Research On The Fractional Lévy Processes

fractional Brownian motion is a gaussian process which is satisfy the follow-ing conditions, and where H is a Hurst parameter, called index or fraction of fractional Brownian motion.Fractional Brownian motion has excellent analysis properties, such as self-similarity, durability and continuous sample function. So it has attracted lots of researchers’attentions, and plays an important role in solving the problem of engineering technology and other science.Fractional Levy process is the generalized form of the fractional Brownian motion. But the fractional Levy process did not exactly the same with the nature of the fractional Brownian motion. For example the fractional Levy processes nor-mally do not have self-similarity. People have studied some properties of the linear fraction Levy processes and real harmonizable fractional Levy processes. Such as sample paths and semi-martingale properties. Especially, Bender, Lindner and Schicks have studied the finite variation of improper Riemann integral of Md, when the index satisfy 0< d< 0.5. They have found and proofed 11 equivalent condi-tions when Md is the semi-martingale.On the basis of this,with the complicated calculation, we found and proved the 11 equivalent conditions are still holds when the index is 0.5< d< 1 in this Daper.