A stochastic maximum principle for risk-sensitive mean-field type control

Boualem Djehiche, Hamidou Tembine, Raul Tempone

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the state and the control but also on the mean of the distribution of the state. Our result extends the risk-sensitive SMP (without mean-field coupling) of Lim and Zhou (2005), derived for feedback (or Markov) type optimal controls, to optimal control problems for non-Markovian dynamics which may be time-inconsistent in the sense that the Bellman optimality principle does not hold. In our approach to the risk-sensitive SMP, the smoothness assumption on the value-function imposed in Lim and Zhou (2005) needs not be satisfied. For a general action space a Peng's type SMP is derived, specifying the necessary conditions for optimality. Two examples are carried out to illustrate the proposed risk-sensitive mean-field type SMP under linear stochastic dynamics with exponential quadratic cost function. Explicit solutions are given for both mean-field free and mean-field models.

Original languageEnglish (US)
Article number7047683
Pages (from-to)2640-2649
Number of pages10
JournalIEEE Transactions on Automatic Control
Volume60
Issue number10
DOIs
StatePublished - Oct 1 2015

Keywords

  • Time inconsistent stochastic control
  • logarithmic transformation
  • maximum principle
  • mean-field SDE
  • risk-sensitive control

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering

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