A second-order stochastic maximum principle for generalized mean-field singular control problem

Hancheng Guo; Jie Xiong

doi:10.3934/mcrf.2018018

Article Contents

2018, Volume 8, Issue 2: 451-473. Doi: 10.3934/mcrf.2018018

This issue Previous Article Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients Next Article Stability and output feedback control for singular Markovian jump delayed systems

A second-order stochastic maximum principle for generalized mean-field singular control problem

Hancheng Guo^, and
Jie Xiong

Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, 999078, China

* Corresponding author: Hancheng Guo
* Corresponding author: Hancheng Guo

Received: March 31, 2017

Revised: September 30, 2017

Published: May 2018

Research supported partially by FDCT 025/2016/A1.

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Abstract

In this paper, we study the generalized mean-field stochastic control problem when the usual stochastic maximum principle (SMP) is not applicable due to the singularity of the Hamiltonian function. In this case, we derive a second order SMP. We introduce the adjoint process by the generalized mean-field backward stochastic differential equation. The keys in the proofs are the expansion of the cost functional in terms of a perturbation parameter, and the use of the range theorem for vector-valued measures.

Keywords:

Stochastic maximum principle,
mean-field control problem,
singular control,
Fréchet derivative,
range theorem of vector-valued measures.

Mathematics Subject Classification: Primary: 93E20, 93E03, 60H30; Secondary: 70G70.

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