• Bernoulli
  • Volume 15, Number 3 (2009), 846-870.

Stochastic differential equations driven by fractional Brownian motions

Yu-Juan Jien and Jin Ma

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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 $H∈(0, 1)$. 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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Bernoulli, Volume 15, Number 3 (2009), 846-870.

First available in Project Euclid: 28 August 2009

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anticipating stochastic calculus fractional Brownian motions Girsanov transformations Skorokhod integrals


Jien, Yu-Juan; Ma, Jin. Stochastic differential equations driven by fractional Brownian motions. Bernoulli 15 (2009), no. 3, 846--870. doi:10.3150/08-BEJ169.

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