Backward stochastic Volterra integral equations in $L^q$ spaces and duality principle with forward stochastic Volterra integral equations

We consider a backward stochastic Volterra integral equation [BSVIE] in the Banach space $E=L^q(\mathbf{S},\mathrm{\Sigma },\mu )$, where $\mu$ is $\sigma$-finte measure. The stochastic integral is defined with respect to an infinite dimensional Wiener process. Under appropriate assumptions, the existence and uniqueness of an adapted solution of BSVIE are being proofed by using martingale representation theorem in Banach space $E$ and Banach fixed-point theorem. Some properties of the solution are also discussed. We prove a duality principle between BSVIEs and forward It\^o Volterra stochastic integral equations with respect to a cylindrical Wiener process.