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.
19
questions
2
votes
1answer
95 views
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 ...
0
votes
0answers
34 views
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 ...
1
vote
1answer
61 views
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_\...
1
vote
1answer
52 views
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 $\...
0
votes
0answers
50 views
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{ ...
1
vote
1answer
82 views
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 ...
2
votes
1answer
509 views
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 ...
4
votes
1answer
92 views
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:
$$
\mathbb{P}_x(X_{\tau_D^-}\in \partial ...
2
votes
2answers
405 views
Monotone convergence theorem for stochastic integrals
I was wondering if there exists an equivalent of a monotone convergence theorem for stochastic integrals. I looked into plenty of books and papers, but I haven't found anything useful. I would expect ...
3
votes
0answers
75 views
A dependent and discrete version of the Komlós-Major-Tusnády theorem
The well-known Komlós-Major-Tusnády approximation gives sharp speed of convergence of a uniform empirical process to a Brownian bridge. Here I am considering how to approach a similar problem with ...
2
votes
0answers
97 views
Modulus of continuity of Lévy process as jump size tends to zero
While reading Kallenberg's "Foundations of Modern Probability Theory", 2nd edition, the following question regarding an argument in the proof of Lemma 15.19 occurred to me.
Let $X_n(t)$ be a sequence ...
3
votes
0answers
81 views
Orlicz spaces and $\phi$-functions
A $\phi$-function $f$ is usually defined as a continuous function $f=\mathbb R_+ \to \mathbb R_+$ such that:
(1) $f$ is nondecreasing.
(2) $f(0)=0$ and $f(x)>0$ for all $x>0$.
(3) $\lim_{x\to ...
2
votes
1answer
195 views
Itô Formula for Hilbert space-valued Lévy processes
I know there are Itô formulas for cylindrical Brownian motions with values in a Hilbert space and Itô formulas for Lévy processes in $\mathbb{R}^d$. My question is:
does there exist an Itô formula ...
2
votes
1answer
269 views
Version of Donsker-Invariance-Principle
Assume we have a Levy process $(X_t)_{t\geq 0}$ with a finite second moment for all $t>0$. For simplicity, say $\operatorname{Var}\left[X_1\right]=1$. Let $\tilde{X}_t:=X_t-t\cdot E\left[X_1\right]$...
2
votes
2answers
389 views
Lévy measure and Lévy process
A Lévy measure $\nu$ on $\mathbb R^{d}$ is a measure satisfying
$$\nu\{0\} = 0, \ \int_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.$$
A Lévy process can be characterized by triples $(b, A, \...
2
votes
0answers
254 views
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 ...
-1
votes
1answer
130 views
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 ...
3
votes
1answer
124 views
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 ...
2
votes
0answers
223 views
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 ...
