title: TEMPERED FRACTIONAL CALCULUS.
abstract: Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series.
authors.affiliation: Department of Statistics and Probability, Michigan State University, East Lansing MI 48823 mcubed@stt.msu.edu URL: http://www.stt.msu.edu/users/mcubed/
authors.forename: Mark M
authors.initials: MM
authors.lastname: Meerschaert
authors.affiliation: Department of Statistics and Probability, Michigan State University, East Lansing MI 48823 sabzika2@stt.msu.edu.
authors.forename: Farzad
authors.initials: F
authors.lastname: Sabzikar
authors.affiliation: School of Sciences, Jimei University, Xiamen, China, 361500 cjhdzdz@163.com.
authors.forename: Jinghua
authors.initials: J
authors.lastname: Chen
keywords: Fractional calculus; anomalous diffusion; random walk