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

Alain Bensoussan; Jens Frehse; Jens Vogelgesang

doi:10.3934/dcds.2010.27.1375

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2010, Volume 27, Issue 4: 1375-1389. Doi: 10.3934/dcds.2010.27.1375

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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
2.
Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, NRW, 53115 Bonn
3.
Harvard Mathematics Department, One Oxford Street, Cambridge, MA 02138

Received: October 31, 2009

Revised: January 31, 2010

Published: November 2010

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Abstract

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.

Keywords:

Stochastic games,
Bellman equation,
nonlinear elliptic and parabolic equations,
stochastic differential games,
Hamiltonians.

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

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