Sufficient conditions of optimality for mean-field stochastic control problems

This paper studies the optimal control problem for stochastic differential equations (SDEs) of mean-field type, in which the coefficients depend on the state of the solution process as well as 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. It is shown that the necessary conditions of optimality (Buckdahn et al., Appl. Math. Optim., vol. 64, pp. 197-216, 2011), along with some convexity/concavity conditions, constitute sufficient conditions of optimality for such problems. As an illustrating example, we apply the result to the linear quadratic stochastic optimal control problem of mean-field type.