This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space $\mathcal{G}^{*}$ of Potthoff--Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed. We introduce a new volatility modulation method through the Wick product and discuss its relation to the pointwise-multiplied volatility model.

Keywords: stochastic integral; Volterra process; volatility modulation; white noise analysis; Malliavin derivative; Skorohod integral