Path-Dependent Backward Stochastic Volterra Integral Equations with Jumps, Differentiability and Duality Principle

We study existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations (BSVIEs, in short) with jumps, where path-dependence means the dependence of the free term and generator of a path of a càdlàg process. Furthermore, we prove path-differentiability of such a solution and establish the duality principle between a linear path-dependent forward stochastic Volterra integral equation (FSVIE, in short) with jumps and a linear path-dependent BSVIE with jumps. As a result of the duality principle we get a comparison theorem and derive a class of dynamic coherent risk measures based on path-dependent BSVIEs with jumps