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    A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control

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    Type
    Article
    Authors
    Djehiche, Boualem
    Tembine, Hamidou
    Tempone, Raul cc
    KAUST Department
    Applied Mathematics and Computational Science Program
    Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2015-02-24
    Online Publication Date
    2015-02-24
    Print Publication Date
    2015-10
    Permanent link to this record
    http://hdl.handle.net/10754/622508
    
    Metadata
    Show full item record
    Abstract
    In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the state and the control but also on the mean of the distribution of the state. Our result extends the risk-sensitive SMP (without mean-field coupling) of Lim and Zhou (2005), derived for feedback (or Markov) type optimal controls, to optimal control problems for non-Markovian dynamics which may be time-inconsistent in the sense that the Bellman optimality principle does not hold. In our approach to the risk-sensitive SMP, the smoothness assumption on the value-function imposed in Lim and Zhou (2005) needs not be satisfied. For a general action space a Peng's type SMP is derived, specifying the necessary conditions for optimality. Two examples are carried out to illustrate the proposed risk-sensitive mean-field type SMP under linear stochastic dynamics with exponential quadratic cost function. Explicit solutions are given for both mean-field free and mean-field models.
    Citation
    Djehiche B, Tembine H, Tempone R (2015) A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control. IEEE Transactions on Automatic Control 60: 2640–2649. Available: http://dx.doi.org/10.1109/TAC.2015.2406973.
    Publisher
    Institute of Electrical and Electronics Engineers (IEEE)
    Journal
    IEEE Transactions on Automatic Control
    DOI
    10.1109/TAC.2015.2406973
    arXiv
    1404.1441
    ae974a485f413a2113503eed53cd6c53
    10.1109/TAC.2015.2406973
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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