Rough path limits of the Wong-Zakai type with a modified drift term. (English) Zbl 1169.60011
Authors’ abstract: “The Wong–Zakai theorem asserts that ODEs driven by “reasonable” (e.g. piecewise linear) approximations of Brownian motion converge to the corresponding Stratonovich stochastic differential equation. With the aid of rough path analysis, we study “non-reasonable” approximations and go beyond a well-known criterion of N. Ikeda and S. H. Watanabe [Stochastic differential equations and diffusion processes. 2nd ed. North Holland (1989; Zbl 0684.60040)] in the sense that our result applies to perturbations on all levels, exhibiting additional drift terms involving any iterated Lie brackets of the driving vector fields. In particular, this applies to the approximations by E. J. McShane [Proc. 6th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1970, No. 3, 263–294 (1972; Zbl 0283.60061)] and H. Sussmann [Stochastic analysis, Proc. Conf. Honor Moshe Zakai 65th Birthday, Haifa/Isr. 1991, 475-493 (1991; Zbl 0733.60082)]. Our approach is not restricted to Brownian driving signals. At last, these ideas can be used to prove optimality of certain rough path estimates.”
Reviewer: Roman Urban (Wrocław)
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60J25 | Continuous-time Markov processes on general state spaces |
| 60J60 | Diffusion processes |
| 34F05 | Ordinary differential equations and systems with randomness |
References:
| [1] | Bass, R. F.; Hambly, B.; Lyons, T., Extending the Wong-Zakai theorem to reversible Markov processes, J. Eur. Math. Soc., 4, 3, 237-269 (2002) · Zbl 1010.60070 |
| [2] | Cass, T.; Friz, P.; Victoir, N., Non-degeneracy of Wiener functionals arising from rough differential equations, Trans. Amer. Math. Soc., 361, 3359-3371 (2009) · Zbl 1175.60034 |
| [3] | Coutin, L.; Lejay, A., Semi-martingales and rough paths theory, Electron. J. Probab., 10, 23, 761-785 (2005), (electronic) · Zbl 1109.60035 |
| [4] | Coutin, L.; Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields, 122, 1, 108-140 (2002) · Zbl 1047.60029 |
| [5] | Friz, P.; Victoir, N., Approximations of the Brownian rough path with applications to stochastic analysis, Ann. Inst. H. Poincaré Probab. Statist., 41, 4, 703-724 (2005) · Zbl 1080.60021 |
| [6] | Friz, P.; Victoir, N., A note on the notion of geometric rough paths, Probab. Theory Related Fields, 136, 3, 395-416 (2006) · Zbl 1108.34052 |
| [7] | Friz, P.; Victoir, N., The Burkholder-Davis-Gundy inequality for enhanced martingales, (Donati-Martin, C.; Émery, M.; Rouault, A.; Stricker, C., Séminaire de Probabilités XLI. Séminaire de Probabilités XLI, Lecture Notes in Math. (2008), Springer) · Zbl 1156.60027 |
| [8] | Friz, P.; Victoir, N., Euler estimates for rough differential equations, J. Differential Equations, 244, 2, 388-412 (2008) · Zbl 1140.60037 |
| [9] | Friz, P.; Victoir, N., On uniformly subelliptic operators and stochastic area, Probab. Theory Related Fields, 142, 3-4, 475-523 (2008) · Zbl 1151.31009 |
| [10] | P. Friz, N. Victoir, Differential equations driven by Gaussian signals I. Ann. Inst. H. Poincaré Probab. Statist. (2009), in press; preprint arXiv:0707.0313v1 [math.PR]; P. Friz, N. Victoir, Differential equations driven by Gaussian signals I. Ann. Inst. H. Poincaré Probab. Statist. (2009), in press; preprint arXiv:0707.0313v1 [math.PR] · Zbl 1202.60058 |
| [11] | Friz, P.; Victoir, N., Multidimensional Stochastic Processes as Rough Paths. Theory and Applications (2009), Cambridge University Press, forthcoming |
| [12] | Gyöngy, I.; Michaletzky, G., On Wong-Zakai approximations with \(δ\)-martingales, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460, 2041, 309-324 (2004), stochastic analysis with applications to mathematical finance · Zbl 1055.60056 |
| [13] | Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1989), North-Holland: North-Holland Amsterdam · Zbl 0684.60040 |
| [14] | Lejay, A., Stochastic differential equations driven by a process generated by divergence form operators, ESAIM Probab. Stat., 10, 356-379 (2006) · Zbl 1181.60085 |
| [15] | Lejay, A.; Lyons, T., On the importance of the Lévy area for studying the limits of functions of converging stochastic processes. Application to homogenization, Current Trends in Potential Theory, 7 (2003) |
| [16] | Lejay, A.; Victoir, N., On \((p, q)\)-rough paths, J. Differential Equations, 225, 1, 103-133 (2006) · Zbl 1097.60048 |
| [17] | Lyons, T., Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14, 2, 215-310 (1998) · Zbl 0923.34056 |
| [18] | Lyons, T. J.; Caruana, M.; Lévy, T., Differential Equations Driven by Rough Paths, Lecture Notes in Math., vol. 1908 (2007), Springer: Springer Berlin, Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6-24, 2004, with an introduction concerning the Summer School by Jean Picard · Zbl 1176.60002 |
| [19] | McShane, E. J., Stochastic differential equations and models of random processes, (Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, Calif., 1970/1971. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, Calif., 1970/1971, Probab. Theory, vol. III (1972), Univ. of California Press: Univ. of California Press Berkeley), 263-294 · Zbl 0283.60061 |
| [20] | Rogers, L. C.G.; Williams, D., Diffusions, Markov Processes, and Martingales, vol. 1 (2000), Cambridge Mathematical Library, Cambridge University Press: Cambridge Mathematical Library, Cambridge University Press Cambridge, Foundations, reprint of the second edition, 1994 · Zbl 0949.60003 |
| [21] | Stroock, D. W., Diffusion semigroups corresponding to uniformly elliptic divergence form operators, (Séminaire de Probabilités, XXII. Séminaire de Probabilités, XXII, Lecture Notes in Math., vol. 1321 (1988), Springer: Springer Berlin), 316-347 · Zbl 0651.47031 |
| [22] | Sussmann, H. J., Limits of the Wong-Zakai type with a modified drift term, (Stochastic Analysis (1991), Academic Press: Academic Press Boston, MA), 475-493 · Zbl 0733.60082 |
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