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August 2009 Stochastic differential equations driven by fractional Brownian motions
Yu-Juan Jien, Jin Ma
Bernoulli 15(3): 846-870 (August 2009). DOI: 10.3150/08-BEJ169

Abstract

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.

Citation

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Yu-Juan Jien. Jin Ma. "Stochastic differential equations driven by fractional Brownian motions." Bernoulli 15 (3) 846 - 870, August 2009. https://doi.org/10.3150/08-BEJ169

Information

Published: August 2009
First available in Project Euclid: 28 August 2009

zbMATH: 1214.60024
MathSciNet: MR2555202
Digital Object Identifier: 10.3150/08-BEJ169

Keywords: anticipating stochastic calculus , fractional Brownian motions , Girsanov transformations , Skorokhod integrals

Rights: Copyright © 2009 Bernoulli Society for Mathematical Statistics and Probability

Vol.15 • No. 3 • August 2009
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