Well-posedness of fully coupled McKean-Vlasov FBSDEs with jumps under full-tuple law dependence

Abstract

We prove existence, uniqueness, and stability for fully coupled McKean-Vlasov forward-backward SDEs with jumps whose drift, diffusion, jump, and driver coefficients may depend Lipschitz-continuously, in quadratic Wasserstein distance, on the joint law of the full solution tuple $Θ=(X,Y,Z,U)$: forward state, backward variable, Brownian integrand, and $L^2(ν)$-valued jump integrand. The terminal function may depend Lipschitz-continuously on $X_T$ and its law. The system is driven by a Brownian motion and an independent compensated Poisson random measure with arbitrary $σ$-finite intensity, so infinite jump activity is admitted. Both the Lipschitz and monotonicity hypotheses are imposed only along diagonal tuple-law pairs $(Θ,\mathrm{Law}(Θ))$; we show that expected diagonal monotonicity is strictly weaker than pointwise monotonicity. Under a jump-extended $G$-monotonicity condition we establish an a priori continuous-dependence estimate, uniqueness, and existence on every prescribed finite horizon, by monotone continuation in the coupling strength from a small-coupling base case. A mean-field dealer-market example realises the $U$-law dependence non-perturbatively: its law interaction is monotone at every interaction strength, and its mark measure has infinite activity.