I am struggling with the derivation of the Lévy-measure of a Gamma-process $X_t$ with law $p_t(x)= \frac{\lambda^{ct}}{\Gamma(ct)}x^{ct-1}e^{-\lambda x}1_{\lbrace x>0 \rbrace }$. The paper I am reading says it is shown "easily" via the characteristic function and using the Levy-Khintchine formula:

$\Phi(s)=E[e^{isX_1}] = exp( ias - \frac{1}{2}\sigma^2s^2 + \int_{\mathbb{R}}(1-e^{isx} - is\mathbb{1}_{|x|<1})d\nu(x))$

I computed via the pdf of the Gamma-distribution $\Phi(s) = E[e^{isX_t}]= (1-\frac{is}{\lambda})^{-ct}$

But how is the Lévy-measure then derived to be $\nu(x) = \frac{ce^{-\lambda s}}{x}1_{\lbrace 1 > 0\rbrace}$?


Hint: Perhaps first switch to the MGF (replacing i*s by s for sufficiently small s). Then take the logarithms of your candidate MGF (first Phi), and that of the Gamma process (second Phi). Differentiate the two expressions, knowing that you can replace integration and differentiation; then integral with respect to the Levy measure is very easy...Compare the two expressions and you are done.


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