Parabolic Bellman-Systems with Mean Field Dependence

We consider the necessary conditions for Nash-points of Vlasov-McKean functionals Ji[v]=∫Qmfi(·,m,v)dxdt (i= 1 ,.. , N). The corresponding payoffs fⁱ depend on the controls v and, in addition, on the field variable m= m(v). The necessary conditions lead to a coupled forward-backward system of nonlinear parabolic equations, motivated by stochastic differential games. The payoffs may have a critical nonlinearity of quadratic growth and any polynomial growth w.r.t. m is allowed as long as it can be dominated by the controls in a certain sense. We show existence and regularity of solutions to these mean-field-dependent Bellman systems by a purely analytical approach, no tools from stochastics are needed.