Diagonal elliptic Bellman systems to stochastic differential games with discount control and noncompact coupling

Abstract

We consider Bellman systems to stochastic differential games with quadratic cost functionals and a discount factor which may be influenced by the players. This leads to a diagonal elliptic system − Δu = H(x, u, ∇u) subject to boundary conditions where the Hamiltonian grows quadratically in grad u and contains a discount term uF0(u, ∇u). We mainly consider the two-dimensional case and 2 or 3 players. Under appropriate conditions we obtain the existence of regular solutions. Examples of cost functionals are presented, where no regularity theory is available up to now.

Keywords

• non-linear partial differential equations
• diagonal systems
• bellman equation
• hamiltonian
• stochastic differential games
• discount control