Georgia Institute of Technology College of Sciences

Fractional calculus (FC) provides a powerful formalism for the modeling of systems whose underlying dynamics is governed by Lévy stochastic processes. In this talk we focus on two applications of FC: (1) non-diffusive transport, and (2) option pricing in finance. Regarding (1), starting from the continuous time random walk model for general Lévy jump distribution functions with memory, we construct effective non-diffusive transport models for the spatiotemporal evolution of the probability density function of particle displacements in the long-wavelength, time-asymptotic limit. Of particular interest is the development of models in finite-size-domains and those incorporating tempered Lévy processes. For the second application, we discuss fractional models of option prices in markets with jumps. Financial instruments that derive their value from assets following FMLS, CGMY, and KoBoL Lévy processes satisfy fractional diffusion equations (FDEs). We discuss accurate, efficient methods for the numerical integration of these FDEs, and apply them to price barrier options. The numerical methods are based on the finite difference discretization of the regularized fractional derivatives in the Grunwald-Letnikov representation.