### Abstract

In this paper, we study a class of risk-sensitive mean-field stochastic differential games. Under regularity assumptions, we use results from standard risk-sensitive differential game theory to show that the mean-field value of the exponentiated cost functional coincides with the value function of a Hamilton-Jacobi-Bellman-Fleming (HJBF) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations and HJBF equations.

Original language | English (US) |
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Title of host publication | Proceedings of the 18th IFAC World Congress |

Publisher | IFAC Secretariat |

Pages | 3222-3227 |

Number of pages | 6 |

Edition | 1 PART 1 |

ISBN (Print) | 9783902661937 |

DOIs | |

State | Published - Jan 1 2011 |

### Publication series

Name | IFAC Proceedings Volumes (IFAC-PapersOnline) |
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Number | 1 PART 1 |

Volume | 44 |

ISSN (Print) | 1474-6670 |

### Keywords

- Mean-field analysis
- Risk-sensitive games
- Stochastic differential games

### ASJC Scopus subject areas

- Control and Systems Engineering

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## Cite this

*Proceedings of the 18th IFAC World Congress*(1 PART 1 ed., pp. 3222-3227). (IFAC Proceedings Volumes (IFAC-PapersOnline); Vol. 44, No. 1 PART 1). IFAC Secretariat. https://doi.org/10.3182/20110828-6-IT-1002.02247