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For a real c\`{a}dl\`{a}g function $f$ defined on a compact interval, itstruncated variation at the level $c>0$ is the infimum of total variations offunctions uniformly approximating $f$ with accuracy $c/2$ and (in opposite tothe total variation) is always finite. In this paper, we discuss exponentialintegrability and concentration properties of the truncated variation offractional Brownian motions, diffusions and L\'{e}vy processes. We develop aspecial technique based on chaining approach and using it we prove Gaussianconcentration of the truncated variation for certain class of diffusions.Further, we give sufficient and necessary condition for the existence ofexponential moment of order $\alpha>0$ of truncated variation of L\'{e}vyprocess in terms of its L\'{e}vy triplet.

Integrability and concentration of the truncated variation for the sample paths of fractional Brownian motions, diffusions and L\'{e}vy processes