Bellman Systems with Mean Field Dependent Dynamics |
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Authors: | Alain BENSOUSSAN Miroslav BUL'{I}v{C}EK and Jens FREHSE |
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Institution: | 1.Jindal School of Management,International Center for Decision and Risk Analysis University of Texas,Dallas,USA;2.Department SEEM City University of Hong Kong, Hung Hom, Kowloon,Hong Kong,China;3.Mathematical Institute, Faculty of Mathematics and Physics,Charles University,Praha 8,Czech Republic;4.Department of Applied Analysis,Institute for Applied Mathematics,Bonn,Germany |
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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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Keywords: | Stochastic games Bellman equation Mean field equation Nonlinearelliptic equations Weak solution Maximum principle |
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