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I am currently using Appelbaum but it does not go into too much detail how we deal with the part with small jumps.

Can someone please recommend a good text book? I have Bertoin at my disposal, but I do not see to find anything in the index.

Also an intro of martingale spaces is appreciated. Appelbaum mentioned Rudin.(which rudin's book is it?) Is there any other?

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up vote 1 down vote accepted

Lévy Processes and Infinitely Divisible Distributions by Ken-Iti Sato is a great source to Lévy processes and characterizations of such. It describes both the Lévy-Khintchine formula and the Lévy-Itô decomposition.

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