Time Change Representation of Stochastic Integrals

By the Dambis--Dubins--Schwarz theorem, any stochastic integral M:=∫₀^· HₛdWₛ of Brownian motion can be written as a time-changed Brownian motion, i.e., M=(W_Tₜ)t₊ for some Brownian motion (W_θ)θ₊ and some time change (Tₜ)t₊. In [J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin--Heidelberg, 1987] and [O. Kallenberg, Stochastic Process. Appl., 40 (1992), pp.~199--223] it is shown that in this statement Brownian motion can be replaced with (symmetric) α-stable Levy motion}. Using the cumulant process of a semimartingale, we give new short proofs. Moreover, we show that the statement cannot be extended to any other Levy processes.