 |
|
|
| | | | | |
|
|
|
|
|
Time reversal for drifted fractional Brownian motion with Hurst index H>1/2
|
Sebastien Darses, University Paris 6
Bruno Saussereau, University of Franche-Comte
|
Abstract
Let X be a drifted fractional Brownian motion with Hurst index H>1/2. We prove that there exists a fractional backward representation of X, i.e. the time reversed process is a drifted
fractional Brownian motion, which continuously extends the one obtained in the theory of time reversal of Brownian diffusions when H=1/2. We then apply our result to stochastic differential equations driven by a fractional Brownian motion.
|
Full text: PDF
Pages: 1181-1211
Published on: September 7, 2007
|
Bibliography
- Alòs, Elisa; Nualart, David. Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75 (2003), no. 3, 129--152. MR1978896 (2004b:60138)
- Carmona, Philippe; Coutin, Laure; Montseny, Gérard. Stochastic integration with respect to fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 1, 27--68. MR1959841 (2003m:60095)
- Cresson, Jacky; Darses, Sébastien. Plongement stochastique des systèmes lagrangiens. (French) [Stochastic embedding of Lagrangian systems] C. R. Math. Acad. Sci. Paris 342 (2006), no. 5, 333--336. MR2201959 (2006i:37115)
- Darses S., Nourdin I. : Stochastic derivatives for fractional diffusions. Ann. Prob. To appear.
- Decreusefond, L. Stochastic integration with respect to Volterra processes. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 2, 123--149. MR2124078 (2005m:60117)
- Decreusefond, L.; Üstünel, A. S. Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999), no. 2, 177--214. MR1677455 (2000b:60133)
- Föllmer, H. Time reversal on Wiener space. Stochastic processes---mathematics and physics (Bielefeld, 1984), 119--129, Lecture Notes in Math., 1158, Springer, Berlin, 1986. MR0838561 (88a:60140)
- Garsia, A. M.; Rodemich, E.; Rumsey, H., Jr. A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 1970/1971 565--578. MR0267632 (42 #2534)
- Haussmann, U. G.; Pardoux, É. Time reversal of diffusions. Ann. Probab. 14 (1986), no. 4, 1188--1205. MR0866342 (88a:60142)
- Millet, A.; Nualart, D.; Sanz, M. Integration by parts and time reversal for diffusion processes. Ann. Probab. 17 (1989), no. 1, 208--238. MR0972782 (90m:60088)
- Moret, Sílvia; Nualart, David. Onsager-Machlup functional for the fractional Brownian motion. Probab. Theory Related Fields 124 (2002), no. 2, 227--260. MR1936018 (2003m:60156)
- Nelson, Edward. Dynamical theories of Brownian motion. Princeton University Press, Princeton, N.J. 1967 iii+142 pp. MR0214150 (35 #5001)
- Nualart, D.: The Malliavin Calculus and Related Topics. Springer Verlag (1996).
- Nualart, David. 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. MR2037156 (2004m:60119)
- Nualart, David; Ouknine, Youssef. Regularization of differential equations by fractional noise. Stochastic Process. Appl. 102 (2002), no. 1, 103--116. MR1934157 (2004b:60151)
- Nualart, David; Ru ac scanu, Aurel. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002), no. 1, 55--81. MR1893308 (2003f:60105)
- Pardoux, É. Grossissement d'une filtration et retournement du temps d'une diffusion. (French) [Enlargement of a filtration and time reversal of a diffusion] Séminaire de Probabilités, XX, 1984/85, 48--55, Lecture Notes in Math., 1204, Springer, Berlin, 1986. MR0942014 (89e:60150)
|
|
|
|
|
|
|
|
| | | | |
Electronic Journal of Probability. ISSN: 1083-6489 |
|
Context