Stochastic differential equations driven by processes generated by divergence form operators. I: A Wong-Zakai theorem. (English) Zbl 1181.60085
Summary: 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.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60J60 | Diffusion processes |
Keywords:
rough paths; stochastic differential equations; stochastic process generated by divergence-form operators; Dirichlet process; approximation of trajectoriesReferences:
| [1] | D.G. Aronson , Non-negative solutions of linear parabolic equation . Ann. Scuola Norm. Sup. Pisa 22 ( 1968 ) 607 - 693 . Numdam | Zbl 0182.13802 · Zbl 0182.13802 |
| [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 . Zbl 1010.60070 · Zbl 1010.60070 · doi:10.1007/s100970200040 |
| [3] | K.-T. Chen , Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula . Ann. of Math. 65 ( 1957 ) 163 - 178 . Zbl 0077.25301 · Zbl 0077.25301 · doi:10.2307/1969671 |
| [4] | L. Coutin and A. Lejay , Semi-martingales and rough paths theory . Electron. J. Probab. 10 ( 2005 ) 761 - 785 . Zbl 1109.60035 · Zbl 1109.60035 |
| [5] | L. Coutin and Z. Qian , Stochastic analysis, rough path analysis and fractional Brownian motions . Probab. Theory Related Fields 122 ( 2002 ) 108 - 140 . Zbl 1047.60029 · Zbl 1047.60029 · doi:10.1007/s004400100158 |
| [6] | F. Coquet and L. Słomiński , On the convergence of Dirichlet processes . Bernoulli 5 ( 1999 ) 615 - 639 . Article | Zbl 0953.60001 · Zbl 0953.60001 · doi:10.2307/3318693 |
| [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 1086.31007 · Zbl 1086.31007 · doi:10.1023/B:POTA.0000021332.71490.c6 |
| [8] | H. Föllmer , Calcul d’Itô sans probabilités , in Séminaire de Probabilités, XV. Lect. Notes Math. 850 ( 1981 ) 143 - 150 . Springer, Berlin. Numdam | Zbl 0461.60074 · Zbl 0461.60074 |
| [9] | H. Föllmer , Dirichlet processes , in Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lect. Notes Math. 851 ( 1981 ) 476 - 478 . Springer, Berlin. Zbl 0462.60046 · Zbl 0462.60046 |
| [10] | M. Fukushima , Y. Oshima and M. Takeda , Dirichlet Forms and Symmetric Markov Process . De Gruyter ( 1994 ). MR 1303354 | Zbl 0838.31001 · Zbl 0838.31001 |
| [11] | F. Flandoli and F. Russo , Generalized integration and stochastic ODEs . Ann. Probab. 30 ( 2002 ) 270 - 292 . Article | Zbl 1022.60054 · Zbl 1022.60054 · doi:10.1214/aop/1020107768 |
| [12] | P. Friz and N. Victoir , A note on the notion of geometric rough paths . To appear in Probab. Theory Related Fields ( 2006 ). MR 2257130 | Zbl 1108.34052 · Zbl 1108.34052 · doi:10.1007/s00440-005-0487-7 |
| [13] | B.M. Hambly and T.J. Lyons , Stochastic area for Brownian motion on the Sierpinski gasket . Ann. Probab. 26 ( 1998 ) 132 - 148 . Article | Zbl 0936.60073 · Zbl 0936.60073 · doi:10.1214/aop/1022855414 |
| [14] | N. Ikeda and S. Watanabe , Stochastic Differential Equations and Diffusion Processes . North Holland, 2\({\text{nd}}\) edition ( 1989 ). MR 1011252 | Zbl 0684.60040 · Zbl 0684.60040 |
| [15] | H. Kunita , Stochastic flows and stochastic differential equations . Cambridge University Press ( 1990 ). MR 1070361 | Zbl 0743.60052 · Zbl 0743.60052 |
| [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/\(\sim \)lejay/. |
| [17] | A. Lejay , An introduction to rough paths , in Séminaire de probabilités, XXXVII. Lect. Notes Math. 1832 ( 2003 ) 1 - 59 , Springer, Berlin. Zbl 1041.60051 · Zbl 1041.60051 |
| [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 . Zbl 1075.60070 · Zbl 1075.60070 · doi:10.1016/j.spa.2003.09.012 |
| [19] | A. Lejay , Stochastic Differential Equations driven by processes generated by divergence form operators II: Convergence results . Institut Élie Cartan de Nancy (preprint), 2003. · Zbl 1185.60061 |
| [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 . Article | Zbl 1016.60071 · Zbl 1016.60071 · doi:10.1214/aop/1023481002 |
| [22] | T. Lyons and Z. Qian , System Control and Rough Paths . Oxford Mathematical Monographs. Oxford University Press ( 2002 ). MR 2036784 | Zbl 1029.93001 · Zbl 1029.93001 · doi:10.1093/acprof:oso/9780198506485.001.0001 |
| [23] | T.J. Lyons and L. Stoica , The limits of stochastic integrals of differential forms . Ann. Probab. 27 ( 1999 ) 1 - 49 . Article | Zbl 0969.60078 · Zbl 0969.60078 · doi:10.1214/aop/1022677253 |
| [24] | T.J. Lyons , Differential equations driven by rough signals . Rev. Mat. Iberoamericana 14 ( 1998 ) 215 - 310 . Zbl 0923.34056 · Zbl 0923.34056 · doi:10.4171/RMI/240 |
| [25] | A. Lejay and N. Victoir , On \((p,q)\)-rough paths . J. Differential Equations 225 ( 2006 ) 103 - 133 . Zbl 1097.60048 · Zbl 1097.60048 · doi:10.1016/j.jde.2006.01.018 |
| [26] | Z. Ma and M. Röckner , Introduction to the Theory of (Non-Symmetric) Dirichlet Forms . Universitext. Springer-Verlag ( 1991 ). Zbl 0826.31001 · Zbl 0826.31001 |
| [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 ). Article | Zbl 0283.60061 · Zbl 0283.60061 |
| [28] | A. Rozkosz , Stochastic Representation of Diffusions Corresponding to Divergence Form Operators . Stochastic Process. Appl. 63 ( 1996 ) 11 - 33 . Zbl 0870.60073 · Zbl 0870.60073 · doi:10.1016/0304-4149(96)00059-2 |
| [29] | A. Rozkosz , On Dirichlet processes associated with second order divergence form operators . Potential Anal. 14 ( 2001 ) 123 - 148 . Zbl 0974.60064 · Zbl 0974.60064 · doi:10.1023/A:1008778601226 |
| [30] | A. Rozkosz and L. Slomiński , Extended Convergence of Dirichlet Processes . Stochastics Stochastics Rep. 65 ( 1998 ) 79 - 109 . Zbl 0917.60076 · Zbl 0917.60076 |
| [31] | D. Revuz and M. Yor , Continuous Martingales and Brownian Motion . Springer-Verlag ( 1990 ). Zbl 0731.60002 · Zbl 0731.60002 |
| [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 0651.47031 · Zbl 0651.47031 |
| [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. arXiv | Zbl 1002.60060 · Zbl 1002.60060 · doi:10.4171/RMI/296 |
| [35] | E. Wong and M. Zakai , On the convergence of ordinary integrals to stochastic integrals . Ann. Math. Statist. 36 ( 1965 ) 1560 - 1564 . Article | Zbl 0138.11201 · Zbl 0138.11201 · doi:10.1214/aoms/1177699916 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
