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Optimal stopping under ambiguity

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  • Frank Riedel

    ()
    (Institute of Mathematical Economics, Bielefeld University)

Abstract

We consider optimal stopping problems for ambiguity averse decision makers with multiple priors. In general, backward induction fails. If, however, the class of priors is time-consistent, we establish a generalization of the classical theory of optimal stopping. To this end, we develop first steps of a martingale theory for multiple priors. We define minimax (super)martingales, provide a Doob-Meyer decomposition, and characterize minimax martingales. This allows us to extend the standard backward induction procedure to ambiguous, time-consistent preferences. The value function is the smallest process that is a minimax supermartingale and dominates the payoff process. It is optimal to stop when the current payoff is equal to the value function. Moving on, we study the infinite horizon case. We show that the value process satisfies the same backward recursion (Bellman equation) as in the finite horizon case. The finite horizon solutions converge to the infinite horizon solution. Finally, we characterize completely the set of time-consistent multiple priors in the binomial tree. We solve two classes of examples: the so-called independent and indistinguishable case (the parking problem) and the case of American Options (Cox-Ross-Rubinstein model).

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File URL: http://www.imw.uni-bielefeld.de/papers/files/imw-wp-390.pdf
File Function: First Version, 2007
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Bibliographic Info

Paper provided by Bielefeld University, Center for Mathematical Economics in its series Working Papers with number 390.

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Length: 37 pages
Date of creation: Mar 2007
Date of revision:
Handle: RePEc:bie:wpaper:390

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Keywords: optimal stopping; ambiguity; uncertainty aversion;

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  1. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
  2. Frank Riedel, 2003. "Dynamic Coherent Risk Measures," Working Papers 03004, Stanford University, Department of Economics.
  3. Eichberger, J. & Kelsey, D., 1993. "Uncertainty Aversion and Dynamic Consistency," Discussion Papers 93-08, Department of Economics, University of Birmingham.
  4. repec:hal:cesptp:halshs-00177057 is not listed on IDEAS
  5. Massimo Marinacci & Fabio Maccheroni & Alain Chateauneuf & Jean-Marc Tallon, 2003. "Monotone Continuous Multiple Priors," ICER Working Papers - Applied Mathematics Series 30-2003, ICER - International Centre for Economic Research.
  6. Larry G. Epstein & Martin Schneider, 2001. "Recursive Multiple-Priors," RCER Working Papers 485, University of Rochester - Center for Economic Research (RCER).
  7. H. Föllmer & Y.M. Kabanov, 1997. "Optional decomposition and Lagrange multipliers," Finance and Stochastics, Springer, vol. 2(1), pages 69-81.
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