Tempered fractional Brownian motion (TFBM) modifies the power law kernel in the moving average representation of a fractional Brownian motion (FBM), adding an exponential tempering. It also has a harmonizable representation. The increments of TFBM are stationary, and the autocovariance of the resulting tempered fractional Gaussian noise (TFGN) has semi-long range dependence, in which the autocorrelations decay like a power law over a moderate length scale, but eventually fall off more rapidly. TFBM can be represented as the linear combination of tempered fractional derivative (or tempered fractional integral) of the indicator functions. This representation and the classical Ito isometry provides to characterize the class of all deterministic functions for which the stochastic integral with respect to TFBM is well defined. Replacing the Gaussian random measure (Brownian motion) in the moving average or harmonizable representation of TFBM by a stable random measure, a linear tempered fractional stable motion (LTFSM), or a real harmonizable tempered fractional stable motion (HTFSM), respectively. Unlike the Gaussian case, LTFSM and HTFSM are two completely different processes. Existence, basic properties, sample path behavior, and dependence structure of both processes will be described.
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