## Bernoulli

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

### Stochastic differential equations driven by fractional Brownian motions

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 28 August 2009

https://projecteuclid.org/euclid.bj/1251463284

Digital Object Identifier
doi:10.3150/08-BEJ169

Mathematical Reviews number (MathSciNet)
MR2555202

Zentralblatt MATH identifier
1214.60024

#### Citation

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. https://projecteuclid.org/euclid.bj/1251463284

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