# 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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### Show that $\int_{\mathbb R^p} x \nu(dx)=0$ in a question related to a certain convergence of measures

Consider the sequence of estochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{I}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...

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### Supremum process of a Cauchy RV

I've asked the same question on stats.stackexchange a week ago to no avail, so here we go again:
Suppose $X_i$ are $\mathrm{Cauchy}(0,~\gamma)$ IID RV's. Does an expression exist for the CDF of the ...

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### A question related to the jumps of a Levy process

The Lévy–Khintchine formula says that any Lévy process, $X=(X(t), t \geq 0)$, has a specific form for its characteristic function. More precisely, for all $t \geq 0$, $u \in \mathbb R^d$:
$$
\mathbb{E}...

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### Stationary Distribution of Langevin Dynamics driven by Lévy Process

Let $f\geq 0$ be a Lipschitz function and let $(L_t)_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$, possibly multivariate). Consider the process given by $$dX_t=-\nabla f(X_t)dt+\...

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### Step in the derivation of the total idle time distribution of an M/G/1 queue

I'm trying to work my way through the proof of Thm. 1.11 in Kyprianou's Introductory Lectures on Fluctuations of Levy Processes with Applications but really struggle to understand the following step. ...

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### A Lévy process is a semimartingale proof

I have to prove that a Lévy process is a semimartingale.
In general we say that $X$ is a semimartingale if it is an adapted process such that, for each
$t ≥ 0$,
$$X (t) = X (0) + M(t) + C(t)$$
where $...

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### The inverse gaussian process

I need help. I'm studying Lévy processes and one of the examples is the inverse gaussian process.
Let $(B_t)_{t\geq 0}$ a Brownian motion and define the first passage time
$\tau_s=inf\{t\geq 0: B_t+ct&...

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### Probability that a Lévy process "closely" follows a predefined trajectory

For a Brownian motion $(B_t)_{t\geq 0}$ it is well-known [Thm 38, David Freedman, Brownian motion and diffusion], that if $f:[0,1] \to \Bbb R$ is a continuous function with $f(0)=0$ then for $\...

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### The Lévy process jumps

I have two questions.
Let $(X_t)_{t\geq 0}$ be a Lévy process with Lévy measure $\nu$. The jump process $\Delta X=\left(\Delta X_t\right)_{t\geq 0}$ is defined by
$\Delta X_t=X_t-X_{t-}$, for every $t\...

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### How to show that $\int x \,d\nu = 0$ using a pseudo-weak convergence of measures?

I have a sequence of $p$-dimensional infinitely divisible random vectors $S_n'$, such that $S_n' \Longrightarrow X$, as $n \to \infty$.
Suppose the following assumptions
The characteristic functions ...

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### Interpretation of the Lévy measure of an infinitely divisible random vector

We know that a random vector $X$ is infinitely divisible (ID) if for all $n \in \mathbb N$, there exist $X_1^n,..., X_{n}^n$ i.i.d. random vectors such that:
\begin{equation}
X = X_1^n + ...+ X_n^...

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### Exceedance distribution of Levy process

Consider a Levy process $L(t)$ with linear drift $-1$, no Brownian motion component, and Poisson jumps at rate 2 with size Uniform($0, 1$), and with $L(0)=0$. This process has zero mean drift.
Let $\...

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### Lévy measure and jump behaviour of the corresponding Lévy process

Let $(X_t)_{t \ge 0}$ be a Lévy process on $\mathbb R$ with Lévy measure $\nu$.
Define the jump counting measure $N(t, A) = \lvert\{s \in [0, t] \mathrel: \Delta X_s \in A\}\rvert$
where $\Delta X_s$ ...

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### Approximation of a random sum of random variables (infinitely divisible distribution) by a triangular array

We know that a Poisson distribution can be approximated by a binomial distribution. More exactly, let $(X_{jn})_{1\leq j \leq n}$ be a i.i.d. triangular array such that
$$P[X_{jn}= 1 ] = p_n = 1- P[X_{...

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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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### Blumenthal 0-1 law

Let us define the following two stopping time $\tau_B=\inf\{t\geq 0: X_t\in B\}, \tau'_B=\inf\{t> 0: X_t\in B\}$, where $\tau_B$ is entrance time and $\tau'_B$ is hitting time. It is clear $\tau_B=\...

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### How do we need to argue in this step of the Itō-Lévy-Khintchine decomposition?

Let
$E$ be a $\mathbb R$-Banach space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$-...

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### Proof of the Lévy–Itō decomposition in this paper

Let
$E$ be a normed $\mathbb R$-vector space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$...

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### Expectation of killed subordinator at first-passage time

I am reading Fluctuations of Levy Processes with Applications by A.E. Kyprianou and I am having struggles understanding a part in the proof of theorem 5.6. Let $Y$ be a subordinator and $\mathbf{e}$ ...

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### Characterization of Poisson random measure in terms of Laplace transform

Let $(E,\mathcal E)$ be a measurable space and $\mu$ be a measure on $(E,\mathcal E)$.
A random measure $\pi$ on $(E,\mathcal E)$ is called Poisson with intensity $\mu$ if
$\pi(B)\sim\operatorname{...

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### What is the sufficient and necessary condition for Blumenthal-Gettor index = 0?

This question comes from the following paper 1961(Blumenthal)
Let us consider a Levy process $X$ whose Levy triplet is $(a,s,\nu)$. According the above paper, Blumenthal-Gettor index is given by $$\...

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### Is a Levy diffusion square integrable with respect to the associated compensated Poisson measure?

Let $X_t$ be a one dimensional Levy diffusion of the form
$$dX_t = \mu(t, X_t) \, dt + \sigma(t, X_t) \, dW_t + \int_{\mathbb R} \, c(t, z) \, \overline N (dt, dz)$$
with $c > -1 + \delta$ for some ...

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### Existence and uniqueness for Levy BSDE with random terminal time

Consider the following setup. Suppose we have:
$(\Omega, \mathcal F, \mathbb P)$ a probability space,
$\mu, \sigma: \mathbb R_+ \to \mathbb R$ Lipschitz continuous functions with $\sigma(t)
\neq 0$ ...

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### Can we show that the characteristic function of an infinitely divisible probability measure has no zeros

Let $E$ be a normed $\mathbb R$-vector space, $\mu$ be a probability measure on $\mathcal B(E)$ and $\varphi_\mu$ denote the characteristic function$^1$ of $\mu$.
Assume $\mu$ is infinitely divisible, ...

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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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### How can we show this estimate for the convolution of two probability measures?

Let $(\delta_k)_{k\in\mathbb N}\subseteq(0,\infty)$ be nonincreasing with $\delta_k\xrightarrow{k\to\infty}0$ and $(\varepsilon_k)_{k\in\mathbb N}\subseteq(0,\infty)$ with $\sum_{k\in\mathbb N}\...

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### If $(\exp(\mu_n))_{n\in\mathbb N}$ is weakly convergent, is the normalized sequence convergent as well?

Let $E$ be a metric space and $\mathcal M(E)$ denoote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$.
I would like to know ...

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### Invariant measures of Levy S.D.Es

Suppose we call a real valued stochastic process $\{Z_t\}$ to be distributed as ${\cal S}\alpha{\cal S}(\sigma)$ if each of the characteristic functions is $\phi_{Z_t}(u) = \exp\left\{-t\vert \sigma u ...

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### Characterization of the generator of a Lévy process using martingale problems

Let $(X_t)_{t\ge0}$ be a real-valued Lévy process. Note that $$\mu_t:=\mathcal L(X_t)\;\;\;\text{for }t\ge0$$ is a continuous convolution semigroup$^1$. Let $$\tau_x:\mathbb R\to\mathbb R\;,\;\;\;y\...

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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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### Existence of a distinguished continuous version of the logarithm of a continuous function

Let $E$ be a $\mathbb R$-Banach space and $\varphi\in C^0(E,\mathbb C\setminus\{0\})$ with $\varphi(0)=1$.
I want to show that there is an unique $\psi\in C^0(E,\mathbb C)$ with $\psi(0)=0$ and $$\...

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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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### If $\mu$ is an infinitely divisible probability measure on $[0,\infty)$, then the Lévy measure of $\mu$ is the vague limit of $n\mu^{*1/n}$

If $\nu$ is a finite measure on $(\mathbb R,\mathcal B(\mathbb R))$, let $\nu^{\ast k}$ denote the $k$-fold convolution¹ of $\nu$ with itself for $k\in\mathbb N_0$, $$\exp(\nu)\mathrel{:=}\sum_{k=0}^\...

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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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### If $L_t=\sum_{i=1}^{N_t}Y_i$ is a compound Poisson process, then $\left|\left\{s\in[0,t]:\Delta L_s\in B\right\}\right|=\sum_{i=1}^{N_t}1_B(Y_i)$

Let $H$ be a $\mathbb R$-Hilbert space, $\mu$ be a finite measure on $\mathcal B(H)$ with $\mu(\{0\})=0$ and $(L_t)_{t\ge0}$ be a $H$-valued càdlàg Lévy process on a probability space $(\Omega,\...

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### An integral involving Levy process with no positive jumps

Let $L_t$ be a Levy process that has no positive jumps, but is not strictly decreasing, i.e
$$
L_t = \gamma t + \sigma B_t + J_t,
$$
where $B_t$ is a Brownian motion, $J_t$ is a pure jump process with ...

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### Existence of the differential entropy for infinitely divisible laws

Let $X$ be an absolutely continuous (i.e. its law is absolutely continuous with respect to the Lebesgue measure) random variable with probability density $p$. Its differential entropy is given by
$$h(...

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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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### Identity for stable Lévy subordinator

I want a proof or a reference for the identity
$$
\int_0^\infty \frac{s^{n-1}}{\Gamma(n)} p_\beta(s,x)\,ds =\frac{x^{n\beta-1}}{\Gamma(\beta n)},\quad x>0, \,n\in\mathbb N,
$$
where $x\mapsto p_\...

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### Probability of exiting on the boundary for a monotone Lévy-type process

Let the continuous function $\ell:\mathbb R \times(0,\infty)\to[0,\infty)$ be a Lévy-type kernel, such that
$$
\sup_{x}\int_0^\infty \min\{1,y\}\ell( x, y)\,dy<\infty,
$$
and suppose that $\...

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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 ...