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Predictable forward performance processes in complete markets

I would like to thank Thaleia Zariphopoulou and Xunyu Zhou for motivating my interest in this topic and for their invaluable feedback on my work. I am grateful for helpful comments and suggestions from Samuel Cohen, Sigrid Källblad, Gechun Liang, Moris Strub, and the referees. I acknowledge support through a start-up grant at the University of Miami. Part of this research was performed while I was visiting the Institute for Mathematical and Statistical Innovation (IMSI), which is supported by the National Science Foundation (Grant No. DMS-1929348).

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  • We establish existence of Predictable Forward Performance Processes (PFPPs) in conditionally 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.

    Mathematics Subject Classification: 91G10, 91G80, 60H30.

    Citation:

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  • [1]

    Angoshtari, B., Zariphopoulou, T. and Zhou, X. Y., Predictable forward performance processes: The binomial case, SIAM Journal on Control and Optimization, 2020, 58(1): 327−347.

    doi: 10.1137/18M1188653.

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    He, X. D., Strub, M. S. and Zariphopoulou, T., Forward rank-dependent performance criteria: Time-consistent investment under probability distortion, Mathematical Finance, 2021, 31(2): 683−721.

    doi: 10.1111/mafi.12298.

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    Källblad, S., Black’s inverse investment problem and forward criteria with consumption, SIAM Journal on Financial Mathematics, 2020, 11(2): 494−525.

    doi: 10.1137/17M1143812.

    [5] Liang, G., Strub, M. S. and Wang, Y., Predictable forward performance processes: Infrequent evaluation and robo-advising applications, arXiv: 2110.08900, 2021.
    [6] Mostovyi, O., Sîrbu, M. and Zariphopoulou, T., On the analyticity of the value function in optimal investment and stochastically dominant markets, arXiv: 2002.01084, 2020.
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    Musiela, M. and Zariphopoulou, T., Portfolio choice under dynamic investment performance criteria, Quantitative Finance, 2009, 9(2): 161−170.

    doi: 10.1080/14697680802624997.

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    Musiela, M. and Zariphopoulou, T., Portfolio choice under space-time monotone performance criteria, SIAM Journal on Financial Mathematics, 2010, 1(1): 326−365.

    doi: 10.1137/080745250.

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    Musiela, M. and Zariphopoulou, T., Initial investment choice and optimal future allocations under time-monotone performance criteria, International Journal of Theoretical and Applied Finance, 2011, 14(1): 61−81.

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    Strub, M. S. and Zhou, X. Y., Evolution of the Arrow–Pratt measure of risk-tolerance for predictable forward utility processes, Finance and Stochastics, 2021, 25(2): 331−358.

    doi: 10.1007/s00780-020-00444-1.

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