Skip to main content

Questions tagged [levy-processes]

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

Filter by
Sorted by
Tagged with
2 votes
1 answer
42 views

Levy Process is a Feller Processs

I want to show that every Levy Process is a Feller Process. Let $ X=(X_t)_{t\geq 0}$ be a Levy Process and $\mu_t(dy):=P\circ X^{-1}(dy)$ . I found a proof where it is shown that the transition ...
kays44's user avatar
  • 71
0 votes
0 answers
36 views

What's so special about the Levy distribution?

The distribution with the PDF below is a particular case of the Levy distribution. $$f(x) = \sqrt{\frac{1}{\pi}}.\frac{e^{-\frac{1}{x}}}{x^{1.5}}$$ Someone said it was their favorite distribution. ...
Rohit Pandey's user avatar
  • 6,943
1 vote
0 answers
41 views

Is a randomly restarted Markov process again Markov?

Let $E$ be a topological space and $(X_t)_{t\ge0}$ be a right-continuous time-homogeneous Markov process with transition semigroup $(P_t)_{t\ge0}$ on a filtered probability space $(\Omega,\mathcal A,(\...
0xbadf00d's user avatar
  • 13.9k
3 votes
2 answers
58 views

What is the transition semigroup of toroidally wrapped Brownian motion?

More generally, let $(B_t)_{t\ge0}$ be an $\mathbb R^d$-valued Lévy process and $W:=\iota(B)$, where $$\iota:\mathbb R^d\to[0,1)^d\;,\;\;\;x\mapsto x-\lfloor x\rfloor$$ (the floor function is applied ...
0xbadf00d's user avatar
  • 13.9k
2 votes
0 answers
23 views

The probability of first passage within finite time

Let $\left(X_t\right)_{t\ge0}$ be a L'evy process with $X_0=0$ which drifts to $-\infty$. Let $T>0$. For $a>0$, define $\tau_a=\inf\{t\ge0: X_t>a\}$. What kind of assumption on $X$ should be ...
user377704's user avatar
0 votes
1 answer
24 views

Relation between the parameters of the stable distribution and the stable process.

For the purpose of this question, the stable distribution with parameters $(\alpha, \beta, c, \mu)$ is the distribution with characteristic function, $$ \phi(z) = \exp\left[iz\mu-|ct|^\alpha\left(1-i\...
Tim Hargreaves's user avatar
1 vote
0 answers
32 views

Geometric Levy motion

Which Levy processes $(L_t)$ admit a geometric motion, i.e., an almost sure global solution of $$ dX_t = X_tdL_t\\ X_0=1 $$ (for some sensible but unspecified notion of "solution")? If the ...
Bananach's user avatar
  • 7,983
3 votes
0 answers
91 views

Is it possible to solve for a function of a differential in an integral equation?

I am attempting to solve an equation for $\nu(x)$. The exact equation is part of my PhD research and is needlessly complex for the core of my issue, I will give a simplified example instead: $\int_{R\...
Flow-Vector's user avatar
0 votes
2 answers
53 views

Non-deterministic bounded Lévy process

Does there exist a non-deterministic $\mathbb{R}$- valued Lévy process $(X_t)_{t \in [0, \infty)}$ such that there exist a $t_0 \in (0, \infty) $ and $R>0$ such that $$ P(0 < X_{t_0} < R)=1 $$...
JackYo's user avatar
  • 179
1 vote
0 answers
41 views

translation invariance of expectation value of hit counting variable for Lévy process

Let $(X_t)_{t \in [0, \infty)}$ a $\mathbb{R}$- valued Markov process, $s, a, u >0$, $I(a) := \{[k \cdot a, (k+1) \cdot a] \ : \ k \in \mathbb{Z} \} $ the family of $a$-integral tiles covering $\...
JackYo's user avatar
  • 179
3 votes
1 answer
82 views

Stochastic continuity in the definition of a Lévy process

A Lévy process is defined as a stochastic process, $X = (X_t)_{t\geq 0}$, with the properties: L1. $X_0 = 0$ a.s. L2. $X$ has independent increments L3. $X$ has stationary increments L4. $X$ is ...
piers's user avatar
  • 31
0 votes
0 answers
16 views

What is the steady state distribution of this Poisson process with non-constant rate?

I am looking for the steady state distribution of the following Poisson process: $$d x(t) = -k_1(x(t)-k_2)dt + k_3dN(t)$$ where $k_1$, $k_2$ and $k_3$ are constants and the rate $\lambda(x)$ of the ...
user1031129's user avatar
0 votes
0 answers
72 views

Help evaluating Gamma integral with negative exponent

The integral is: $$\frac{1}{\Gamma{(1-\alpha)}}\int^\infty_0(1-e^{-\lambda y})e^{-y}y^{-(1+\alpha)} dy$$ where $\alpha\in(0,1)$ and $y,\lambda>0$. I've tried using IBP, in which I get: $$\frac{1}{\...
krill_445's user avatar
0 votes
0 answers
13 views

Product Property for Lévy Processes

The product property for random variables (RVs) [1] (Page 20) provides a way to transform symmetric $\alpha$-stable ($S\alpha S$) random variables to ones with different indices. In particular, let $X$...
Tolga Birdal's user avatar
2 votes
0 answers
70 views

Integral from inverse Gaussian process

By Mathematica I know that $$\frac{1}{\sqrt{2\pi}}\int^{\infty}_{0}(1-e^{i\theta x})x^{-3/2}e^{-\frac{b^{2}x}{2}}dx = -b+\sqrt{b^{2}-2i\theta}$$ but unfortunately I cannot compute it analytically. ...
Mozqrt's user avatar
  • 41
0 votes
1 answer
72 views

Intuitive explanation for Feller processes

We know that a Markov process is a stochastic process in which the future evolution of the system depends solely on its current state and is independent of its past states given the present state. ...
K252's user avatar
  • 53
0 votes
0 answers
36 views

Derivative of the quadratic variation of Levy process

Let $L(t)$ be a n-dimensional Levy process having the decomposition $$ L(t) = \int_{B} x \widetilde{N}(t,dx) $$ where $B=\{ |x|<1 \}$ and $\widetilde{N}(dt,dx) = N(dt,dx) - \nu(dx)dt$ is the ...
MrIncandenza's user avatar
0 votes
0 answers
16 views

Image measure of alpha-stable Levy measure under coordinate projection

Let $\nu$ be a symmetric $\alpha$-stable Levy measure on $\mathbb{R}^d$, $\alpha \in (0,2]$ (if you like, let $d=2$). Since the density of $\nu$ is radial, the image measure of $\nu$ under $x \mapsto \...
PDEprobabilist's user avatar
3 votes
0 answers
68 views

Uniform integrability with respect to a family of measures

Let $(\mu_n)_{n \in \mathbb N}$ be a sequence of Levy measures such that: $$\lim_{n \to \infty}\mu_n(E) = \mu(E), \quad (\forall\,\, E \,\, \mu-\hbox{Continuity set}, 0 \notin \overline{E})$$ where $\...
PSE's user avatar
  • 544
1 vote
1 answer
225 views

Explanation of notation on Levy-Ito decomposition in a paper by El Fatini and Boukanjime?

In El Fatini and Boukanjime "Stochastic analysis of a two delayed epidemic model incorporating Lévy processes with a general non-linear transmission" paper, the first equation of (3) is ...
Math's user avatar
  • 129
0 votes
0 answers
27 views

A Levy process which is not a compound Poisson drifting to $-\infty$

I have read a few books about L'evy processes and tried to find a concrete example such that $\{X_t\}$ is a L'evy process (but not a compound Poisson) drifting to $-\infty$ and $0$ is not regular for $...
user377704's user avatar
1 vote
0 answers
30 views

Are all discontinuous levy processes sparse?

As described by the title, I would like to know if all discontinuous Levy processes (or jump Levy process, such as Poisson Process, Cauchy process, Gamma Process, etc.) are sparse processes? The ...
Username's user avatar
0 votes
1 answer
58 views

Explanation of assumptions on the Levy measure

I have a question about standard assumptions that are made on the Levy measure when defining an infinitely divisible distribution on $\mathbb{R}$. The characteristic function is of the form: $$ \hat{\...
MMM's user avatar
  • 751
1 vote
1 answer
39 views

Condition guaranteeing integrability

Let us consider a Levy process $X_t$ with the characteristic function of the form: $$ \hat{\mu}(t) = \text{exp}\biggl( i\gamma t - \frac{\sigma^2}{2} t^2 + \int_{\mathbb{R}} (e^{itx} - 1 - itx \mathbb{...
MMM's user avatar
  • 751
2 votes
1 answer
212 views

A problem related to the convergence of infinitely divisible distributions

We have a classic theorem about convergence of infinitely divisible (ID) distributions. Before, remember that we say that the ID random vector $X$ has a Lévy-Khintchine representation, for a suitable ...
André Goulart's user avatar
2 votes
1 answer
45 views

If $\int_{\mathbb{R}} \min(1,x^2)\nu(dx)<\infty$ then $\nu(A)< \infty$ for all A if $0$ is in the interior of $A^c$ [closed]

I am currently reading "Fluctuations of Lévy Processes with Applications" by Kyprianou, where he mentions (p.36 at the top) that if $\nu$ is a measure on $\mathbb{R}\setminus\{0\}$ ...
bumbummath's user avatar
0 votes
0 answers
112 views

Symmetric $\alpha$-stable distributions

Consider the expression appearing in the Levy-Kchinchine formula: $$\varphi(t) = e^{ita - \sigma^2/(2t)+\int_{-\infty}^\infty [e^{itx} - 1 -itx \mathbf{1}_{|x|<1}]\,{\rm d}\nu(x)},$$ but with $a =0,...
Robert Barg's user avatar
1 vote
1 answer
117 views

Random measures in category theory

Let $(\Omega, \mathcal F, \mathbb P)$ be probability space and $(X, \mathcal M)$ be measurable space. We define random measure $\mu$ as function $$\mu:\Omega \times M \to [0, \infty]$$ such that \...
Esgeriath's user avatar
  • 2,376
0 votes
1 answer
35 views

Problem regarding Lévy processes and p-integrability

I am quite new to Lévy processes and stumbled upon this problem and I cant seem to figure it out. If $(L_t)_{t\geq}$ is a Lévy process and $p>0$. How would one show that $$ L_t \in \mathcal{L}^P \...
John7789's user avatar
0 votes
0 answers
41 views

Transformation that keeps Poisson process invariant in Lie groups

As we know that the Brownian motion is invariant under orthogonal transformation , *see article of Bernt Oksendal for more details : **https://doi.org/10.1007/BF01045159**.*page 215 corollary 1. My ...
Anas Cobain's user avatar
0 votes
1 answer
44 views

Infinite divisibility of joint distribution assuming that of the marginals

Assume $F$ is a multivariate dsitribution on $\mathbb R^n$ such that for all $i=1,\ldots,n$ its marginals $F_i$ are infinitely-divisible on $\mathbb R$. Is $F$ necessarily infinitely-divisible?
Mr_3_7's user avatar
  • 321
3 votes
1 answer
257 views

Infinitely Divisible Distribution and Lévy-Khinchin representation

I am studying the book Probability Theory by A. Klenke (3rd Edition). I do not understand the construction of a process from which to obtain the Lévy-Khinchin representation for an infinitely ...
Enrico's user avatar
  • 563
0 votes
1 answer
32 views

Question about an inequality for Levy Processes

Let $(X_t)_t$ be a standard Levy process. We assume $t\mapsto X_t$ is continuous in probability or a.s. cadlag. In this case, why does the continuity of $x\mapsto e^{i\xi \cdot x}$ implies that there ...
nomadicmathematician's user avatar
0 votes
2 answers
18 views

A question about a form of Markov ineqauality

I am studying Levy processes, and met the argument that if $X_t$ is a.s. cadlag then $X_t$ is continuous in probability. The proof goes by showing that $$\lim_{u\to t} P(|X_u - X_t|>\epsilon)=\lim_{...
nomadicmathematician's user avatar
1 vote
1 answer
124 views

Compound Poisson Distribution and its Expected Value

I am trying to understand the proof of Theorem 16.14 of Probability Theory by A. Klenke (3rd version) about the Levy-Khinchin formula. I would like to know how to prove this: $$E[X]=\int x e^{-v(\...
Enrico's user avatar
  • 563
1 vote
0 answers
70 views

Show that poisson compound process is levy

Why is this process a Levy process Hello. I am currently studying Levy processes. In particular, I am trying to understand the proof of the following Theorem: " Let $b \in \mathbb{R}^d$, $M \in \...
LinearAlgebruh's user avatar
1 vote
0 answers
17 views

on unimodality for Exponential Levy process

let $X_t$ be a levy process. I wonder is there any conditions for $Y_t=exp(X_t)$ to be unimodal? Are the conditions for $X_t$ to be unimodal still work for $Y_t$?
Rourou's user avatar
  • 11
0 votes
1 answer
86 views

PDF of stable distribution in terms of the Fox $H$-function for the case $\alpha > 1$.

The PDF of Stable Distribution in terms of a Fox H-function for the case $α≤1$ is available at https://reference.wolfram.com/language/ref/FoxH.html (Examples\Applications) in the form: $S(x;\alpha,\...
student's user avatar
  • 111
0 votes
1 answer
84 views

Expectation of waiting time ratios

Question: At a restaurant's bathroom, men and women arrive at the same rate, on average 20 minutes. Assume the number of people who waiting in the queue is Poisson distributed. Men spend on average 60 ...
Kapes Mate's user avatar
  • 1,424
1 vote
0 answers
137 views

Hamilton-Jacobi-Bellman equation for Levy processes

I am trying to understand a optimal investment/stochastic control Problem and derive the HJB equation for following Wealth Process $dX^{\phi}(t)=\int_{0}^{t} X^{\phi}(s-)(r+\phi(s)(\mu-r))ds+\int_{0}^{...
ez43eg's user avatar
  • 63
3 votes
0 answers
70 views

Domain of the infinitesimal generator of subordinators

Let $(X_t)$ be a subordinator (not killed). Since $(X_t)$ is a non-decreasing Levy process, we have the corresponding infinitesimal generator: \begin{equation} Af(x)=\delta f'(x)+\int_{0}^{\infty}(f(x+...
user377704's user avatar
0 votes
0 answers
115 views

What is the cardinality of the number of jumps in a Lévy process on a given time interval?

If a Lévy process can be defined as a right continuous process with existing left limits, then according to the following linked theorem (Prove that the number of jump discontinuities is countable for ...
Kapes Mate's user avatar
  • 1,424
1 vote
0 answers
17 views

Interpret conditional expextation of stochastic process

I am trying to understand what the follwing means $$ v(x) = \sup_\tau \mathbb{E}_x(f(X_\tau)) = \sup_\tau \mathbb{E}(f(X_\tau) | X_0 = x) = \sup_\tau \frac{\mathbb{E}(1_{X_0 =x} f(X_\tau))}{P(X_0 = x)}...
Constantin Höing's user avatar
1 vote
1 answer
61 views

Exercise on Levy measure

I have to prove that a Borel measure $\mu$ on $\mathbb{R}^+$ such that $$ \int_0^\infty (z\wedge z^2)\mu(dz)<\infty $$ is a Levy measure. MY ATTEMPT: We need to check that $\int_0^\infty(1\wedge z^...
Pefok's user avatar
  • 664
1 vote
0 answers
86 views

Poisson Process/Lévy process

I want to prove that a Poisson process fulfils the continuity in probability property (https://en.wikipedia.org/wiki/L%C3%A9vy_process#Mathematical_definition) $lim_{h \rightarrow 0} \mathbb{P}(|N_{t+...
galaxy--'s user avatar
  • 475
1 vote
0 answers
55 views

Finding finite dimensional distribution of Levy process from the characteristic function

This is part of a statement from From Levy type processes to Parabolic SPDEs by Rene SChilling. Corollary 2.4 states that the finite dimensional distributions $P(X_{t_1}\in dx_1 ,\dots, X_{t_n}\in ...
nomadicmathematician's user avatar
5 votes
1 answer
296 views

Show that a Levy measure $\nu$ (which arises from a convergence of Infinitely Divisible random vectors) is such that $\int x d\nu(x)=0$

Let $(X_{jn})_{1\leq j \leq n}$ be a triangular array of $p-$dimensional random vectors (row independent). Suppose $X_{jn} \sim \mu_{jn}$ and  1. $\,\, E X_{jn}= \int_{\mathbb R^p} x d \mu_{jn}=0$   2....
PSE's user avatar
  • 544
0 votes
1 answer
66 views

Show that a Levy measure (which arises from a convergence of Infinitely Divisible random vectors) has infinite total mass

Given an array of probability measures $(\mu_{jn})_{1\leq j \leq n}$, all defined on borelians of $\mathbb R^p$. Let $S_n'$ a sequence of $p-$dimensional random vectors with $S_n' \Longrightarrow X$ ...
PSE's user avatar
  • 544
1 vote
0 answers
20 views

Fractional integral related to stable process on half-space

I'm working with the isotropic $\alpha$-stable Lévy process in $\mathbb{R}^d$ $(\alpha \in (0,1) \text{ and } d \geq 2)$. I know that the distribution of the first hitting of this process into the ...
Sonny Medina's user avatar
1 vote
1 answer
41 views

If $\nu$ is a Levy measure, how to show that $\nu( (-\epsilon,\epsilon)^c ) <\infty$

We know that the Levy measure on borelians is a measure $\nu$ such that \begin{equation} \int \frac{|x|^2}{1+ |x|^2} \nu (dx) < \infty \end{equation} We can show that this is equivalent to: \...
user346624's user avatar

1
2 3 4 5
8