| Abstract: |
| In the present paper we discuss a new type of mean-field coupled forward-backward stochastic differential equations (MFFBSDEs). The novelty consists in the fact that the coefficients of both the forward as well as the backward SDEs depend not only on the controlled solution processes $(X_t,Y_t,Z_t)$ at the current time $t$, but also on the law of the paths of $(X,Y,u)$ of the solution process and the process by which it is controlled.
The existence for such a MFFBSDE which is fully coupled through the law of the paths of $(X,Y)$ in the coefficients of both the forward and the backward equations is proved under rather general assumptions. The main part of the work is devoted to the study of Pontryagin`s maximal principle for such a MFFBSDE. The dependence of the coefficients of the law of the paths of the solution processes and their control makes that a completely new and interesting criterion for the optimality of a stochastic control for the MFFBSDE is obtained. Furthermore, we show that this necessary optimality condition is, under the assumption of convexity of the Hamiltonian, also sufficient.
The talk is based on joint work with Rainer Buckdahn (UBO, France), Juan Li (SDU, China), Junsong Li (SDU, China). |
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