## Electronic Journal of Probability

### Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions

#### Abstract

We study the Taylor expansion for the solution of a differential equation driven by a multi-dimensional Hölder path with exponent  $H> 1/2$. We derive a convergence criterion that enables us to write the solution as an infinite sum of iterated integrals on a non empty interval. We apply our deterministic results to stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H > 1/2$. We also study the convergence in L2 of the stochastic Taylor expansion by using L2 estimates of iterated integrals and Borel-Cantelli type arguments.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 51, 21 pp.

Dates
Accepted: 6 July 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062373

Digital Object Identifier
doi:10.1214/EJP.v17-2136

Mathematical Reviews number (MathSciNet)
MR2955043

Zentralblatt MATH identifier
1252.60052

Rights

#### Citation

Baudoin, Fabrice; Zhang, Xuejing. Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions. Electron. J. Probab. 17 (2012), paper no. 51, 21 pp. doi:10.1214/EJP.v17-2136. https://projecteuclid.org/euclid.ejp/1465062373

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