Let $Y(t)$, $0 \leq t \leq 1$, be a Lévy process. Denote by $\{ \Delta Y_i \}$ the jumps of the process. I would like to show that the set $\{i: |\Delta Y_i| > r\}$ is finite almost surely. However, I do not entirely understand what should be shown and how to approach this problem. […]

Let $(X_t)_{t \geq 0}$ a Lévy process and $\varepsilon>0$. Is there anything known about the asymptotics of the probability $$\mathbb{P}(|X_t| > \varepsilon)$$ as $t \to 0$? Obviously, by the stochastic continuity, this probability converges to $0$ – but how fast? I tried to apply Markov’s inequality (assuming that the corresponding moment exists); then I get […]

In another post an inequality referred to as “Etemadi’s Inequality” is mentioned twice – in the original post as well as in the answer. However, the contexts of usage are such as to raise the question whether the inequality intended by the users (ziT and saz, respectively) is the inequality that goes by the same […]

If one defines a Levy process as a stochastic process $(X_t)_{t\geq0}$ that has independent and stationary increments with (a.s.) cadlag paths (hence a def. withouth stochastic continuity). How can I conclude that this process has a.s. no jumps at fixed times (i.e. show that the proba of a positive jump at time $t$ for $t$ […]

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