Fully Coupled Nonlinear Mean-Field FBSΔEs: Solvability and an LQ Buffer-Adjustment Illustration

Abstract

This paper establishes sufficient conditions for finite-horizon solvability of fully coupled nonlinear mean-field forward--backward stochastic difference equations with dependence on unconditional first moments. The forward recursion uses conditional projections of the next backward state and its product with the innovation. Building on existing domination--monotonicity methods, we formulate deterministic matrix combinations in centered and mean coordinates and prove a continuation estimate uniform in the homotopy parameter. Global Lipschitz continuity and one active domination--coercivity direction yield a unique square-integrable adapted solution, an a priori bound, and coefficient stability. The active parameter can be normalized without imposing an additional smallness restriction on the original coefficients. A sign transformation handles the opposite monotonicity orientation. A nonlinear example with saturating state and mean interactions verifies the assumptions, including degenerate domination directions. A scalar mean-field LQ buffer-adjustment model illustrates the theorem: its Hamiltonian solution characterizes the unique open-loop optimizer. An exact finite scenario-tree calculation checks this characterization against direct quadratic optimization.