Bellmann's equation for stochastic games with discount control

Abstract
The above problem leads to diagonal elliptic or parabolic systems with a 
right hand side growing quadratically in the gradient of the value 
functions. Discount control means that the discount factors exp(-at) in 
the cost functionals may be depend an the control functions v ,i.e.  
a=a(v). This creates a nonlinearity on the right hand side of the system 
which is not covered by known techniques from partial differential 
equations. We discuss this situation and present some new existence and 
regularity results for such Bellmann systems.