Lévy processes

Let $\alpha \in (0,2)$ and let $X$ be a Lévy process where $b=\sigma =0$ in the Lévy-Khintchine formula and the Lévy measure is $m(d x)=c|x|^{-1-\alpha } d x$. Show that if $a>0$ and $Y_{t}=a^{1 / \alpha } X_{a t}$, then $Y$ has the same law as $X$. The process $X$ is known as a symmetric stable process of index $\alpha$.

Suppose $W_{t}=\left(W_{t}^{1}, W_{t}^{2}\right)$ is a two-dimensional Brownian motion started at 0 . Let $\tau_{s}=$ $\inf \left\{ t>0: W_{t}^{1}>s\right\}$. Prove that $W_{\tau_{t}}^{2}$ is a Lévy process and determine the Lévy measure.

Hint: Use scaling to make a guess.

Let $W$ be a one-dimensional Brownian motion and let $L^{0}$ be the local time at 0 . Let $T_{t}$ be the inverse of $L^{0}$, that is, $T_{t}=\inf \left\{ s: L_{s}^{0} \geq t\right\}$. Show $T_{t}$ is a Lévy process and determine the Lévy measure.

Hint: Use scaling to get started.

Let $W_{t}$ be a one-dimensional Brownian motion, $L_{t}^{y}$ the local time at level $y$, and $T_{t}$ the inverse local time at 0 , that is, $T_{t}=\inf \left\{ s: L_{s}^{0} \geq t\right\}$. Let $x>0$ be fixed. Prove that $L_{T_{t}}^{x}$ is a Lévy process.

Let $X$ be a Lévy process with Lévy measure $m$. Prove that if $A$ and $B$ are disjoint closed sets, then

for each $x$, where $B-y=\left\{ z-y: z \in B\right\}$. This is the Lévy system formula in the case of Lévy processes. There is an analogous formula for Hunt processes.

A stable subordinator $X$ of order $\alpha \in (0,1)$ is a Lévy process whose characteristic function is given by (42.6), where $b=\sigma^{2}=0$ and $m(d x)=c \; \mathbb {1}_{(x>0)}|x|^{-\alpha -1} d x$. Suppose $X$ is a stable subordinator of index $\alpha$ and $W$ is a Brownian motion. Show that, up to a deterministic time change, the process $Z_{t}=W_{X_{t}}$ is a symmetric stable process of index $2 \alpha$.

Hint: Start by using scaling.

Let $Z_{t}$ be a symmetric stable process of order $\alpha \in (0,2)$. Show that if $\varepsilon >0$, then