A Stochastic Maximum Principle for General Mean-Field Systems

In this paper we study the optimal control problem for a class of general mean-field stochastic differential equations, in which the coefficients depend, nonlinearly, on both the state process as well as of its law. In particular, we assume that the control set is a general open set that is not nece...

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Published in:Applied mathematics & optimization Vol. 74; no. 3; pp. 507 - 534
Main Authors: Buckdahn, Rainer, Li, Juan, Ma, Jin
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2016
Springer Nature B.V
Subjects:
Adjoints
Applications of mathematics
Brownian motion
Calculus of Variations and Optimal Control; Optimization
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Control
DIFFERENTIAL EQUATIONS
Dynamic programming
Euclidean space
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Maximum principle
MEAN-FIELD THEORY
NONLINEAR PROBLEMS
Numerical and Computational Physics
OPTIMAL CONTROL
Optimization
Random variables
Regularity
Simulation
Stochastic control theory
STOCHASTIC PROCESSES
Stochasticity
Studies
Systems Theory
Theoretical
VARIATIONAL METHODS
93E20
60H10
Mean-field SDE
60H30
Stochastic control
Maximum principle
91B28
McKean–Vlasov equation
ISSN:0095-4616, 1432-0606
Online Access:Get full text
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Summary:In this paper we study the optimal control problem for a class of general mean-field stochastic differential equations, in which the coefficients depend, nonlinearly, on both the state process as well as of its law. In particular, we assume that the control set is a general open set that is not necessary convex, and the coefficients are only continuous on the control variable without any further regularity or convexity. We validate the approach of Peng (SIAM J Control Optim 2(4):966–979, 1990 ) by considering the second order variational equations and the corresponding second order adjoint process in this setting, and we extend the Stochastic Maximum Principle of Buckdahn et al. (Appl Math Optim 64(2):197–216, 2011 ) to this general case.
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ISSN:0095-4616
1432-0606
DOI:10.1007/s00245-016-9394-9