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Predictable Forward Performance Processes in Complete Markets

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  • Bahman Angoshtari

Abstract

We establish existence of Predictable Forward Performance Processes (PFPPs) in complete markets, which has been previously shown only in the binomial setting. Our market model can be a discrete-time or a continuous-time model, and the investment horizon can be finite or infinite. We show that the main step in construction of PFPPs is solving a one-period problem involving an integral equation, which is the counterpart of the functional equation found in the binomial case. Although this integral equation has been partially studied in the existing literature, we provide a new solution method using the Fourier transform for tempered distributions. We also provide closed-form solutions for PFPPs with inverse marginal functions that are completely monotonic and establish uniqueness of PFPPs within this class. We apply our results to two special cases. The first one is the binomial market and is included to relate our work to the existing literature. The second example considers a generalized Black-Scholes model which, to the best of our knowledge, is a new result.

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  • Bahman Angoshtari, 2022. "Predictable Forward Performance Processes in Complete Markets," Papers 2206.03608, arXiv.org, revised Sep 2022.
    • Handle: RePEc:arx:papers:2206.03608
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    References listed on IDEAS

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    1. Xue Dong He & Moris S. Strub & Thaleia Zariphopoulou, 2021. "Forward rank‐dependent performance criteria: Time‐consistent investment under probability distortion," Mathematical Finance, Wiley Blackwell, vol. 31(2), pages 683-721, April.
    2. M. Musiela & T. Zariphopoulou, 2011. "Initial Investment Choice And Optimal Future Allocations Under Time-Monotone Performance Criteria," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(01), pages 61-81.
    3. Gechun Liang & Moris S. Strub & Yuwei Wang, 2021. "Predictable Forward Performance Processes: Infrequent Evaluation and Applications to Human-Machine Interactions," Papers 2110.08900, arXiv.org, revised Dec 2023.
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