### 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) |
---|---|

Title of host publication | Proceedings of the 18th IFAC World Congress |

Pages | 3222-3227 |

Number of pages | 6 |

Volume | 18 |

Edition | PART 1 |

DOIs | |

State | Published - 2011 |

Event | 18th IFAC World Congress - Milano, Italy Duration: Aug 28 2011 → Sep 2 2011 |

### Other

Other | 18th IFAC World Congress |
---|---|

Country | Italy |

City | Milano |

Period | 8/28/11 → 9/2/11 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Control and Systems Engineering

### Cite this

*Proceedings of the 18th IFAC World Congress*(PART 1 ed., Vol. 18, pp. 3222-3227) https://doi.org/10.3182/20110828-6-IT-1002.02247

**Risk-sensitive mean-field stochastic differential games.** / Hamidou, Tembine; Zhu, Quanyan; Başar, Tamer.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 18th IFAC World Congress.*PART 1 edn, vol. 18, pp. 3222-3227, 18th IFAC World Congress, Milano, Italy, 8/28/11. https://doi.org/10.3182/20110828-6-IT-1002.02247

}

TY - GEN

T1 - Risk-sensitive mean-field stochastic differential games

AU - Hamidou, Tembine

AU - Zhu, Quanyan

AU - Başar, Tamer

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

KW - Mean-field analysis

KW - Risk-sensitive games

KW - Stochastic differential games

UR - http://www.scopus.com/inward/record.url?scp=84866747480&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866747480&partnerID=8YFLogxK

U2 - 10.3182/20110828-6-IT-1002.02247

DO - 10.3182/20110828-6-IT-1002.02247

M3 - Conference contribution

AN - SCOPUS:84866747480

SN - 9783902661937

VL - 18

SP - 3222

EP - 3227

BT - Proceedings of the 18th IFAC World Congress

ER -