IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2008.00963.html
   My bibliography  Save this paper

Existence and uniqueness of recursive utilities without boundedness

Author

Listed:
  • Timothy M. Christensen

Abstract

This paper derives primitive, easily verifiable sufficient conditions for existence and uniqueness of (stochastic) recursive utilities for several important classes of preferences. In order to accommodate models commonly used in practice, we allow both the state-space and per-period utilities to be unbounded. For many of the models we study, existence and uniqueness is established under a single, primitive "thin tail" condition on the distribution of growth in per-period utilities. We present several applications to robust preferences, models of ambiguity aversion and learning about hidden states, and Epstein-Zin preferences.

Suggested Citation

  • Timothy M. Christensen, 2020. "Existence and uniqueness of recursive utilities without boundedness," Papers 2008.00963, arXiv.org, revised Aug 2021.
  • Handle: RePEc:arx:papers:2008.00963
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2008.00963
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Andriy Norets, 2010. "Continuity and differentiability of expected value functions in dynamic discrete choice models," Quantitative Economics, Econometric Society, vol. 1(2), pages 305-322, November.
    2. Rust, John, 1987. "Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher," Econometrica, Econometric Society, vol. 55(5), pages 999-1033, September.
    3. Hansen, Lars Peter & Sargent, Thomas J., 2007. "Recursive robust estimation and control without commitment," Journal of Economic Theory, Elsevier, vol. 136(1), pages 1-27, September.
    4. Bidder, R.M. & Smith, M.E., 2018. "Doubts and variability: A robust perspective on exotic consumption series," Journal of Economic Theory, Elsevier, vol. 175(C), pages 689-712.
    5. Fabrice Collard & Sujoy Mukerji & Kevin Sheppard & Jean‐Marc Tallon, 2018. "Ambiguity and the historical equity premium," Quantitative Economics, Econometric Society, vol. 9(2), pages 945-993, July.
    6. Lars Peter Hansen & Thomas J Sargent, 2014. "Robust Control and Model Uncertainty," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 5, pages 145-154, World Scientific Publishing Co. Pte. Ltd..
    7. Andrew Ang & Allan Timmermann, 2012. "Regime Changes and Financial Markets," Annual Review of Financial Economics, Annual Reviews, vol. 4(1), pages 313-337, October.
    8. Frank Schorfheide & Dongho Song & Amir Yaron, 2018. "Identifying Long‐Run Risks: A Bayesian Mixed‐Frequency Approach," Econometrica, Econometric Society, vol. 86(2), pages 617-654, March.
    9. Lars Peter Hansen & José A. Scheinkman, 2009. "Long-Term Risk: An Operator Approach," Econometrica, Econometric Society, vol. 77(1), pages 177-234, January.
    10. Klibanoff, Peter & Marinacci, Massimo & Mukerji, Sujoy, 2009. "Recursive smooth ambiguity preferences," Journal of Economic Theory, Elsevier, vol. 144(3), pages 930-976, May.
    11. Larry G. Epstein & Stanley E. Zin, 2013. "Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 12, pages 207-239, World Scientific Publishing Co. Pte. Ltd..
    12. Nengjiu Ju & Jianjun Miao, 2012. "Ambiguity, Learning, and Asset Returns," Econometrica, Econometric Society, vol. 80(2), pages 559-591, March.
    13. Lars Peter Hansen & Thomas J Sargent, 2014. "Robust Control and Model Misspecification," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 6, pages 155-216, World Scientific Publishing Co. Pte. Ltd..
    14. Lars Peter Hansen & Thomas J Sargent, 2014. "Doubts or Variability?," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 7, pages 217-256, World Scientific Publishing Co. Pte. Ltd..
    15. Jaroslav Borovička & John Stachurski, 2020. "Necessary and Sufficient Conditions for Existence and Uniqueness of Recursive Utilities," Journal of Finance, American Finance Association, vol. 75(3), pages 1457-1493, June.
    16. Jason R. Blevins, 2014. "Nonparametric identification of dynamic decision processes with discrete and continuous choices," Quantitative Economics, Econometric Society, vol. 5(3), pages 531-554, November.
    17. Cecchetti, Stephen G & Lam, Pok-sang & Mark, Nelson C, 1990. "Mean Reversion in Equilibrium Asset Prices," American Economic Review, American Economic Association, vol. 80(3), pages 398-418, June.
    18. Jessica A. Wachter, 2013. "Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility?," Journal of Finance, American Finance Association, vol. 68(3), pages 987-1035, June.
    19. Mariano M. Croce & Martin Lettau & Sydney C. Ludvigson, 2015. "Investor Information, Long-Run Risk, and the Term Structure of Equity," Review of Financial Studies, Society for Financial Studies, vol. 28(3), pages 706-742.
    20. Ravi Bansal & Amir Yaron, 2004. "Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles," Journal of Finance, American Finance Association, vol. 59(4), pages 1481-1509, August.
    21. Lars Peter Hansen & Thomas J Sargent, 2014. "Fragile Beliefs and the Price of Uncertainty," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 9, pages 293-330, World Scientific Publishing Co. Pte. Ltd..
    22. Pok-sang Lam & Stephen G. Cecchetti & Nelson C. Mark, 2000. "Asset Pricing with Distorted Beliefs: Are Equity Returns Too Good to Be True?," American Economic Review, American Economic Association, vol. 90(4), pages 787-805, September.
    23. Timothy Christensen, 2014. "Nonparametric Stochastic Discount Factor Decomposition," Papers 1412.4428, arXiv.org, revised May 2017.
    24. Lars Peter Hansen & John C. Heaton & Nan Li, 2008. "Consumption Strikes Back? Measuring Long-Run Risk," Journal of Political Economy, University of Chicago Press, vol. 116(2), pages 260-302, April.
    25. Timothy M. Christensen, 2017. "Nonparametric Stochastic Discount Factor Decomposition," Econometrica, Econometric Society, vol. 85, pages 1501-1536, September.
    26. J. Rust & J. F. Traub & H. Wozniakowski, 2002. "Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case?," Econometrica, Econometric Society, vol. 70(1), pages 285-329, January.
    27. Marinacci, Massimo & Montrucchio, Luigi, 2010. "Unique solutions for stochastic recursive utilities," Journal of Economic Theory, Elsevier, vol. 145(5), pages 1776-1804, September.
    28. Fabrice Collard & Sujoy Mukerji & Kevin Sheppard & Jean‐Marc Tallon, 2018. "Ambiguity and the historical equity premium," Quantitative Economics, Econometric Society, vol. 9(2), pages 945-993, July.
    29. Pierre Collin-Dufresne & Michael Johannes & Lars A. Lochstoer, 2016. "Parameter Learning in General Equilibrium: The Asset Pricing Implications," American Economic Review, American Economic Association, vol. 106(3), pages 664-698, March.
    30. Ravi Bansal & Dana Kiku & Ivan Shaliastovich & Amir Yaron, 2014. "Volatility, the Macroeconomy, and Asset Prices," Journal of Finance, American Finance Association, vol. 69(6), pages 2471-2511, December.
    31. Bloise, Gaetano & Vailakis, Yiannis, 2018. "Convex dynamic programming with (bounded) recursive utility," Journal of Economic Theory, Elsevier, vol. 173(C), pages 118-141.
    32. Lars Peter Hansen & Jose A. Scheinkman, 2012. "Recursive utility in a Markov environment with stochastic growth," Working Papers 1380, Princeton University, Department of Economics, Econometric Research Program..
    33. Peter Arcidiacono & Robert A. Miller, 2011. "Conditional Choice Probability Estimation of Dynamic Discrete Choice Models With Unobserved Heterogeneity," Econometrica, Econometric Society, vol. 79(6), pages 1823-1867, November.
    34. Fernando Alvarez & Urban J. Jermann, 2005. "Using Asset Prices to Measure the Persistence of the Marginal Utility of Wealth," Econometrica, Econometric Society, vol. 73(6), pages 1977-2016, November.
    35. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-384, March.
    36. Christensen, Timothy M., 2015. "Nonparametric Identification Of Positive Eigenfunctions," Econometric Theory, Cambridge University Press, vol. 31(6), pages 1310-1330, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Borovička, Jaroslav & Stachurski, John, 2021. "Stability of equilibrium asset pricing models: A necessary and sufficient condition," Journal of Economic Theory, Elsevier, vol. 193(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Christensen, Timothy M., 2022. "Existence and uniqueness of recursive utilities without boundedness," Journal of Economic Theory, Elsevier, vol. 200(C).
    2. Nengjiu Ju & Jianjun Miao, 2012. "Ambiguity, Learning, and Asset Returns," Econometrica, Econometric Society, vol. 80(2), pages 559-591, March.
    3. Jaroslav Borovička & John Stachurski, 2020. "Necessary and Sufficient Conditions for Existence and Uniqueness of Recursive Utilities," Journal of Finance, American Finance Association, vol. 75(3), pages 1457-1493, June.
    4. Jianjun Miao & Bin Wei & Hao Zhou, 2019. "Ambiguity Aversion and the Variance Premium," Quarterly Journal of Finance (QJF), World Scientific Publishing Co. Pte. Ltd., vol. 9(02), pages 1-36, June.
    5. Hansen, Lars Peter & Sargent, Thomas J., 2011. "Robustness and ambiguity in continuous time," Journal of Economic Theory, Elsevier, vol. 146(3), pages 1195-1223, May.
    6. Bidder, R.M. & Smith, M.E., 2018. "Doubts and variability: A robust perspective on exotic consumption series," Journal of Economic Theory, Elsevier, vol. 175(C), pages 689-712.
    7. Zhang, Jian & Kong, Dongmin & Liu, Hening & Wu, Ji, 2019. "Asset pricing with time varying pessimism and rare disasters," International Review of Economics & Finance, Elsevier, vol. 60(C), pages 165-175.
    8. Sumru Altug & Cem Cakmakli & Fabrice Collard & Sujoy Mukerji & Han Ozsoylev, 2020. "Ambiguous Business Cycles: A Quantitative Assessment," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 38, pages 220-237, October.
    9. Backus, David & Ferriere, Axelle & Zin, Stanley, 2015. "Risk and ambiguity in models of business cycles," Journal of Monetary Economics, Elsevier, vol. 69(C), pages 42-63.
    10. Borovička, Jaroslav & Stachurski, John, 2021. "Stability of equilibrium asset pricing models: A necessary and sufficient condition," Journal of Economic Theory, Elsevier, vol. 193(C).
    11. Guanlong Ren & John Stachurski, 2018. "Dynamic Programming with Recursive Preferences: Optimality and Applications," Papers 1812.05748, arXiv.org, revised Jun 2020.
    12. Max Gillman & Michal Kejak & Michal Pakoš, 2015. "Learning about Rare Disasters: Implications For Consumption and Asset Prices," Review of Finance, European Finance Association, vol. 19(3), pages 1053-1104.
    13. Szőke, Bálint, 2022. "Estimating robustness," Journal of Economic Theory, Elsevier, vol. 199(C).
    14. Michael Johannes & Lars Lochstoer & Pierre Collin-Dufresne, 2015. "Parameter Learning in General Equilibrium: The Asset Pricing Implications," 2015 Meeting Papers 647, Society for Economic Dynamics.
    15. Sumru Altug & Cem Cakmakli & Fabrice Collard & Sujoy Mukerji & Han Ozsoylev, 2020. "Ambiguous Business Cycles: A Quantitative Assessment," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 38, pages 220-237, October.
    16. A. Ronald Gallant & Mohammad Jahan-Parvar & Hening Liu, 2015. "Measuring Ambiguity Aversion," Finance and Economics Discussion Series 2015-105, Board of Governors of the Federal Reserve System (U.S.).
    17. Daniele Pennesi, 2013. "Asset Prices in an Ambiguous Economy," Carlo Alberto Notebooks 315, Collegio Carlo Alberto.
    18. Pierre Collin-Dufresne & Michael Johannes & Lars A. Lochstoer, 2013. "Parameter Learning in General Equilibrium: The Asset Pricing Implications," NBER Working Papers 19705, National Bureau of Economic Research, Inc.
    19. Hansen, Lars Peter, 2013. "Uncertainty Outside and Inside Economic Models," Nobel Prize in Economics documents 2013-7, Nobel Prize Committee.
    20. Borovička, Jaroslav & Hansen, Lars Peter, 2014. "Examining macroeconomic models through the lens of asset pricing," Journal of Econometrics, Elsevier, vol. 183(1), pages 67-90.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2008.00963. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: http://arxiv.org/ .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.