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We define two new classes of stochastic processes, called tempered fractionalL\'{e}vy process of the first and second kinds (TFLP and TFLP $I\!I$,respectively). TFLP and TFLP $I\!I$ make up very broad finite-variance,generally non-Gaussian families of transient anomalous diffusion models thatare constructed by exponentially tempering the power law kernel in the movingaverage representation of a fractional L\'{e}vy process. Accordingly, theincrement processes of TFLP and TFLP $I\!I$ display semi-long range dependence.We establish the sample path properties of TFLP and TFLP $I\!I$. We further usea flexible framework of tempered fractional derivatives and integrals todevelop the theory of stochastic integration with respect to TFLP and TFLP$I\!I$, which may not be semimartingales depending on the value of the memoryparameter and choice of marginal distribution.

On fractional L\'evy processes: tempering, sample path properties and stochastic integration