Previous |  Up |  Next

Article

Keywords:
fractional Brownian motion; Girsanov theorem; weak solutions
Summary:
Existence of a weak solution to the $n$-dimensional system of stochastic differential equations driven by a fractional Brownian motion with the Hurst parameter $H\in (0,1)\setminus \{\frac 12\}$ is shown for a time-dependent but state-independent diffusion and a drift that may by split into a regular part and a singular one which, however, satisfies the hypotheses of the Girsanov Theorem. In particular, a stochastic nonlinear oscillator driven by a fractional noise is considered.
References:
[1] Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001), 766-801. DOI 10.1214/aop/1008956692 | MR 1849177
[2] Boufoussi, B., Ouknine, Y.: On a SDE driven by a fractional Brownian motion and with monotone drift. Elect. Comm. Probab. 8 (2003), 122-134. MR 2042751 | Zbl 1060.60060
[3] Cheridito, P., Nualart, D.: Stochastic integral of divergence type with respect to fBm with Hurst parametr $H\in(0,\frac12)$. Ann. I. H. Poincaré Probab. Stat. 41 (2005), 1049-1081. DOI 10.1016/j.anihpb.2004.09.004 | MR 2172209
[4] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambdridge University Press, Cambridge (1992). MR 1207136 | Zbl 0761.60052
[5] Decreusefond, L., Üstunel, A. S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999), 177-214. DOI 10.1023/A:1008634027843 | MR 1677455
[6] Denis, L., Erraoni, M., Ouknine, Y.: Existence and uniqueness for solutions of one dimensional SDE's driven by an additive fractional noise. Stoch. Stoch. Rep. 76 (2004), 409-427. DOI 10.1080/10451120412331299336 | MR 2096729
[7] Duncan, T. E., Maslowski, B., Pasik-Duncan, B.: Semilinear stochastic equations in a Hilbert space with a fractional Brownian motion. (to appear) in SIAM J. Math. Anal. MR 2481295
[8] Duncan, T. E., Maslowski, B., Pasik-Duncan, B.: Linear stochastic equations in a Hilbert space with a fractional Brownian motion, Control Theory Applications in Financial Engineering and Manufacturing, Chapter 11, 201-222. Springer-Verlag, New York (2006). MR 2353483
[9] Fernique, X.: Régularité des trajectoires des fonctions aléatoires gaussiennes. École d'Été de Probabilités de Saint-Flour IV--1974, LNM 480, Springer-Verlag, Berlin (1975), 1-96. MR 0413238 | Zbl 0331.60025
[10] Friedman, A.: Stochastic Differential Equations and Applications, vol. I. AP, New York (1975). MR 0494490
[11] Hu, Y.: Integral transformations and anticipative calculus for fractional Brownian motions. Mem. Amer. Math. Soc. 175 (2005). MR 2130224 | Zbl 1072.60044
[12] Hu, Y., Nualart, D.: Differential equations driven by Hölder continuous functions of order greater than $1/2$. Stochastic analysis and applications, 399-413, Springer, Berlin (2007). MR 2397797 | Zbl 1144.34038
[13] Karatzas, I., Shreve, S. E.: Brownian Motion and Stochastic Calculus. Springer-Verlag, New York (1988). MR 0917065 | Zbl 0638.60065
[14] Kufner, A., John, O., Fučík, S.: Function Spaces. Academia, Praha (1977). MR 0482102
[15] Kurzweil, J.: Ordinary Differential Equations. Elsevier, Amsterdam (1986). MR 0929466 | Zbl 0667.34002
[16] Lyons, T.: Differential Equations Driven by Rough Signals. Rev. Mat. Iberoamericana 14 (1998), 215-310. DOI 10.4171/RMI/240 | MR 1654527 | Zbl 0923.34056
[17] Lyons, T.: Differential Equations Driven by Rough Signals (I): an extension of an inequality of L. C. Young. Math. Res. Lett. 1 (1994), 451-464. DOI 10.4310/MRL.1994.v1.n4.a5 | MR 1302388 | Zbl 0835.34004
[18] Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202 (2003), 277-305. DOI 10.1016/S0022-1236(02)00065-4 | MR 1994773 | Zbl 1027.60060
[19] Mémin, J., Mishura, Y., Valkeila, E.: Inequalities for the moments of Wiener integrals with respect to fractional Brownian motions. Stat. Prob. Lett. 51 (2001), 197-206. DOI 10.1016/S0167-7152(00)00157-7 | MR 1822771
[20] Mishura, Y., Nualart, D.: Weak solutions for stochastic differential equations with additive fractional noise. Stat. Probab. Lett. 70 (2004), 253-261. DOI 10.1016/j.spl.2004.10.011 | MR 2125162
[21] Nourdin, I., Simon, T.: On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion. Statist. Probab. Lett. 76 (2006), 907-912. DOI 10.1016/j.spl.2005.10.021 | MR 2268434 | Zbl 1091.60008
[22] Nualart, D., Rǎşcanu, A.: Differential Equations driven by Fractional Brownian Motion. Collect. Math. 53 (2002), 55-81. MR 1893308
[23] Nualart, D., Ouknine, Y.: Regularization of differential equations by fractional noise. Stochastic Process. Appl. 102 (2002), 103-116. DOI 10.1016/S0304-4149(02)00155-2 | MR 1934157 | Zbl 1075.60536
[24] Nualart, D., Ouknine, Y.: Stochastic differential equations with additive fractional noise and locally unbounded drift. Stochastic inequalities and applications, 353-365, Birkhäuser, Basel (2003). MR 2073441 | Zbl 1039.60061
[25] Nualart, D.: Stochastic integration with respect to fractional Brownian motion and applications. Stochastic models (Mexico City, 2002), 3-39, Contemp. Math., 336, Amer. Math. Soc., Providence, RI (2003). DOI 10.1090/conm/336/06025 | MR 2037156 | Zbl 1063.60080
[26] Zähle, M.: Stochastic differential equations with fractal noise. Math. Nachr. 278 (2005), 1097-1106. DOI 10.1002/mana.200310295 | MR 2150381
Partner of
EuDML logo