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Extended Mean Field Control Problems: Stochastic Maximum Principle and Transport Perspective

Beatrice Acciaio · Julio Backhoff‐Veraguas · Rene A. Carmona

69 citations0/0 files storedSIAM Journal on Control and Optimization

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Stochastic processes and financial applications Optimization and Variational Analysis Geometric Analysis and Curvature Flows

Abstract

We study mean field stochastic control problems where the cost function and the state dynamics depend upon the joint distribution of the controlled state and the control process. We prove suitable versions of the Pontryagin stochastic maximum principle, both in necessary and in sufficient forms, which extend the known conditions to this general framework. We suggest a variational approach for a weak formulation of these control problems. We show a natural connection between this weak formulation and optimal transport on path space, which inspires a novel discretization scheme.

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References

2001Optimal execution of portfolio transactions 2013Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations 2010A Maximum Principle for SDEs of Mean-Field Type 2009Mean-field backward stochastic differential equations: A limit approach 2011A General Stochastic Maximum Principle for SDEs of Mean-field Type 2015Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics 2012Control of McKean–Vlasov dynamics versus mean field games 2017Mean field game of controls and an application to trade crowding

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