  Optimal control problems of forward-backward stochastic Volterra integral
equations (FBSVIEs, in short) with closed control regions are formulated and
studied. Instead of using spike variation method as one may imagine, here we
turn to treat the non-convexity of the control regions by borrowing some tools
in set-valued analysis and adapting them into our stochastic control systems. A
duality principle between linear backward stochastic Volterra integral
equations and linear stochastic Fredholm-Volterra integral equations with
conditional expectation are derived, which extends and improves the
corresponding results in [25], [30]. Some first order necessary optimality
conditions for optimal controls of FBSVIEs are established. In contrast with
existed common routines to treat the non-convexity of stochastic control
problems, here only one adjoint system and one-order differentiability
requirements of the coefficients are needed.
