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This paper is concerned with an optimal control problem for aforward-backward stochastic differential equation (FBSDE, for short) with arecursive cost functional determined by a backward stochastic Volterra integralequation (BSVIE, for short). It is found that such an optimal control problemis time-inconsistent in general, even if the cost functional is reduced to aclassical Bolza type one as in Peng [50], Lim-Zhou [41], and Yong [74].Therefore, instead of finding a global optimal control (which istime-inconsistent), we will look for a time-consistent and locally optimalequilibrium strategy, which can be constructed via the solution of anassociated equilibrium Hamilton-Jacobi-Bellman (HJB, for short) equation. Averification theorem for the local optimality of the equilibrium strategy isproved by means of the generalized Feynman-Kac formula for BSVIEs and somestability estimates of the representation for parabolic partial differentialequations (PDEs, for short). Under certain conditions, it is proved that theequilibrium HJB equation, which is a nonlocal PDE, admits a unique classicalsolution. As special cases and applications, the linear-quadratic problems, amean-variance model, a social planner problem with heterogeneous Epstein-Zinutilities, and a Stackelberg game are briefly investigated. It turns out thatour framework can cover not only the optimal control problems for FBSDEsstudied in [50,41,74], and so on, but also the problems of the generaldiscounting and some nonlinear appearance of conditional expectations for theterminal state, studied in Yong [75,77] and Bj\"{o}rk-Khapko-Murgoci [7].

Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutions