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Existence and Uniqueness of a Fixed Point for the Bellman Operator in Deterministic Dynamic Programming

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  • Takashi Kamihigashi

    (Research Institute for Economics & Business Administration (RIEB), Kobe University, Japan)

Abstract

We study existence and uniqueness of a fixed point for the Bellman operator in deterministic dynamic programming. Without any topological assumption, we show that the Bellman operator has a unique fixed point in a restricted domain, that this fixed point is the value function, and that the value function can be computed by value iteration.

Suggested Citation

  • Takashi Kamihigashi, 2012. "Existence and Uniqueness of a Fixed Point for the Bellman Operator in Deterministic Dynamic Programming," Discussion Paper Series DP2012-05, Research Institute for Economics & Business Administration, Kobe University.
    • Handle: RePEc:kob:dpaper:dp2012-05
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    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/DP2012-05.pdf
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    References listed on IDEAS

    as
    1. Le Van, Cuong & Morhaim, Lisa, 2002. "Optimal Growth Models with Bounded or Unbounded Returns: A Unifying Approach," Journal of Economic Theory, Elsevier, vol. 105(1), pages 158-187, July.
    2. Takashi Kamihigashi, 2008. "On the principle of optimality for nonstationary deterministic dynamic programming," International Journal of Economic Theory, The International Society for Economic Theory, vol. 4(4), pages 519-525, December.
    3. V. Filipe Martins-da-Rocha & Yiannis Vailakis, 2010. "Existence and Uniqueness of a Fixed Point for Local Contractions," Econometrica, Econometric Society, vol. 78(3), pages 1127-1141, May.
    4. Takashi Kamihigashi, 2011. "Existence and Uniqueness of a Fixed Point for the Bellman Operator in Deterministic Dynamic Programming," Discussion Paper Series DP2011-23, Research Institute for Economics & Business Administration, Kobe University.
    5. Michel, Philippe, 1990. "Some Clarifications on the Transversality Condition," Econometrica, Econometric Society, vol. 58(3), pages 705-723, May.
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    7. Juan Rincón-Zapatero & Carlos Rodríguez-Palmero, 2007. "Recursive utility with unbounded aggregators," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(2), pages 381-391, November.
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    Cited by:

    1. Takashi Kamihigashi, 2011. "Existence and Uniqueness of a Fixed Point for the Bellman Operator in Deterministic Dynamic Programming," Discussion Paper Series DP2011-23, Research Institute for Economics & Business Administration, Kobe University.
    2. Takashi Kamihigashi & Cuong Le Van, 2015. "Necessary and Sufficient Conditions for a Solution of the Bellman Equation to be the Value Function: A General Principle," Post-Print halshs-01159177, HAL.
    3. Philippe Bich & Jean-Pierre Drugeon & Lisa Morhaim, 2015. "On Aggregators and Dynamic Programming," Post-Print halshs-01169552, HAL.
    4. Philippe Bich & Jean-Pierre Drugeon & Lisa Morhaim, 2015. "On Aggregators and Dynamic Programming," Documents de travail du Centre d'Economie de la Sorbonne 15053, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    5. Philippe Bich & Jean-Pierre Drugeon & Lisa Morhaim, 2015. "On Aggregators and Dynamic Programming," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01169552, HAL.

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    More about this item

    Keywords

    Dynamic programming; Bellman operator; Value function; Fixed point;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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