  This papers develops a stochastic integration theory with respect to
volatility modulated L\'{e}vy-driven Volterra (VMLV) processes. It extends
recent results in the literature to allow for stochastic volatility and pure
jump processes in the integrator. The new integration operator is based on
Malliavin calculus and describes an anticipative integral. Fundamental
properties of the integral are derived and important applications are given.
