# Why $\lambda\equiv\nu(\mathbb{R})$ for compound poisson process?

I've seen this notation $\lambda\equiv\nu(\mathbb{R})$ in the book of Tankov and Cont for compound poisson process. I thought before that $\lambda$ (jump intensity) can be choosen independently of jump measure $\nu$. What does this notation imply?

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## 1 Answer

Jump intensity can be chosen independently of jump measure

This looks like an abuse of terminology in this context. You should clearly distinguish between the jump measure and the normalized jump measure.

If the total jump intensity is finite, it is sometimes convenient to think of a normalized (probability) measure $\nu / \nu(\mathbb{R})$, which represents the conditional distribution of an individual jump. In this case this is true, we can choose $\lambda := \nu(\mathbb{R})$ and the normalized measure $\nu / \nu(\mathbb{R})$ independently. However, in other situations it is more natural not to normalize $\nu$ - and it is indeed impossible if $\nu(\mathbb{R}) = \infty$. So, shortly speaking, this is just the definition of $\lambda$.

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