Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is $$ \nu \left( dx\right) = A \sum_{n=1} ^{\infty} p^n \delta_{-n}\left( dx \right) + Bx^{\beta-1}\left( 1+x \right)^{-\alpha -\beta}e^{-\lambda x } \mathbf{1}_{\left ]0,+\infty \right[}\left( x\right)dx.$$

I'd like to know how to show that a price process $S_t = S_0 \exp\left( r t + X_t \right)$ of a risky actif under interest rate $r >0$ is well defined and admits first and second order moments.

Someone could help on it, please?

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    $\begingroup$ This question was cross-posted here and here, where it has been answered. I'm going to close the question on Quant.SE. $\endgroup$ – chrisaycock Feb 24 '13 at 23:50