A General Stochastic Maximum Principle for SDEs of Mean-field Type

Journal Article · Sat Oct 15 04:00:00 EDT 2011 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S00245-011-9136-Y· OSTI ID:22043878

Buckdahn, Rainer ^[1]; Djehiche, Boualem ^[2]

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1. Universite de Bretagne Occidentale, Departement de Mathematiques (France)
2. Royal Institute of Technology, Department of Mathematics (Sweden)

We study the optimal control for stochastic differential equations (SDEs) of mean-field type, in which the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. For a general action space a Peng's-type stochastic maximum principle (Peng, S.: SIAM J. Control Optim. 2(4), 966-979, 1990) is derived, specifying the necessary conditions for optimality. This maximum principle differs from the classical one in the sense that here the first order adjoint equation turns out to be a linear mean-field backward SDE, while the second order adjoint equation remains the same as in Peng's stochastic maximum principle.

• OSTI ID: 22043878
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 2 Vol. 64; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
DIFFERENTIAL EQUATIONS
MATHEMATICAL SOLUTIONS
MEAN-FIELD THEORY
OPTIMAL CONTROL
SPACE
STOCHASTIC PROCESSES