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Representation formulas for limit values of long run stochastic optimal controls

Author

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  • Li, Jin
  • Quincampoix, Marc
  • Renault, Jérôme
  • Buckdahn, Rainer

Abstract

A classical problem in stochastic ergodic control consists of studying the limit behavior of the optimal value of a discounted integral in infinite horizon (the so called Abel mean of an integral cost) as the discount factor $\lambda$ tends to zero or the value defined with a Cesàro mean of an integral cost when the horizon $T$ tends to $+ \infty$. We investigate the possible limits in the norm of uniform convergence topology of values defined through Abel means or Ceàro means when $ \lambda \to 0^+ $ and $T \to + \infty $, respectively. Here we give two types of new representation formulas for the accumulation points of the values when the averaging parameter converges. We show that there is only one possible accumulation point which is the same for Abel means or Cesàro means. The first type of representation formula is based on probability measures on the product of the state space and the control state space, which are limits of occupational measures. The second type of representation formula is based on measures which are the projection of invariant measure on the space of relaxed controls. We also give a result comparing the both sets of measures involved in both classes of representation formulas. An important consequence of the representation formulas is the existence of the limit value when one has the equicontinuity property of Abel or Cesàro mean values. This is the case, for example, for nonexpansive stochastic control systems. In the end some insightful examples are given which help to better understand the results.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Li, Jin & Quincampoix, Marc & Renault, Jérôme & Buckdahn, Rainer, 2019. "Representation formulas for limit values of long run stochastic optimal controls," TSE Working Papers 19-1007, Toulouse School of Economics (TSE).
  • Handle: RePEc:tse:wpaper:122930
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    References listed on IDEAS

    as
    1. Renault, Jérôme & Venel, Xavier, 2017. "A distance for probability spaces, and long-term values in Markov Decision Processes and Repeated Games," TSE Working Papers 17-748, Toulouse School of Economics (TSE).
    2. Dan Goreac & Oana-Silvia Serea, 2012. "Some Applications of Linear Programming Formulations in Stochastic Control," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 572-593, November.
    3. Jérôme Renault & Xavier Venel, 2017. "Long-Term Values in Markov Decision Processes and Repeated Games, and a New Distance for Probability Spaces," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 349-376, May.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Stochastic nonexpansivity condition; limit value; stochastic optimal control;
    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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