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

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  • 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.

    Mathematics Subject Classification: Primary: 35J60, 35K55; Secondary: 35J55, 35B65.


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