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Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case

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  • Juan Pablo RincÛn-Zapatero
  • Carlos RodrÌguez-Palmero

Abstract

We study the problem of the existence and uniqueness of solutions to the Bellman equation in the presence of unbounded returns. We introduce a new approach based both on consideration of a metric on the space of all continuous functions over the state space, and on the application of some metric fixed point theorems. With appropriate conditions we prove uniqueness of solutions with respect to the whole space of continuous functions. Furthermore, the paper provides new sufficient conditions for the existence of solutions that can be applied to fairly general models. It is also proven that the fixed point coincides with the value function and that it can be approached by successive iterations of the Bellman operator. Copyright The Econometric Society 2003.

Suggested Citation

  • 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, September.
  • Handle: RePEc:ecm:emetrp:v:71:y:2003:i:5:p:1519-1555
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