ANTICIPATIVE CALCULUS FOR LÉVY PROCESSES AND STOCHASTIC DIFFERENTIAL EQUATIONS

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