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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.

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File URL: http://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/DP2012-05.pdf
File Function: First version, 2012
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Bibliographic Info

Paper provided by Research Institute for Economics & Business Administration, Kobe University in its series Discussion Paper Series with number DP2012-05.

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Length: 14 pages
Date of creation: Feb 2012
Date of revision:
Handle: RePEc:kob:dpaper:dp2012-05

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Keywords: Dynamic programming; Bellman operator; Value function; Fixed point;

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  1. Juan Rincón-Zapatero & Carlos Rodríguez-Palmero, 2007. "Recursive utility with unbounded aggregators," Economic Theory, Springer, vol. 33(2), pages 381-391, November.
  2. Vailakis, Yiannis & Martins-da-Rocha, Victor Filipe, 2008. "Existence and Uniqueness of a Fixed-Point for Local Contractions," Economics Working Papers (Ensaios Economicos da EPGE) 677, FGV/EPGE Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil).
  3. Michel, Philippe, 1990. "Some Clarifications on the Transversality Condition," Econometrica, Econometric Society, vol. 58(3), pages 705-23, May.
  4. 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.
  5. Takashi Kamihigashi, 2007. "On the Principle of Optimality for Nonstationary Deterministic Dynamic Programming," Discussion Paper Series 200, Research Institute for Economics & Business Administration, Kobe University.
  6. Juan Pablo Rincón-Zapatero & Carlos Rodríguez-Palmero, 2009. "Corrigendum to "Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case" Econometrica, Vol. 71, No. 5 (September, 2003), 1519-1555," Econometrica, Econometric Society, vol. 77(1), pages 317-318, 01.
  7. Le Van, C. & Morhaim, L., 2000. "Optimal Growth Models with Bounded or Unbounded Returns : a Unifying Approach," Papiers d'Economie Mathématique et Applications 2000.64, Université Panthéon-Sorbonne (Paris 1).
  8. Juan Pablo RincÛn-Zapatero & Carlos RodrÌguez-Palmero, 2003. "Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case," Econometrica, Econometric Society, vol. 71(5), pages 1519-1555, 09.
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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.

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