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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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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Investigating the properties of a stochastic process with infinitely divisible finite distributions

We know that if $X = [X_t , t \geq 0 ]$ is a Levy process, then any marginal distribution has the charasteristic function - ch. f. of $X_t$ - given by $$\varphi_t(r)=e^{t \phi(r)}$$ whith: $$\phi(r) =...
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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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Levy Ito decomposition

I was having some difficulties understanding the Levy-Ito decomposition, so I summarised some results into one result. Could anyone please tell me if the following makes sense, i.e. is mathematically ...
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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 ...
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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 ...
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escape points of Levy processes

Suppose $D$ is a domain in $\mathbb{R}^d$, $x\in D$. $X_t$ is a Levy process with Lévy triplet ${\displaystyle (b,0,\mu )}$ . Can one give a brief proof for: $$ \mathbb{P}_x(X_{\tau_D^-}\in \partial ...
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2 votes
2 answers
637 views

Monotone convergence theorem for stochastic integrals

I was wondering if there exists an equivalent of a monotone convergence theorem for stochastic integrals. I looked into plenty of books and papers, but I haven't found anything useful. I would expect ...
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A dependent and discrete version of the Komlós-Major-Tusnády theorem

The well-known Komlós-Major-Tusnády approximation gives sharp speed of convergence of a uniform empirical process to a Brownian bridge. Here I am considering how to approach a similar problem with ...
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Modulus of continuity of Lévy process as jump size tends to zero

While reading Kallenberg's "Foundations of Modern Probability Theory", 2nd edition, the following question regarding an argument in the proof of Lemma 15.19 occurred to me. Let $X_n(t)$ be a sequence ...
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Orlicz spaces and $\phi$-functions

A $\phi$-function $f$ is usually defined as a continuous function $f=\mathbb R_+ \to \mathbb R_+$ such that: (1) $f$ is nondecreasing. (2) $f(0)=0$ and $f(x)>0$ for all $x>0$. (3) $\lim_{x\to ...
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Itô Formula for Hilbert space-valued Lévy processes

I know there are Itô formulas for cylindrical Brownian motions with values in a Hilbert space and Itô formulas for Lévy processes in $\mathbb{R}^d$. My question is: does there exist an Itô formula ...
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Version of Donsker-Invariance-Principle

Assume we have a Levy process $(X_t)_{t\geq 0}$ with a finite second moment for all $t>0$. For simplicity, say $\operatorname{Var}\left[X_1\right]=1$. Let $\tilde{X}_t:=X_t-t\cdot E\left[X_1\right]$...
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Lévy measure and Lévy process

A Lévy measure $\nu$ on $\mathbb R^{d}$ is a measure satisfying $$\nu\{0\} = 0, \ \int_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.$$ A Lévy process can be characterized by triples $(b, A, \...
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Radon-Nikodym for continuous time processes

Likelihood theory for statistical inference concerning stochastic processes in continuous time are well used. How ever i've found no real literature concerning the fundamentals. What is know from ...
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Definition: Grigelionis Process?ch [closed]

Background I've been reading this article and it keeps referring to "Grigelionis processes", which apparently generalize Levy processes. However the paper does not define these object clearly and ...
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Besov regularity of càdlàg functions?

Let $D(\mathbb{R})$ be the space of functions from $\mathbb{R}$ to $\mathbb{R}$ that are right continuous with left limits (also referred to as càdlàg functions). $D(\mathbb{R})$ is often called the ...
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What is the Blumenthal-Getoor index of Student's distributions?

For infinitely divisible random variables, Blumenthal and Getoor introduced in [1] an index that allow to study for instance the local Hölder regularity of Lévy processes. For a symmetric infinitely ...
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