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Construction of a class of forward performance processes in stochastic factor models, and an extension of Widder’s theorem

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  • Levon Avanesyan

    (Princeton University)

  • Mykhaylo Shkolnikov

    (Princeton University)

  • Ronnie Sircar

    (Princeton University)

Abstract

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. Given multiple traded assets, the prices of which depend on multiple observable stochastic factors, we construct a large class of forward performance processes, as well as the corresponding optimal portfolios, with power-utility initial data and for stock–factor correlation matrices with eigenvalue equality (EVE) structure, which we introduce here. This is done by solving the associated nonlinear parabolic partial differential equations (PDEs) posed in the “wrong” time direction. Along the way, we establish on domains an explicit form of the generalised Widder theorem of Nadtochiy and Tehranchi (Math. Finance 27:438–470, 2015, Theorem 3.12) and rely for that on the Laplace inversion in time of the solutions to suitable linear parabolic PDEs posed in the “right” time direction.

Suggested Citation

  • Levon Avanesyan & Mykhaylo Shkolnikov & Ronnie Sircar, 2020. "Construction of a class of forward performance processes in stochastic factor models, and an extension of Widder’s theorem," Finance and Stochastics, Springer, vol. 24(4), pages 981-1011, October.
    • Handle: RePEc:spr:finsto:v:24:y:2020:i:4:d:10.1007_s00780-020-00436-1
    DOI: 10.1007/s00780-020-00436-1
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    References listed on IDEAS

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    1. Thaleia Zariphopoulou, 2001. "A solution approach to valuation with unhedgeable risks," Finance and Stochastics, Springer, vol. 5(1), pages 61-82.
    2. Sigrid Källblad & Jan Obłój & Thaleia Zariphopoulou, 2018. "Dynamically consistent investment under model uncertainty: the robust forward criteria," Finance and Stochastics, Springer, vol. 22(4), pages 879-918, October.
    3. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    4. Gechun Liang & Thaleia Zariphopoulou, 2015. "Representation of homothetic forward performance processes in stochastic factor models via ergodic and infinite horizon BSDE," Papers 1511.04863, arXiv.org, revised Nov 2016.
    5. N. El Karoui & Mohamed M'Rad, 2010. "Stochastic Utilities With a Given Optimal Portfolio : Approach by Stochastic Flows," Papers 1004.5192, arXiv.org, revised Apr 2013.
    6. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    7. Henderson, Vicky & Hobson, David, 2007. "Horizon-unbiased utility functions," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1621-1641, November.
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    Citations

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    Cited by:

    1. Wenyuan Wang & Kaixin Yan & Xiang Yu, 2024. "Optimal portfolio under ratio-type periodic evaluation in incomplete markets with stochastic factors," Papers 2401.14672, arXiv.org.
    2. Maxim Bichuch & Jean‐Pierre Fouque, 2023. "Optimal investment with correlated stochastic volatility factors," Mathematical Finance, Wiley Blackwell, vol. 33(2), pages 342-369, April.
    3. 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.
    4. Jean-Pierre Fouque & Ruimeng Hu & Ronnie Sircar, 2021. "Sub- and Super-solution Approach to Accuracy Analysis of Portfolio Optimization Asymptotics in Multiscale Stochastic Factor Market," Papers 2106.11510, arXiv.org, revised Oct 2021.
    5. Levon Avanesyan & Ronnie Sircar, 2020. "Power mixture forward performance processes," Papers 2012.10847, arXiv.org.
    6. Goncalo dos Reis & Vadim Platonov, 2020. "Forward utility and market adjustments in relative investment-consumption games of many players," Papers 2012.01235, arXiv.org, revised Mar 2022.

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    More about this item

    Keywords

    Factor models; Forward performance processes; Generalised Widder theorem; Hamilton–Jacobi–Bellman equations; Ill-posed partial differential equations; Incomplete markets; Merton problem; Optimal portfolio selection; Positive eigenfunctions; Time-consistency;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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