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

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.