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Rank-Dependent Predictable Forward Performance Processes

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  • Bahman Angoshtari
  • Shida Duan

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.

Suggested Citation

  • Bahman Angoshtari & Shida Duan, 2024. "Rank-Dependent Predictable Forward Performance Processes," Papers 2403.16228, arXiv.org.
    • Handle: RePEc:arx:papers:2403.16228
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    References listed on IDEAS

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    1. Gechun Liang & Yifan Sun & Thaleia Zariphopoulou, 2023. "Representation of forward performance criteria with random endowment via FBSDE and application to forward optimized certainty equivalent," Papers 2401.00103, arXiv.org.
    2. Jianming Xia & Xun Yu Zhou, 2016. "Arrow–Debreu Equilibria For Rank-Dependent Utilities," Mathematical Finance, Wiley Blackwell, vol. 26(3), pages 558-588, July.
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    5. Hanqing Jin & Jianming Xia & Xun Yu Zhou, 2019. "Arrow–Debreu equilibria for rank‐dependent utilities with heterogeneous probability weighting," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 898-927, July.
    6. Gechun Liang & Moris S. Strub & Yuwei Wang, 2023. "Predictable Relative Forward Performance Processes: Multi-Agent and Mean Field Games for Portfolio Management," Papers 2311.04841, arXiv.org, revised Dec 2023.
    7. Zuo Quan Xu, 2013. "A New Characterization of Comonotonicity and its Application in Behavioral Finance," Papers 1311.6080, arXiv.org, revised Jun 2014.
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