In this paper we study a stochastic differential equation driven by a fractional Brownian motion with a discontinuous coefficient. We also give an approximation to the solution of the equation. This is a first step to define a fractional version of the skew Brownian motion. (2000): Primary 60H15; secondary 65C30. Mathematics Subject Classifications

We construct a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals for any Hurst parameter H , and we derive a rate of convergence, which becomes better when H approaches 1/2. The construction is based on the Mandelbrot-van Ness stochastic integral representation of fractional Brownian motion and on a strong transport process approximation of Brownian motion. The objective of this method is to facilitate simulation.

Abstract. We present strong approximations with rate of convergence for the solution of a stochastic differential equation of the formis fractional Brownian motion with Hurst index H, and we assume existence of a unique solution with Doss-Sussmann representation. The results are based on a strong approximation of B H by means of transport processes of Garzón et al (2009 [11]). If σ is bounded away from 0, an approximation is obtained by a general Lipschitz dependence result of Römisch and Wakolbinger (1985 [25]). Without that assumption on σ, that method does not work, and we proceed by means of Euler schemes on the Doss-Sussmann representation to obtain another approximation, whose proof is the bulk of the paper.

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We give a strong approximation of Rosenblatt process via transport processes and we give the rate of convergence.2010 AMS Classification Numbers: 60B10, 60F05, 60H05.