Parabolic Bellman-Systems with Mean Field Dependence

详细信息    查看全文

• 作者：Alain Bensoussan ; Dominic Breit ; Jens Frehse
• 刊名：Applied Mathematics and Optimization
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：73
• 期：3
• 页码：419-432
• 全文大小：434 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Calculus of Variations and Optimal Control
  Systems Theory and Control
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  
• 出版者：Springer New York
• ISSN：1432-0606
• 卷排序：73

文摘

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.KeywordsNonlinear parabolic systemsBellman equationsStochastic differential gamesMean field dependence