Time Change Representation of Stochastic Integrals

By the Dambis–Dubins–Schwarz theorem, any stochastic integral $M:=\int_0^\cdot H_sdW_s$ of Brownian motion can be written as a time-changed Brownian motion, i.e., $M=({\widehat{W}}_{\widehat{T_t}})_{t\in\mathbf{R}_+}$ for some Brownian motion $({\widehat{W}}_\theta)_{\theta\in\mathbf{R}_+}$ and some time change $({\widehat{T_t}})_{t\in\mathbf{R}_+}$. 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) $\alpha$-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.