Stochastic differential equations driven by fractional Brownian motions
Yu-Juan Jien and Jin Ma
Source: Bernoulli Volume 15, Number 3
(2009), 846-870.
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.
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Keywords: anticipating stochastic calculus; fractional Brownian motions; Girsanov transformations; Skorokhod integrals
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.bj/1251463284
Digital Object Identifier: doi:10.3150/08-BEJ169
Mathematical Reviews number (MathSciNet): MR2555202
Zentralblatt MATH identifier: 05815958
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