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Good rough path sequences and applications to anticipating stochastic calculus. (English) Zbl 1132.60053
This article is concerned with anticipative Stratonovich stochastic differential equations \[ d Y_t = V_0(Y_t) d t + \sum_{i=1}^d V_i(Y_t)\circ d B^i_t, \quad Y_0 = y_0 \] driven by some stochastic process which will be lifted to a rough path in a Lie group. Neither adaptedness of the initial point \(y_0\) and the \(C^1\) vector field \(V = (V_0,\dots, V_d)\) nor commuting conditions for the latter are assumed. Under simple conditions on the stochastic process, the authors prove that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. It is shown that this condition is satisfied by Brownian motion.
As applications, the authors obtain rather flexible results for these solutions such as support theorems, large deviation principles and Wong-Zakai approximations for SDEs driven by Brownian motion along anticipating vectorfields. The authors suggest that this perspective may unify many results on anticipative SDEs.

60H99 Stochastic analysis
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Ahn, H. and Kohatsu-Higa, A. (1995). The Euler scheme for anticipating stochastic differential equations. Stochastics Stochastics Rep. 54 247–269. · Zbl 0857.60053
[2] Caballero, M. E., Fernández, B. and Nualart, D. (1995). Smoothness of distributions for solutions of anticipating stochastic differential equations. Stochastics Stochastics Rep. 52 303–322. · Zbl 0864.60040
[3] Caballero, M. E., Fernández, B. and Nualart, D. (1997). Composition of skeletons and support theorems. In Stochastic Differential and Difference Equations (I. Csiszár and Gy. Michaletzky, eds.) 21–33. Birkhäuser, Boston. · Zbl 0888.60045
[4] Friz, P. (2005). Continuity of the Itô-map for Hölder rough paths with applications to the support theorem in Hölder norm. In Probability and Partial Differential Equations in Modern Applied Mathematics (E. C. Waymire and J. Duan, eds.) 117–135. Springer, New York. · Zbl 1090.60038
[5] Friz, P., Lyons, T. and Stroock, D. (2006). Lévy’s area under conditioning. Ann. Inst. H. Poincare Probab. Statist. 42 89–101. · Zbl 1099.60054 · doi:10.1016/j.anihpb.2005.02.003 · numdam:AIHPB_2006__42_1_89_0 · eudml:77889
[6] Friz, P. and Victoir, N. (2005). Approximations of the Brownian rough path with applications to stochastic analysis. Ann. Inst. H. Poincare Probab. Statist. 41 703–724. · Zbl 1080.60021 · doi:10.1016/j.anihpb.2004.05.003 · numdam:AIHPB_2005__41_4_703_0 · eudml:77863
[7] Friz, P. and Victoir, N. (2006). A note on the notion of geometric rough path. Probab. Theory Related Fields 136 395–416. · Zbl 1108.34052 · doi:10.1007/s00440-005-0487-7
[8] Kohatsu-Higa, A. and León, J. A. (1997). Anticipating stochastic differential equations of Stratonovich type. Appl. Math. Optim. 36 263–289. · Zbl 0904.60040 · doi:10.1007/s002459900063
[9] Kohatsu-Higa, A., León, J. A. and Nualart, D. (1997). Stochastic differential equations with random coefficients. Bernoulli 3 233–245. · Zbl 0885.60049 · doi:10.2307/3318589
[10] Lejay, A. (2003). Introduction to rough paths. Séminaire de Probabilités XXXVII . Lecture Notes in Math. 1832 1–59. Springer, Berlin. · Zbl 1041.60051
[11] Lejay, A. and Lyons, T. (2003). On the importance of the Lévy area for studying the limits functions of converging stochastic processes. Application to homogenization. Current Trends in Potential Theory 63–84. · Zbl 1199.60292
[12] Lejay, A. and Victoir, N. (2006). On \((p,q)\)-rough paths. J. Differential Equations 225 103–133. · Zbl 1097.60048 · doi:10.1016/j.jde.2006.01.018
[13] Lévy, P. (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris. · Zbl 0034.22603
[14] Lyons, T. (1998). J. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215–310. · Zbl 0923.34056 · doi:10.4171/RMI/240 · eudml:39555
[15] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths . Oxford Univ. Press. · Zbl 1029.93001 · doi:10.1093/acprof:oso/9780198506485.001.0001
[16] Ledoux, M., Qian, Z. and Zhang, T. (2002). Large deviations and support theorem for diffusion processes via rough paths. Stochastic Process. Appl. 102 265–283. · Zbl 1075.60510 · doi:10.1016/S0304-4149(02)00176-X
[17] Millet, A. and Nualart, D. (1992). Support theorems for a class of anticipating stochastic differential equations. Stochastics Stochastics Rep. 39 1–24. · Zbl 0762.60048
[18] Millet, A., Nualart, D. and Sanz, M. (1992). Large deviations for a class of anticipating stochastic differential equations. Ann. Probab. 20 1902–1931. · Zbl 0769.60053 · doi:10.1214/aop/1176989535
[19] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York. · Zbl 0837.60050
[20] Ocone, D. and Pardoux, E. (1989). A generalized Itô–Ventzell formula. Application to a class of anticipating stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 25 39–71. · Zbl 0674.60057 · numdam:AIHPB_1989__25_1_39_0 · eudml:77339
[21] Reutenauer, C. (1993). Free Lie Algebras . Oxford Science Publications. · Zbl 0798.17001
[22] Rovira, C. and Sanz-Solé, M. (1997). Anticipating stochastic differential equations: Regularity of the law. J. Funct. Anal. 143 157–179. · Zbl 0877.60038 · doi:10.1006/jfan.1996.2972
[23] Sanz-Solé, M. and Sarrà, M. (1999). Logarithmic estimates for the density of an anticipating stochastic differential equation. Stochastic Process. Appl. 79 301–321. · Zbl 0962.60044 · doi:10.1016/S0304-4149(98)00092-1
[24] Stroock, D. W. (1993). Probability Theory, an Analytic View . Cambridge Univ. Press. · Zbl 0925.60004
[25] Stroock, D. and Varadhan, S. R. (1972). On the support of diffusion processes with applications to the strong maximum principle. Proc. Sixth Berkeley Sympos. Math. Statist. Probability III . Probability Theory 333–359. Univ. California Press, Berkeley. · Zbl 0255.60056
[26] Young, L. C. (1936). An inequality of Hölder type, connected with Stieltjes integration. Acta Math. 67 251–282. · Zbl 0016.10404 · doi:10.1007/BF02401743
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