# Questions tagged [levy-processes]

Theory and applications of Lévy processes (stochastic processes with stationary and independent increments): e.g. path properties, stochastic differential equations driven by jump-type processes, fluctuation theory of Lévy processes, queuing theory.

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### Compound poisson processes (Construction)

I'm studying compound poisson processes and in "Levy processes and infinitely divisible distributions" there is this theorem (4.3) : To proof that it is a Levy process we have to show that: ...
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### A Levy process is a.s. continuous

I have to proof this: If is a Levy process then for each the sample path is, with probability 1, continuous as s=t. This is the proof: I don't understand the conclusion. Can someone explain to me ...
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### Second moment of stochastic integral wrt Levy Processes

I have a question about the second moment of the integral wrt Levy Processes. Let Z a Levy processe. We know that: And a few page later is written that by differentiation of the characteristic ...
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### A question about the proof of the Levy-Khintchine representation Theorem

I'm studying Infinitely Divisible random variables using this Lecture Notes. And I have a question that is driving me crazy. In the proof of the "only if" part of the Levy-Khintchine ...
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### Is the limit of compound Poisson random variables a compound Poisson r.v.?

Let $Y$ be an infinitely divisible (I.D.) random variable. Let $\nu$ be any measure not necessarily finite: $\nu(\mathbb R)\leq \infty$. Suppose that $Y \sim (0, \nu,0)_0$ according to the notation on ...
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### What are the Lévy processes with specific increments?

It is known that the increment of the Wiener process $W$ is drawn from a Gaussian distribution, i.e. $\Delta W \sim \mathcal{N}(0, \delta t)$. I wonder what are the Lévy processes with increments from ...
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### Is this statement of the Lévy–Khintchine formula ill-posed?

Please take a look at the following statement of the Lévy–Khintchine formula given in Probability Theory: A Comprehensive Course (2nd edition)$^1$: Am I missing something or is this an ill-posed ...
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### Existence of unique convolution semigroups of probability measures on more general spaces then $\mathbb R^d$

Let $E$ be a $\mathbb R$-Banach space, $\mathcal M_1(E)$ (resp. $\mathcal M_1^\infty(E)$) denote the set of probability measures (resp. infinitely divisible probability measures) on $E$, $\varphi_\mu$ ...
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### Is the distribution of a Banach space valued Lévy process uniquely determined by its characteristic function?

Let $E$ be a $\mathbb R$-Banach space. Remember that if $\mu$ is a finite measure on $\mathcal B(E)$ then $$\Phi_\mu:E'\to\mathbb C\;,\;\;\;\varphi\mapsto\int\mu({\rm d}x)e^{{\rm i}\varphi(x)}$$ is ...
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