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On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion. (English) Zbl 1091.60008
Summary: The problem of absolute continuity for a class of SDEs driven by a real fractional Brownian motion of any Hurst index is addressed. First, we give an elementary proof of the fact that when the diffusion coefficient does not vanish, the solution to the SDE has a positive density for all \(t>0\). Second, we extend in our setting the classical entrance-time criterion of N. Bouleau and F. Hirsch [J. Funct. Anal. 69, 229–259 (1986; Zbl 0605.60058)].

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G18 Self-similar stochastic processes
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[1] Baudoin, F.; Coutin, L., Etude en temps petit des solutions d’EDS conduites par des mouvements browniens fractionnaires, C. R. math. acad. sci. Paris, 341, 1, 39-42, (2005) · Zbl 1073.60061
[2] Bouleau, N.; Hirsch, F., Formes de Dirichlet générales et densité des variables aléatoires réelles sur l’espace de Wiener, J. funct. anal., 69, 2, 229-259, (1986) · Zbl 0605.60058
[3] Bouleau, N.; Hirsch, F., Dirichlet forms and analysis on Wiener space, (1991), De Gruyter Berlin · Zbl 0748.60046
[4] Coutin, L.; Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motion, Probab. theory related fields, 122, 1, 108-140, (2002) · Zbl 1047.60029
[5] Doss, H., Liens entre équations différentielles stochastiques et ordinaires, Ann. inst. H. Poincaré probab. statist., 13, 1, 99-125, (1977) · Zbl 0359.60087
[6] Feyel, D., de la Pradelle, A., 2005. Curvilinear integral along enriched paths. Prepublication. · Zbl 1110.60031
[7] Gradinaru, M.; Nourdin, I.; Russo, F.; Vallois, P., m-order integrals and Itô’s formula for non-semimartingale processes; the case of a fractional Brownian motion with any Hurst index, Ann. inst. H. Poincaré probab. statist., 41, 781-806, (2005) · Zbl 1083.60045
[8] Nourdin, I., 2004. Ph.D. Thesis, Nancy.
[9] Nourdin, I., Simon, T., 2005. On the absolute continuity of drifted Lévy processes. Ann. Probab., to appear.
[10] Nualart, D., The Malliavin calculus and related topics, (1995), Springer Berlin · Zbl 0837.60050
[11] Nualart, D., Ouknine, Y., 2003. Stochastic differential equations with additive fractional noise and locally unbounded drift. In: Stochastic Inequalities and Applications. Birkhäuser, Basel. Progr. Probab. 56, 353-365. · Zbl 1039.60061
[12] Russo, F.; Vallois, P., Forward, backward and symmetric stochastic integration, Probab. theory related fields, 97, 4, 403-421, (1993) · Zbl 0792.60046
[13] Sussmann, H.J., On the gap between deterministic and stochastic ordinary differential equations, Ann. probab., 6, 19-41, (1978) · Zbl 0391.60056
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