I will introduce the concept of forward rank-dependent performance processes, extending the original notion to forward criteria that incorporate probability distortions and, at the same time, accommodate “real-time” incoming market information. 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, I will 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. I will then fully characterize the viable class of probability distortion processes, providing a bifurcation-type result. This will also characterize the candidate optimal wealth process, whose structure motivates the introduction of a new, distorted measure and a related dynamic market. I will, 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 provides a direct construction method for forward rank-dependent criteria with dynamic incoming information. Furthermore, a direct by-product of our work are new results on the so-called dynamic utilities and time-inconsistent problems in the classical (backward) setting. Indeed, it turns out that open questions in the latter setting can be directly addressed by framing the classical problem as a forward one under suitable information rescaling.
