Applied Mathematics & Optimization

, Volume 73, Issue 3, pp 419–432 | Cite as

Parabolic Bellman-Systems with Mean Field Dependence

  • Alain Bensoussan
  • Dominic BreitEmail author
  • Jens Frehse


We consider the necessary conditions for Nash-points of Vlasov-McKean functionals \(\mathcal {J}^{i}[\mathbf{v}]=\int _{Q}mf^{i}(\cdot ,m,\mathbf{v})\,dx\,dt\) (\(i=1,...,N\)). The corresponding payoffs \(f^{i}\) depend on the controls \(\mathbf{v}\) and, in addition, on the field variable \(m=m(\mathbf{v})\). 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.


Nonlinear parabolic systems Bellman equations Stochastic differential games Mean field dependence 

Mathematics Subject Classification

35B50 35K55 35B65 49L20 



The first author acknowledges support from the National Science Foundation, Grant DMS-1303775 and the Research Grant Council of Hong Kong Special Administrative Region, Grant GRF 500113.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Naveen Jindal School of Management, University of Texas at DallasRichardsonUSA
  2. 2.Department of SEEMCity University of Hong KongKowloon TongHong Kong
  3. 3.Department of MathematicsHeriot-Watt UniversityEdinburghUK
  4. 4.Department of Applied MathematicsUniversity of BonnBonnGermany

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