stochastic Process. Appl.63 (1996) 11–33. Zbl0870.60073
- A. Rozkosz, Weak Convergence of Diffusions Corresponding to Divergence Form Operator. Stochastics Stochastics Rep.57 (1996) 129–157. Zbl0885.60067
- A. Rozkosz and L. Slomiński, Extended Convergence of Dirichlet Processes. Stochastics Stochastics Rep.65 (1998) 1–2, 79–109. Zbl0917.60076
- D.W. Stroock, Diffusion Semigroups Corresponding to Uniformly Elliptic Divergence Form Operator, in Séminaire de Probabilités XXII, Lecture Notes in Math.1321 316–347. Springer-Verlag (1988).

## NotesEmbed ?

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