  In this paper, we consider a linear-quadratic optimal control problem of
mean-field stochastic differential equation with jump diffusion, which is also
called as an MF-LQJ problem. Here, cost functional is allowed to be indefinite.
We use an equivalent cost functional method to deal with the MF-LQJ problem
with indefinite weighting matrices. Some equivalent cost functionals enable us
to establish a bridge between indefinite and positive-definite MF-LQJ problems.
With such a bridge, solvabilities of stochastic Hamiltonian system and Riccati
equations are further characterized. Optimal control of the indefinite MF-LQJ
problem is represented as a state feedback via solutions of Riccati equations.
As a by-product, the method provides a new way to prove the existence and
uniqueness of solution to mean field forward-backward stochastic differential
equation with jump diffusion (MF-FBSDEJ, for short), where existing methods in
literature do not work. Some examples are provided to illustrate our results.
