I asked the following around 5 months ago and I did not get any satisfactory answer. Can anybody help me out now? Can anybody give me a solid reference to the form of Levy theorem used in Chapter 9 to characterize Brownian motions? This article http://individual.utoronto.ca/norma.../Levy_characterization_of_Brownian_motion.pdf is talking about quadratic variations as a central concept. Comparing the covariation definition mentioned in https://en.wikipedia.org/wiki/Quadratic_variation and in the CMP I see some risk that for stochastic processes there are two different definitions (with different, i.e. not equivalent, underlying meaning). Therefore I think that there is a chance that Levy's theorem is used inappropriately in the CMP. Hopefully, there will be somebody who could help me out on this. Thanks in advance! Thx in advance! S.