I would appreciate if someone helps me with introducing a reference explaining the path properties of Levy Processes. In other words, I want to know a good interpretation of the Levy - Khintchine formula.
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1$\begingroup$ Rogers & Williams's 2-volume book would be my first port of call, though I admit I can't remember precisely what they say about Levy-Khintchine $\endgroup$– Yemon ChoiCommented Jun 8, 2012 at 6:47
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$\begingroup$ Either Applebaum's or Kyprianou's book should help you: amazon.com/Processes-Stochastic-Calculus-Cambridge-Mathematics/… amazon.com/… Also Kallemberg's "Foundations of Modern Probability" devotes a chapter to infinitely divisible random variables (i.e Levy Kintchine Formula) $\endgroup$– Felipe OlmosCommented Jun 8, 2012 at 10:03
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$\begingroup$ @ Hassan : I think that Sato's book is the best reference on the subject. Best regards $\endgroup$– The BridgeCommented Jun 8, 2012 at 19:05
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$\begingroup$ @ Yemon Choi, Felipe Olmos, The Bridge: Thanks guys for your help. I already study "Levy Processes & Stochastic Calculus by D. Applebaum", you know this book goes more in details and theory behind Levy - Khintchine... but I'm looking for a reference explaining the properties of paths. For example, I'm working on some application of Levy processes in Finance and insurance so i need some properties of Levy paths... $\endgroup$– user23254Commented Jun 12, 2012 at 4:55
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$\begingroup$ @Hassan: please register so you can merge duplicate accounts $\endgroup$– Yemon ChoiCommented Jun 13, 2012 at 1:43
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Phil Protter also has a book (I don't recall if it discusses path properties). If you just want more information on the Skorokhod space (cadlag functions), P. Billingsley's convergence of probability measures is a good book. If you want path properties of more general stable processes (which are not necessarily Levy, i.e. may not have stationary and indpt increments) then Samorodnitsky and Taqqu have a chapter on path properties.
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$\begingroup$ @ Paul, Thanks man for your help. I'm gonna take a look at the book by" Taqqu". $\endgroup$– user24404Commented Jun 13, 2012 at 0:58
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$\begingroup$ @Hassan before you delve into that book (ST94), let me reiterate that it considers stable processes generally. What many people refer to as THE stable process is called alpha-stable Levy motion in ST94. That and stable subordinators might possibly be the only Levy processes they consider in their path property analysis (chapters 9-12). $\endgroup$ Commented Jun 15, 2012 at 21:45