# Questions tagged [levy-processes]

Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.

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### Level-crossing time of Levy process is larger than a fixed time with positive probability

Let $(X_t)_{t\geq 0}$ be a Lévy process on $\Bbb R$ with characteristic triple $(a, \sigma , \pi)$. For a level $K>0$ define $$\tau_K := \inf\{t>0 : X_t > K\}$$ Question: When does it hold ...
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### Domain of the infinitesimal generator of subordinators

Let $(X_t)$ be a subordinator (not killed). Since $(X_t)$ is a non-decreasing Levy process, we have the corresponding infinitesimal generator: \begin{equation} Af(x)=\delta f'(x)+\int_{0}^{\infty}(f(x+...
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### What is the cardinality of the number of jumps in a Lévy process on a given time interval?

If a Lévy process can be defined as a right continuous process with existing left limits, then according to the following linked theorem (Prove that the number of jump discontinuities is countable for ...
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### Compound Poisson Process infinitely many jumps?

When I look at the following compound Poisson Process, where $(N_{t})_{t\geq0}$ is a Poisson Process with Parameter $\lambda$ and $\xi_{i}$ are $\mathbb{R}$ valued i.i.d random Variables independent ...
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### Lévy characterization of Hilbert-space-valued Wiener process

Let $H$ be a $\mathbb R$-Hilbert space (assume $H=\mathbb R^d$ for some $d\in\mathbb N$, if this is helpful for you to understand the following) and $(X_t)_{t\ge0}$ be an $H$-valued continuous Lévy ...
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### Show that every continuous Lévy process is a Wiener process

Let $H$ be a $\mathbb R$-Hilbert space (assume $H=\mathbb R^d$ for some $d\in\mathbb N$, if this is helpful for you to understand the following) and $(X_t)_{t\ge0}$ be an $H$-valued continuous Lévy ...
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