Forward rank‐dependent performance criteria: Time‐consistent investment under probability distortion

Xue Dong He, Moris S. Strub and Thaleia Zariphopoulou

Mathematical Finance, 2021, vol. 31, issue 2, 683-721

Abstract: We introduce the concept of forward rank‐dependent performance criteria, 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 and provide the following dichotomy: 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. Finally, our results on forward rank‐dependent performance criteria motivate us to revisit the classical (backward) setting. We follow the so‐called dynamic utility approach and derive conditions for existence and a construction of dynamic rank‐dependent utility processes.

Date: 2021
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