  The nested distance builds on the Wasserstein distance to quantify the
difference of stochastic processes, including also the information modelled by
filtrations. The Sinkhorn divergence is a relaxation of the Wasserstein
distance, which can be computed considerably faster. For this reason we employ
the Sinkhorn divergence and take advantage of the related (fixed point)
iteration algorithm. Furthermore, we investigate the transition of the entropy
throughout the stages of the stochastic process and provide an
entropy-regularized nested distance formulation, including a characterization
of its dual. Numerical experiments affirm the computational advantage and
supremacy.
