# The inverse gaussian process

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

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.