Stochastic differential equations driven by fractional Brownian motions

Details Stochastic differential equations driven by fractional Brownian motions Stochastic differential equations driven by fractional Brownian motions

2009-09-04

arxiv.org/abs/0909.0893v1

Bernoulli 2009, Vol. 15, No. 3, 846-870

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/08-BEJ169 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/08-BEJ169

In this paper, we study the existence and uniqueness of a class of stochastic differential equations driven by fractional Brownian motions with arbitrary Hurst parameter $H\in (0,1)$. In particular, the stochastic integrals appearing in the equations are defined in the Skorokhod sense on fractional Wiener spaces, and the coefficients are allowed to be random and even anticipating. The main technique used in this work is an adaptation of the anticipating Girsanov transformation of Buckdahn [Mem. Amer. Math. Soc. 111 (1994)] for the Brownian motion case. By extending a fundamental theorem of Kusuoka [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 567--597] using fractional calculus, we are able to prove that the anticipating Girsanov transformation holds for the fractional Brownian motion case as well. We then use this result to prove the well-posedness of the SDE.

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