Rank-Dependent Predictable Forward Performance Processes

ArXiv ID: 2403.16228 “View on arXiv”

Authors: Unknown

Abstract

Predictable forward performance processes (PFPPs) are stochastic optimal control frameworks for an agent who controls a randomly evolving system but can only prescribe the system dynamics for a short period ahead. This is a common scenario in which a controlling agent frequently re-calibrates her model. We introduce a new class of PFPPs based on rank-dependent utility, generalizing existing models that are based on expected utility theory (EUT). We establish existence of rank-dependent PFPPs under a conditionally complete market and exogenous probability distortion functions which are updated periodically. We show that their construction reduces to solving an integral equation that generalizes the integral equation obtained under EUT in previous studies. We then propose a new approach for solving the integral equation via theory of Volterra equations. We illustrate our result in the special case of conditionally complete Black-Scholes model.

Keywords: Predictable Forward Performance Processes, Rank-Dependent Utility, Stochastic Optimal Control, Volterra Equations, Probability Distortion, Derivatives/Options

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, involving stochastic optimal control, rank-dependent utility, and solving Volterra integral equations, indicating advanced mathematics. However, it presents a theoretical framework with no backtesting, datasets, or implementation details, focusing on existence proofs and analytical constructions rather than empirical validation.

  flowchart TD
    A["Research Goal:<br>Extend Predictable Forward Performance Processes<br>from Expected Utility to Rank-Dependent Utility"] --> B["Modeling Framework"]
    
    subgraph B ["Methodology"]
        B1["Define Rank-Dependent<br>Utility Function"]
        B2["Assume Conditionally<br>Complete Market"]
        B3["Apply Exogenous<br>Probability Distortions"]
    end

    B --> C["Derive Integral Equation"]
    
    subgraph C ["Computational Process"]
        C1["Generalize EUT Equation"]
        C2["Apply Volterra<br>Equation Theory"]
    end

    C --> D["Case Study:<br>Conditionally Complete Black-Scholes Model"]
    D --> E["Key Outcomes"]
    
    subgraph E ["Findings"]
        E1["Existence of<br>Rank-Dependent PFPPs"]
        E2["Solvability via<br>Volterra Framework"]
        E3["Broadened Application<br>beyond EUT"]
    end