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Bellman Systems with Mean Field Dependent Dynamics

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Abstract

The authors deal with nonlinear elliptic and parabolic systems that are the Bellman like systems associated to stochastic differential games with mean field dependent dynamics. The key novelty of the paper is that they allow heavily mean field dependent dynamics. This in particular leads to a system of PDE’s with critical growth, for which it is rare to have an existence and/or regularity result. In the paper, they introduce a structural assumptions that cover many cases in stochastic differential games with mean field dependent dynamics for which they are able to establish the existence of a weak solution. In addition, the authors present here a completely new method for obtaining the maximum/minimum principles for systems with critical growths, which is a starting point for further existence and also qualitative analysis.

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Authors and Affiliations

  1. Jindal School of Management, International Center for Decision and Risk Analysis University of Texas, Dallas, 800 W. Campbell Rd, SM30 Richardson, TX, 75080-3021, USA

    Alain Bensoussan

  2. Department SEEM City University of Hong Kong, Hung Hom, Kowloon, Hong Kong, China

    Alain Bensoussan

  3. Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Miroslav Bulíček

  4. Department of Applied Analysis, Institute for Applied Mathematics, Endenicher Allee 60, 53115, Bonn, Germany

    Jens Frehse

Authors
  1. Alain Bensoussan
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  2. Miroslav Bulíček
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  3. Jens Frehse
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Corresponding author

Correspondence to Alain Bensoussan.

Additional information

This work was supported by the National Science Foundation (Nos.DMS 1303775, DMS 1612880), the Hong Kong SAR Research Grant Council (Nos.GRF 500113, 11303316), Hausdorff Center for Mathematics at University of Bonn and the Czech Science Foundation (No. 16-03230S).

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Bensoussan, A., Bulíček, M. & Frehse, J. Bellman Systems with Mean Field Dependent Dynamics. Chin. Ann. Math. Ser. B 39, 461–486 (2018). https://doi.org/10.1007/s11401-018-0078-4

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  • DOI: https://doi.org/10.1007/s11401-018-0078-4

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