Let $N$ be a Poisson random measure on $(\mathbf R^d\setminus\{0\})\times \mathbf R_+$ with intensity measure $\nu\otimes dt$. Here $dt$ is the Lebesgue measure on $\mathbf R_+$. Let $\tilde N$ be the associated compensated Poisson random measure. Namely, $\tilde N(A\times [0,t]) = N(A\times [0,t]) - t\nu(A)$ for $A\in\mathcal B(\mathbf R^d\setminus\{0\})$ and $t>0$. Given a constant $r>1$, we consider the following pure jump Levy process \begin{equation}\tag{1} L_t = \int_0^t \int_{0<|x|\le r} x \tilde N(dx,dt) + \int_0^t \int_{|x|>r} x N(dx,dt). \end{equation}
Now there are two ways to identify the Levy triplet of $L=\{L_t\}_{t\ge0}$ associated to the indicate function $\mathbf 1_{\{0<|x|\le 1\}}$. But what I am confused is I get two different results!
Solution 1. Rewrite (1) to \begin{equation} L_t = \int_0^t \int_{0<|x|\le 1} x \tilde N(dx,dt) + \int_0^t \int_{|x|>1} x N(dx,dt) - t \int_{1<|x|\le r} x\nu(dx). \end{equation} Hence, the Levy triplet of $L$ associated to $\mathbf 1_{\{0<|x|\le 1\}}$ is \begin{equation} \left( - \int_{1<|x|\le r} x\nu(dx), 0, \nu\right). \end{equation}
Solution 1. Define a new Poisson random measure $M$ by $M(A\times [0,t]) = rN(rA\times [0,t])$, where $rA:=\{rx: x\in A\}$. Then $M$ has intensity $\mu(A) = r\nu(rA)$ and associated compensated Poisson random measure $\tilde M(A\times [0,t]) = r\tilde N(rA\times [0,t])$. Use the change of variable $x = ry$ to rewrite (1) to \begin{equation} \begin{split} L_t &= \int_0^t \int_{0<|y|\le 1} ry \tilde N(rdy,dt) + \int_0^t \int_{|y|>1} ry N(rdy,dt) \\ &= \int_0^t \int_{0<|y|\le 1} y \tilde M(dy,dt) + \int_0^t \int_{|y|>1} y M(dy,dt). \end{split} \end{equation} Hence, the Levy triplet of $L$ associated to $\mathbf 1_{\{0<|x|\le 1\}}$ is \begin{equation} ( 0, 0, r\nu(r\cdot)). \end{equation}
I cannot find anything wrong in the two solutions. But the results are contradictory, there must be somewhere going wrong. I am confused then... Any hints or comments will be appreciated. TIA...
EDIT: Assume $r$ is an integer so that the new measure $M$ is exactly integer-valued and hence a Poisson random measure?