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Recently (~10 years ago), Kuchler&Tappe have set up a new stochastic process called Bilateral Gamma process. This process is defined through its increments:

$$\forall t\geq s, X_t-X_s\sim \Gamma_{BG}(\alpha_+(t-s), \lambda_+, \alpha_-(t-s), \lambda_-)$$

where:

$$\Gamma_{BG}(\alpha_+, \lambda_+, \alpha_-, \lambda_-) =\Gamma(\alpha_+, \lambda_+)*\Gamma( \alpha_-, \lambda_-). $$

My question is maybe naive but, from this definition: how can we see that $X$ is a pure jump process ?

Thank you very much !

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    $\begingroup$ By construction BG is event based from the get go. There are assumed to be two streams of events, the "up events" which cause the price to rise a smidgeon and the "down events" which cause the price to drop a wee bit. If no event then the price does not change. From this they deduced the equations you mention, but I don't think just looking at the equations we can see how the thing works, it is more the other way around. $\endgroup$
    – nbbo2
    Dec 4, 2023 at 12:58
  • $\begingroup$ OK, thank you very much @nbbo2 ! $\endgroup$
    – NancyBoy
    Dec 4, 2023 at 20:05

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