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

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Abstract

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.

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Authors and Affiliations

  1. Département de Mathématiques, Université de Bretagne Occidentale, 6, avenue Victor-le-Gorgeu, B.P. 809, 29285, Brest cedex, France

    Rainer Buckdahn

  2. Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Boualem Djehiche

  3. School of Mathematics and Statistics, Shandong University at Weihai, Weihai, 264209, P.R. China

    Juan Li

Authors
  1. Rainer Buckdahn
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  2. Boualem Djehiche
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  3. Juan Li
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Correspondence to Juan Li.

Additional information

R. Buckdahn work has been done in the frame of the Marie Curie ITN Project “Deterministic and Stochastic Controlled Systems and Applications”, call: F97-PEOPLE-2007-1-1-ITN, No: 213841-2.

J. Li work has been supported by the NSF of P.R. China (Nos. 10701050, 11071144), Independent Innovation Foundation of Shandong University, SRF for ROCS (SEM) and National Basic Research Program of China (973 Program) (No. 2007CB814904).

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Buckdahn, R., Djehiche, B. & Li, J. A General Stochastic Maximum Principle for SDEs of Mean-field Type. Appl Math Optim 64, 197–216 (2011). https://doi.org/10.1007/s00245-011-9136-y

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