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

Author

Listed:
  • Nicole El Karoui

    (LPMA; UMR CNRS 6632; Université Pierre et Marie Curie)

  • Caroline Hillairet

    (LAGA; UMR CNRS 7539; Université Paris 13)

  • Mohamed Mrad

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. ;Classification-JEL: 60H15, 91B16, 91B70, 91G10

Suggested Citation

  • Nicole El Karoui & Caroline Hillairet & Mohamed Mrad, 2017. "Consistent Utility of Investment and Consumption : a forward/backward SPDE viewpoint," Working Papers 2017-74, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2017-74
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    References listed on IDEAS

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    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. Nicole El Karoui & Caroline Hillairet & Mohamed Mrad, 2019. "Construction of an aggregate consistent utility, without Pareto optimality. Application to Long-Term yield curve Modeling," Post-Print hal-01721441, HAL.
    2. Mrad Mohamed, 2020. "Mixture of consistent stochastic utilities, and a priori randomness," Working Papers hal-01728554, HAL.
    3. Nicole El Karoui & Caroline Hillairet & Mohamed Mrad, 2018. "Construction of an aggregate consistent utility, without Pareto optimality. Application to Long-Term yield curve Modeling," Working Papers hal-01721441, HAL.

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    Keywords

    Market-consistent progressive utility of investment and consumption; Forward/backward stochastic partial differential equation (SPDE);

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