# Questions tagged [levy-processes]

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

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### Does partial averaging allow moving increments in and out of an expectation?

Given a Levy process $X$ at different points in time $s$ and $t$, and if I have an expression like this: $$\mathbb{E}[X_t \cdot \mathbb{E}[X_s]]$$ I want to know if I can use partial averaging to ...
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### How to calculate expected value of integral?

How to calculate $E \big[(\int ^{t} _{-\infty} e^{\lambda u} d\tilde{L_\alpha}(u))^A (\int ^{t+h} _{-\infty} e^{\lambda u} d\tilde{L_\alpha}(u))^B\big]$, where \begin{align} \tilde{L}_\alpha (t) = \...
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### Inverting a Laplace transform (for a Lévy process)

Let $\psi(\theta) = c\theta + \frac{\sigma^{2}}{2}\theta^{2} - \frac{\lambda\theta}{\alpha + \theta}.$ For those who are wondering where this function comes from, $\psi$ is the Laplace exponent for a ...
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### Simulating a Cauchy process

My question is how to I simulate sample paths from a Cauchy process? I know this can be done using two Brownian motions, but I am trying to do it from the basics. It's known that if we have a Levy ...
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### The Levy measure of a multivariate alpha-stable Levy motion

I am having difficulties to understand the form of the Levy measure of the multivariate Levy-stable motion. Let me start by defining the one dimensional motion in order to clarify my question. The ...
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### generate random numbers of Lévy distribution

I am going to generate a random step length which is drawn from a Lévy distribution $$\textrm{Lévy } \sim u= t^{-\lambda},\;\; 1<λ ≤3$$ How to (in computer) generate $u$ of such probability? ...
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### Laplace exponent of a standard $\alpha-$stable subordinator

I'm trying to calculate the Laplace exponent of a standar $\alpha-$stable subordinator. An $\alpha-$stable subordinator has Lévy measure $\frac{c}{x^{1+\alpha}}dx,$ where $\alpha\in(0,1)$ and $c$ is ...
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### Does the Levy process stay in any open ball at any fixed time with positive probability?

Question: Let $X$ be a $d$-dimensional Levy process. Then for every $t>0$ and $a>0$, \begin{equation}\tag{1} \mathbf P\{|X_t|<a\}>0\ ? \end{equation} The question comes from the proof ...
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### Generalizing a proof for the density of stopped subordinators

Is it possible to generalize the proof of the statement below to general Lévy kernels $\rho(r)dr$ or even Lévy-type kernels $\rho(x,r)dr$ (in the sense of Lévy Matters III by Böttcher et al.)? ...
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### (In)dependence of solutions to certain SDEs

Consider a Levy process $L$ in $\mathbb R^d$ written in the Levy-Kchinchine decomosition as $$L(t)=bt + W(t) + Z(t),$$ where $bt$ is the drift part, $W(t)$ is the Wiener part and $Z(t)$ is the jump ...