# If $\pi_t$ is the jump measure of a Lévy process and $B_1,\ldots,B_k$ are disjoint, then $\pi_t(B_1),\pi_t(B_k)$ are independent

Let

• $$E$$ be a normed $$\mathbb R$$-vector space;
• $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space;
• $$(\mathcal F_t)_{t\ge0}$$ be a filtration on $$(\Omega,\mathcal A)$$;
• $$(X_t)_{t\ge0}$$ be an $$E$$-valued càdlàg $$(\mathcal F_t)_{t\ge0}$$-Lévy process on $$(\Omega,\mathcal A,\operatorname P)$$;
• $$\tau^B_0:=0$$ and $$\tau^B_n:=\inf\underbrace{\left\{t>\tau^B_{n-1}:\Delta X_t\in B\right\}}_{=:\:I^B_n}$$ for $$n\in\mathbb N$$ and $$B\in\mathcal B(E)$$;
• $$\pi_t(B):=\sum_{\substack{s\in[0,\:t]\\\Delta X_s\ne0}}1_B\left(\Delta X_s(\omega)\right)=\left|\left\{s\in(0,t]:0\ne\Delta X_s\in B\right\}\right|$$ for $$B\in\mathcal B(E)$$ and $$t\ge0$$.

Let $$k\in\mathbb N$$ and $$B_1,\ldots,B_k\in\mathcal B(E)$$ be disjoint. How can we show that $$\left(\pi_1(B_1),\ldots,\pi_k(B_k)\right)$$ is independent? (Maybe we need to assume that $$0\not\in\overline{B_i}$$ for all $$i\in\{1,\ldots,k\}$$.)

Since $$B_1,\ldots,B_k\in\mathcal B(E)$$ are disjoint, it clearly holds $$I^{B_i}_m\cap I^{B_j}_n=\emptyset\tag1$$ for all $$i,j\in\{1,\ldots,k\}$$ and $$m,n\in\mathbb N$$. But how do we need to proceed?

I don't know if it is useful, but we may note the following: If $$B\in\mathcal B(E)$$ with $$0\not\in\mathcal B(E)$$, then $$\pi_t(B)<\infty$$ and $$\pi_t(B)=\sum_{n\in\mathbb N}1_{[0,\:t]}\left(\tau^B_n\right)\tag1$$ (since $$\tau^B_n\xrightarrow{n\to\infty}\infty$$, the sum on the right-hand side is finite) for all $$t\ge0$$. Moreover, we can show that $$\tau^B_1$$ is exponentially distributed and if we define $$\tilde\Omega:=\left\{\tau^B_1<\infty\right\}$$, $$\tilde{\operatorname P}[A]:=\operatorname P\left[A\mid\tilde\Omega\right]$$ for $$A\in\tilde{\mathcal A}:=\left.\mathcal A\right|_{\tilde\Omega}$$ and $$\tilde\tau^B_n:=\left.\tau^B_n\right|_{\tilde\Omega}$$ for $$n\in\mathbb N$$, then $$\left(\tilde\tau^B_n-\tilde\tau^B_{n-1}\right)_{n\in\mathbb N}$$ is independent and identically distributed on $$\left(\tilde\Omega,\tilde{\mathcal A},\tilde{\operatorname P}\right)$$.

• What is $\Delta X_t$?
– 温泽海
Commented Apr 1, 2022 at 12:12