Backward stochastic Volterra integral equations in l^q spaces and their application

We consider a backward stochastic Volterra integral equation [BSVIE] in the Banach space E=l^q (S,∑,μ) , where μ is σ-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. For an application we can consider an optimal control problem in Banach space E where the state process will be defined as forward Ito Volterra stochastic integral equation with respect to a cylindrical Wiener process. There are also applications in finance for example, rate processes in a Heath-Jarrow-Morton model which satisfy a stochastic partial differential equation of first order in Banach space.