## Abstract

We study the optimal control for stochastic differential equations (SDEs) of mean-field type, in which the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. For a general action space a Peng's-type stochastic maximum principle (Peng, S.: SIAM J. Control Optim. 2(4), 966-979, 1990) is derived, specifying the necessary conditions for optimality. This maximum principle differs from the classical one in the sense that here the first order adjoint equation turns out to be a linear mean-field backward SDE, while the second order adjoint equation remains the same as in Peng's stochastic maximum principle.

Original language | English (US) |
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Pages (from-to) | 197-216 |

Number of pages | 20 |

Journal | Applied Mathematics and Optimization |

Volume | 64 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2011 |

## Keywords

- Maximum principle
- McKean-Vlasov equation
- Mean-field SDE
- Stochastic control
- Time inconsistent control

## ASJC Scopus subject areas

- Control and Optimization
- Applied Mathematics