We generalize the definition of the fractional Brownian motion of exponent $H$ to the case where $H$ is no longer a constant, but a function of the time index of the process. This allows us to model non stationary continuous processes, and we show that $H(t)$ and $2-H(t)$ are indeed respectively the local Hölder exponent and the local box and Hausdorff dimension at point $t$. Finally, we propose a simulation method and an estimation procedure for $H(t)$ for our model.