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A maximum principle for SDEs of mean-field type. (English) Zbl 1215.49034

Summary: We study the optimal control of a Stochastic Differential Equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem.

MSC:

49K45 Optimality conditions for problems involving randomness
93E20 Optimal stochastic control
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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