| Abstract: |
| We present 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 and the backward SDEs depend not only on the controlled solution processes (X, Y, Z)
at the current time t, but also on the law of the paths of (X, Y, u) of the solution process and the control
process. The existence of the solution for such a MFFBSDE 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. Concerning the law, we just suppose the continuity under the 2-Wasserstein distance
of the coefficients with respect to the law of (X, Y). The uniqueness is shown under Lipschitz assumptions
and the non anticipativity of the law of X in the forward equation. The main part of the work is devoted
to the study of the Pontryagin maximal principle for such a MFFBSDE. The dependence of the coefficients on
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. In particular, also the
Hamiltonian is novel and quite different from that in the existing literature. Last but not least, under the
assumption of convexity of the Hamiltonian we show that our optimality condition is not only necessary but
also sufficient. |
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