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Consistent Utility of Investment and Consumption : a forward/backward SPDE viewpoint

Author

Listed:
  • Nicole El Karoui

    (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

  • Caroline Hillairet

    (ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - GENES - Groupe des Écoles Nationales d'Économie et Statistique - IP Paris - Institut Polytechnique de Paris)

  • Mohamed Mrad

    (LAGA - Laboratoire Analyse, Géométrie et Applications - UP8 - Université Paris 8 Vincennes-Saint-Denis - UP13 - Université Paris 13 - Institut Galilée - CNRS - Centre National de la Recherche Scientifique)

Abstract

This paper provides an extension of the notion of consistent progressive utilities U to consistent progressive utilities of investment and consumption (U, V). It discusses the notion of market consistency in this forward framework, compared to the classic backward setting with a given terminal utility, and whose value function is an example of such consistent forward utility. To ensure the consistency with the market model or a given set of test processes, we establish a stochastic partial differential equation (SPDE) of Hamilton-Jacobi-Bellman (HJB)-type that U has to satisfy. This SPDE highlights the link between the utility of wealth U and the utility of consumption V, and between the drift and the volatility characteristics of the utility U. By associating with the HJB-SPDE two SDEs, we discuss the existence and the uniqueness of a concave solution. Finally, we provide explicit regularity conditions and characterize the consistent pairs of consistent utilities of investment and consumption. Some examples, such as power utilities, illustrate the theory.

Suggested Citation

  • Nicole El Karoui & Caroline Hillairet & Mohamed Mrad, 2018. "Consistent Utility of Investment and Consumption : a forward/backward SPDE viewpoint ," Post-Print hal-01458419, HAL.
    • Handle: RePEc:hal:journl:hal-01458419
    Note: View the original document on HAL open archive server: https://hal.science/hal-01458419v1
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    References listed on IDEAS

    as
    1. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    2. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "An Intertemporal General Equilibrium Model of Asset Prices," Econometrica, Econometric Society, vol. 53(2), pages 363-384, March.
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    Cited by:

    1. Wing Fung Chong & Gechun Liang, 2024. "Robust forward investment and consumption under drift and volatility uncertainties: A randomization approach," Papers 2410.01378, arXiv.org.
    2. Ng, Kenneth Tsz Hin & Chong, Wing Fung, 2024. "Optimal investment in defined contribution pension schemes with forward utility preferences," Insurance: Mathematics and Economics, Elsevier, vol. 114(C), pages 192-211.
    3. Nicole El Karoui & Mohamed Mrad & Caroline Hillairet, 2020. "Ramsey Rule with Progressive Utility in Long Term Yield Curves Modeling," Post-Print hal-00974815, HAL.
    4. Mohamed Mrad & El Karoui & Caroline Hillairet, 2020. "Ramsey Rule with Progressive Utility in Long Term Yield Curves Modeling," Post-Print hal-04553875, HAL.
    5. Gechun Liang & Moris S. Strub & Yuwei Wang, 2023. "Predictable forward performance processes: Infrequent evaluation and applications to human‐machine interactions," Mathematical Finance, Wiley Blackwell, vol. 33(4), pages 1248-1286, October.
    6. 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.
    7. 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.

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