I just need a good reference and/or some more well-known results (which I don't know yet) for the following situation.
Let $\alpha\in(0,1)$. If I have an $\alpha$-stable subordinator $\mathbf{X}$, i.e. a nondecreasing, pure-jump Levy process with $\mathbf{X}(1)\sim G_{\alpha}$, where $G_{\alpha}$ is a one-sided $\alpha$-stable law. Which properties does $\mathbf{X}^{-1}$ (right-continuous inverse or first-passage time) have?
I heared it is a continuous process? That it is nondecreasing is evident, but at which points is it strictly increasing? And "how" many such points are there?
Thank's a lot!