Before revision:  It is well known that displaced diffusions  exhibit dynamics which are in-between arithmetic Brownian motions (ABM) or geometric Brownian motions (GBM) depending of the choice of the shift parameter.
After revision:  It is well known that displaced diffusions exhibit dynamics which are in-between arithmetic Brownian motions (ABM) and geometric Brownian motions (GBM) depending of the choice of the shift parameter.