We develop an anticipative calculus for Lévy processes with finite second moment. The calculus is based on the chaos expansion of square-integrable random variables in terms of iterated integrals of the compensated Poisson random measure. We define a space of smooth and generalized random variables in terms of such chaos expansions, and introduce anticipative stochastic integration, the Wick product and the so-called S-transform. These concepts serve as tools for studying stochastic differential equations with anticipative initial conditions. We apply the S-transform to find the unique solutions to a class of linear stochastic differential equations. The solutions can be expressed in terms of the Wick product.