  We introduce the concept of forward rank-dependent performance processes,
extending the original notion to forward criteria that incorporate probability
distortions. A fundamental challenge is how to reconcile the time-consistent
nature of forward performance criteria with the time-inconsistency stemming
from probability distortions. For this, we first propose two distinct
definitions, one based on the preservation of performance value and the other
on the time-consistency of policies and, in turn, establish their equivalence.
We then fully characterize the viable class of probability distortion
processes, providing a bifurcation-type result. Specifically, it is either the
case that the probability distortions are degenerate in the sense that the
investor would never invest in the risky assets, or the marginal probability
distortion equals to a normalized power of the quantile function of the pricing
kernel. We also characterize the optimal wealth process, whose structure
motivates the introduction of a new, distorted measure and a related market. We
then build a striking correspondence between the forward rank-dependent
criteria in the original market and forward criteria without probability
distortions in the auxiliary market. This connection also provides a direct
construction method for forward rank-dependent criteria. A byproduct of our
work are some new results on the so-called dynamic utilities and on
time-inconsistent problems in the classical (backward) setting.
