Abstract
We study an extended mean-field control problem with partial observation, where the dynamic of the state is given by a forward-backward stochastic differential equation of McKean-Vlasov type. The cost functional, the state and the observation all depend on the joint distribution of the state and the control process. Our problem is motivated by the recent popular subject of mean-field games and related control problems of McKean-Vlasov type. We first establish a necessary condition in the form of Pontryagin’s maximum principle for optimality. Then a verification theorem is obtained for optimal control under some convex conditions of the Hamiltonian function. The results are also applied to studying linear-quadratic mean-filed control problem in the type of scalar interaction.
Highlights
The stochastic differential equations (SDEs) of McKean-Vlasov type were introduced by Kac [23] in 1956 as a stochastic model for the Vlasov-kinetic equation of plasma
The readers are referred to the monographs of Carmona and Delarue [17] and Bensoussan et al [6] for an overview of McKean-Vlasov type control problems
In view of the wide applications in finance and economics of above extended mean-field control system, the purpose of this paper is to study the maximum principle of the extended mean-field control problem with partial observation, where the state and the observation both depend on the joint distribution of the state and the control process
Summary
The stochastic differential equations (SDEs) of McKean-Vlasov type were introduced by Kac [23] in 1956 as a stochastic model for the Vlasov-kinetic equation of plasma. Buckdahn et al [12] studied mean-filed non-Markovian stochastic optimal control problems with partial observation, where the coefficients depend on the conditional law of the state. We need use the L-derivative w.r.t. probability measure, especially the partial L-derivatives because of the dependence of joint distribution In this case, we can obtain new adjoint equations and variation equations, which are both mean-field FBSDEs (see (3.11) and (3.13)), and we give the existence and uniqueness of solutions for variation equations and adjoint equations, see Theorem 3.1 and Remark 3.8. We give the detailed proofs of some lemmas of Section 3
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