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 Electronic Communications in Probability > Vol. 15(2010) > Paper 37 open journal systems 


Moment estimates for solutions of linear stochastic differential equations driven by analytic fractional Brownian motion

Jeremie M Unterberger, Institut Elie Cartan de Nancy - Universite Henri Poincare - Nancy (France)


Abstract
As a general rule, differential equations driven by a multi-dimensional irregular path Γ are solved by constructing a rough path over Γ. The domain of definition -- and also estimates -- of the solutions depend on upper bounds for the rough path; these general, deterministic estimates are too crude to apply e.g. to the solutions of stochastic differential equations with linear coefficients driven by a Gaussian process with H"older regularity α<1/2. We prove here (by showing convergence of Chen's series) that linear stochastic differential equations driven by analytic fractional Brownian motion [6,7] with arbitrary Hurst index α∈(0,1) may be solved on the closed upper half-plane, and that the solutions have finite variance


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Pages: 411-417

Published on: September 30, 2010
Bibliography
  1. Baudoin, Fabrice; Coutin, Laure. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Process. Appl. 117 (2007), no. 5, 550--574. MR2320949 (2008c:60050)
  2. Friz, Peter K.; Victoir, Nicolas B. Multidimensional stochastic processes as rough paths.Theory and applications.Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010. xiv+656 pp. ISBN: 978-0-521-87607-0 MR2604669
  3. Gubinelli, M. Controlling rough paths. J. Funct. Anal. 216 (2004), no. 1, 86--140. MR2091358 (2005k:60169)
  4. A. Lejay. Global solutions to rough differential equations with unbounded vector fields. Preprint hal available on: http://hal.archives-ouvertes.fr/inria-00451193.
  5. A. Neuenkirch, I. Nourdin, A. Rössler, S. Tindel. Trees and asymptotic developments for fractional stochastic differential equations. Preprint arXiv:math.PR/0611306.
  6. S. Tindel, J. Unterberger. The rough path associated to the multidimensional analytic fBm with any Hurst parameter. Preprint arXiv:0810.1408. To appear in: Collectanea Mathematica.
  7. Unterberger, Jérémie. Stochastic calculus for fractional Brownian motion with Hurst exponent $H>frac 14$: a rough path method by analytic extension. Ann. Probab. 37 (2009), no. 2, 565--614. MR2510017 (2010i:60089)
  8. J. Unterberger. A stochastic calculus for multidimensional fractional Brownian motion with arbitrary Hurst index, Stoch. Proc. Appl. 120 (8), 1444-1472 (2010).
  9. J. Unterberger. Hölder-continuous rough paths by Fourier normal ordering, Comm. Math. Phys. 298 (1), 1--36 (2010).













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Electronic Communications in Probability. ISSN: 1083-589X