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November  2010, 27(4): 1375-1389. doi: 10.3934/dcds.2010.27.1375

Systems of Bellman equations to stochastic differential games with non-compact coupling

1. 

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

2. 

Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, NRW, 53115 Bonn, Germany

3. 

Harvard Mathematics Department, One Oxford Street, Cambridge, MA 02138, United States

Received  November 2009 Revised  February 2010 Published  March 2010

We consider a class of non-linear partial differential systems like

-div$(a(x)\nabla u_{\nu}) +\lambda u_{\nu}=H_{\nu}(x, Du) \, $

with applications for the solution of stochastic differential games with $N$ players, where $N$ is an arbitrary but positive number. The Hamiltonian $H$ of the non-linear system satisfies a quadratic growth condition in $D u$ and contains interactions between the players in the form of non-compact coupling terms $\nabla u_{i} \cdot\nabla u_j$. A $L^{\infty}\cap H^1$-estimate and regularity results are shown, mainly in two-dimensional space. The coupling arises from cyclic non-market interaction of the control variables.

Citation: Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375
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