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I need help. I'm studying Lévy processes and one of the examples is the inverse gaussian process.

Let $(B_t)_{t\geq 0}$ a Brownian motion and define the first passage time

$\tau_s=inf\{t\geq 0: B_t+ct>s\}$.

Then $\tau=(\tau_s)_{s>0}$ is a Lévy process and $\mathbb{E}\left[e^{iz\tau_s}\right]=exp(s\sqrt{c^2-2iz}-c)$.

So my questions are:

  1. What is the the Lévy Khintchine representation, i.e. what is the generating triplet $(a,b,\nu)$ of this proces?
  2. What is the Lévy-Ito decomposition?

Thank you

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We follow the presentation from : "AN INTRODUCTION TO LEVY PROCESSES " ERIK BAURDOUX AND ANTONIS PAPAPANTOLEON

For NIG with chf

$$\phi_{NIG}(u)=e^{iu\mu}\frac{exp(\delta \sqrt{\alpha^{2}-\beta^{2}})}{exp(\delta\sqrt{\alpha^{2}-(\beta+iu)^{2}})},$$

they prove that the Lévy-Ito decomposition is

$$L_{t}=tE[L_{1}]+\int_{0}^{t}\int_{\mathbb{R}}x(\mu^{L}-\nu^{NIG})(ds,dx)$$ where $\mu^{L}$ is a Poisson process and $$\nu^{NIG}(dx)=e^{\beta x}\frac{\delta\alpha}{\pi |x|}K_{1}(\alpha |x|)dx,$$ for $K_{1}$ denotes the Bessel function of the third kind with index $1$. The Lévy-triplet is $(E[L_{1}],0,\nu^{NIG})$, which also gives the Lévy-Khintchine representation.

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