A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one

We will focus - in dimension one - on the SDEs of the type dX(t) = sigma(X-t)dB(t) + b(X-t) dt where B is a fractional Brownian motion. Our principal aim is to describe a simple theory - from our point of view - allowing to study this SDE, and this for any H is an element of (0, 1). We will consider several definitions of solutions and, for each of them, study conditions under which one has existence and/or uniqueness. Finally, we will examine whether or not the canonical scheme associated to our SDE converges, when the integral with respect to fBm is defined using the Russo-Vallois symmetric integral.