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Assume you have a Lévy process X.

Let $N(t,A)$ be defined as the number of jumps in the interval $(0,t]$, such that the jumps size $\Delta X_s \in A$.

It can be shown that if $0 \ne \bar{A}$, then $N(t,A)$ is a Poisson process with intensity $\nu(A)=E(N(1,A))$.

Hence it is easy to see that for these A, we have that $\tilde{N}(t,A)=N(t,A)-t\nu(A)$ is a martingale.

But I have a problem with $\tilde{N}(t,A)$ , when the closure of $A$ might contain $0$. Do we have that $\tilde{N}(t,A)$ is a martingale then, or may it be in some cases, and maybe not in other cases ?

One of my problems is the extension of $\tilde{N}(t,A)$ when $A$ is not bounded from below, because then $N(t,A)$ and $\nu(A)$ may both be infinite ?

In Protter's proof of the Lévy-Itô decomposition he ends up with the integral $\int_{|x|<1}x \tilde{N}(t,dx)$ as an $L^2$-limit of martingales of the form $\int_{\epsilon_n\le|x|<1}x \tilde{N}(t,dx)$. But has he then extended $\tilde{N}(t,A)$?

I guess I am also wondering if $\tilde{N}(t,A)$ is always defined ? It is defined for sets $A$ with closure not containing 0, but is it also defined for all sets $A$, or can it be extended to give meaning for all sets $A$ ?

This problem comes up when I want to do stochastic integration with respect to this measure, and integrate functions like $\int_0^T\int_Ef(t,x)(\omega)\tilde{N}(dt,dx)$. In order to do Itô-calculus we need that $\tilde{N}$ is a "martingale-valued measure"(in the most general case we need it to be a semimartingale I guess(Protter), but a theory that I am reading only uses martingale valued measures).

Did
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1 Answers1

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It seems that when we talk about Poisson random measure, we often assume that the characteristic measure $\nu$ is $\sigma$-finite. Then the random measure $\tilde{N}(t,dx)$ is also $\sigma$-finite. Of course, $\tilde{N}(t,A)$ is not a martingale if $\nu(A)=\infty$. To define the stochastic integral $$\int^T_0 f(t,x)\tilde{N}(dt,A),$$we first consider the case that $f$ is bounded and $\nu(A)<\infty$ and then extend it into more generalized situations. See, e.g., pages 59-62 of Ikeda and Watanabe's Stochastic Differential Equations and Diffusion Processes (North-Holland, 1989).

Tomasz Kania
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