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Thank you. I'm familiar with the fact that for a Levy process $X_t$ and some $s\in \mathbb{R}$ $\bigtriangleup X_{s}=0$ a.s. However, since the number of $s\in \mathbb{R}$ for which a jump occurred might not be measurable I don't see how it answers my question.
Yes, thank you!!! for some reason I was convinced that Levy measure of the joint process should be $\delta_{0}(y_{1})\frac{\alpha}{\Gamma(1-\alpha)}y_{2}^{-\alpha-1}dy_{2}$.
Thanks, but how do I find the Levy triplet of $(N_t,D_t)$ - its Levy measure, drift and Gaussian component(which I know is zero since this is a strictly increasing process)?