# 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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### Continuous-time random walk on $\mathbb{R}$ that never stays still

Consider a walker on the real line $\mathbb{R}$ and two probability density functions $w$ and $j$ defined over $\mathbb{R}$. A walker starts at $0$ and iterates the following: it samples a waiting ...
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### Poisson point process in polar coordinates

Let $D = \mathbb{R^+} \times (\mathbb{R}\backslash \{0\})$ Let $\mu(dt \times dx)$ be a $\sigma$-finite measure on the Borel $\sigma$-algebra $\sigma(D)$. Let $M(dt \times dx)$ be the Poisson random ...
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### Define the convolution root of probability measures on a measurable group

Let $(G,\mathcal G)$ be a measurable group and $\nu^{\ast k}$ denote the $k$th convolution power of a probability measure $\nu$ on $(G,\mathcal G)$ for $k\in\mathbb N$. Remember that a probability ...
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### Quantiles of a Levy process

Let $X = \{ X_t \in {\bf R}, t \geq 0 \}$ be a 1-dimensional (real) Levy process. Suppose further that the distribution of $X_t$ is not concentrated on a grid. (This forces the distribution of $X_t$ ...
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### Characteristic function and moments

Let $X\in L^1(\Omega)$ and $\phi_X$ the corresponding characteristic function. We know that: $\phi_X$ is $n$ times differentiable (at $u=0$) iff $\mathbb{E}[X^n]<\infty$. (This depends a bit on ...
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### Hitting order of sets by a Lévy process

Let $X$ be a transient Lévy process on $\mathbb R$, and $B\subseteq \mathbb R$ a Borel set with first hitting time $T_B = \inf \left\{t>0 : X_t\in B\right\}$. For Borel $A\subseteq B$, can anything ...
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### Polynomial growth of random walks: critical values?

Consider a sequence of i.i.d. random variables $(X_n)_{n\geq 0}$, and set $S_N = \sum_{n=1}^N X_n$. For which $p> 0$ do we have that \begin{equation} \lim \inf \frac{|S_N|}{N^{1/p}} > 0 \text{ ...
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### Is there a name for the sample variance process of a Lévy process?

Let $X_t$ be a Lévy process which is known to have mean zero and finite variance $t \cdot \sigma^2$, but for which the value of $\sigma^2$ is unknown. How do we estimate $\sigma^2$? One approach would ...
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### Simulation of Lévy walk

I have problems to find out how to do discrete simulation of the Lévy walk. I can sum my doubts in a few questions: According to Wikipedia it seems to me that Lévy flight can be produced just by ...