Variation analysis of uncertain stationary independent increment processes

A stationary independent increment process is an uncertain process with stationary and independent increments. This paper aims to calculate the variance of stationary independent increment processes, and gains that, for each fixed time, the variance is a constant multiplying the square of time. Based on this result, it is proved that the total variation of stationary independent increment process with finite variance is bounded almost surely. Besides, the quadratic variation of stationary independent increment process with finite variance is 0 almost surely and in mean.

► Standard deviation of the process is proved to be a constant multiplying the time.
► Total variation of the process is proved to be finite on any bounded interval.
► The quadratic variation of the process is proved to 0 almost surely.