We discuss a stochastic differential equation, as a modelling framework for the turbulent velocity field, that is capable of capturing basic stylized facts of the statistics of velocity increments. In particular, we focus on the evolution of the probability density of velocity increments characterized by a normal inverse Gaussian shape with heavy tails for small scales and aggregational Gaussianity for large scales. In addition, we show that the proposed model is in accordance with Kolmogorov's refined similarity hypotheses.