Motivated by the optimality system associated with controlled (forward) Volterra integral equations (FVIEs, for short), the well-posedness of coupled forward-backward Volterra integral equations (FBVIEs, for short) is studied. A key feature of FBVIEs is that the unknown pair \({(\mathcal {X}(t,s),\mathcal {Y}(t,s))}\) depends on two arguments. By treating t as a parameter and s as a time variable, an FBVIE can be viewed as a system of ordinary differential equations (ODEs, for short) taking values in an infinite-dimensional space \({(\mathcal {X}(\cdot ,s),\mathcal {Y}(\cdot ,s)); s\in [0,T]}\) . To establish well-posedness, a new non-local monotonicity condition is introduced, by which a bridge in infinite-dimensional spaces is constructed. Then by extending the method of continuation developed by Hu and Peng (Probab Theory Relat Fields 103:273–283, 1995), Yong (Probab Theory Relat Fields 107:537–572, 1997) and Peng and Wu (SIAM J Control Optim 37:825–843, 1999) for differential equations, we prove the well-posedness of FBVIEs. The crucial step is to apply the chain rule to the mapping \(t\mapsto \big [\int _\cdot ^T\langle \mathcal {Y}(s,s),\mathcal {X}(s,\cdot )\rangle ds+\langle G(\mathcal {X}(T,T)),\mathcal {X}(T,\cdot )\rangle \big ](t)\) . This is based on the observation that in LQ problems for ODEs, \(\langle X(t),Y(t)\rangle \) represents the value function, whereas for LQ problems governed by FVIEs the value function is given by \(\int _t^T \langle \mathcal {Y}(s,s),\mathcal {X}(s,t)\rangle ds+\langle G\mathcal {X}(T,T),\mathcal {X}(T,t)\rangle \) .
