Solvability of Coupled Forward-Backward Volterra Integral Equations

Motivated by the optimality system associated with controlled (forward) Volterra integral equations (FVIEs, for short), the well-posedness of coupled forward-backward Voterra integral equations (FBVIEs, for short) is studied. The main feature of FBVIEs is that the unknown $\{(X(t,s),Y(t,s))\}$ has two arguments. By taking $t$ as a parameter and $s$ as a (time) variable, one can regard FBVIE as a system of ordinary differential equations (ODEs, for short), with infinite-dimensional space values $\{(X(\cdot,s),Y(\cdot,s));\,s\in[0,T]\}$. To establish the well-posedness of such an FBVIE, a new non-local monotonicity condition is introduced, by which a bridge in infinite-dimensional spaces is constructed. Then by generalizing the method of continuation developed by \cite{Hu-Peng1995,Yong1997,Peng-Wu1999} for differential equations, we have established the well-posedness of FBVIEs.The key is to apply the chain rule to the mapping $t\mapsto\big[\int_\cdot^T\langle Y(s,s),X(s,\cdot)\rangle ds +\langle G(X(T,T)),X(T,\cdot)\rangle\big](t)$.