Bellman systems of stochastic differential games with three players. (English) Zbl 1054.91511
Menaldi, José Luis (ed.) et al., Optimal control and partial differential equations. In honour of Professor Alain Bensoussan’s 60th birthday. Proceedings of the conference, Paris, France, December 4, 2000. Amsterdam: IOS Press; Tokyo: Ohmsha (ISBN 1-58603-096-5; 4-274-90412-1). 3-22 (2001).
Summary: Bellman systems \(-\Delta u+ \alpha u= H(.,\nabla u)\), \(u\in H^{1,2}_0({\mathcal O},\mathbb{R}^3)\), \({\mathcal O}\) a bounded domain of \(\mathbb{R}^n\) are considered. The Hamiltonian has quadratic growth in \(\nabla u\). The technique to obtain estimates in \(H^1({\mathcal O})\), \(C^\alpha({\mathcal O})\) and \(H^{2,q}({\mathcal O})\) for solutions \(u\) is explained for the case that \(H\) satisfies a structure condition of the following type:
\[
|H_\nu(p)|\leq K|p||p_\nu|+ \sum_{\mu\leq\nu} |p_\nu|^2+ K
\]
. Simple stochastic differential games related to the Bellman system are considered.
For the entire collection see [Zbl 1053.49001].
For the entire collection see [Zbl 1053.49001].
MSC:
91A15 | Stochastic games, stochastic differential games |
35J60 | Nonlinear elliptic equations |
93E20 | Optimal stochastic control |
35B65 | Smoothness and regularity of solutions to PDEs |