Abstract
This dissertation advances the theory of Predictable Forward Performance Processes (PFPPs) by developing new models inspired by behavioral finance, focusing on rank-dependent and loss-averse preferences.
Chapter 1 introduces PFPPs in continuous and discrete time, reviews the limitations of expected utility in dynamic settings, and provides background on behavioral models—particularly Cumulative Prospect Theory and its three components: probability distortion, loss aversion, and reference dependence.
Chapter 2 develops Rank-Dependent PFPPs (RDPFPPs) by incorporating probability distortions into the PFPP framework. In conditionally complete markets, their existence reduces to solving a sequence of integral equations. Using Volterra techniques, we construct explicit solutions, including closed-form expressions when inverse marginal functions are completely monotonic. A conditionally complete Black-Scholes market model illustrates the approach.
Chapter 3 addresses PFPPs for loss-averse agents. Instead of S-shaped utilities, we apply the concavification principle to use concave envelopes that preserve loss aversion while avoiding non-concavities. The resulting forward problem leads to a free-boundary Fredholm integral equation of the first kind. We study its ill-posed structure via a characteristic system and Tikhonov regularization, using resolvent analysis and finite-rank kernel approximations to ensure existence and uniqueness. The resulting methods offer constructive, tractable solutions for PFPPs with behavioral features.
Overall, this work bridges forward investment theory with behavioral decision making models, contributing to the broader effort of incorporating empirically relevant preferences into stochastic control frameworks.