Questions tagged [levy-processes]
Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.
237
questions
3
votes
1answer
43 views
Proof that the jump measure of a Lévy process is a Poisson random measure
Let $(X_t)_{t \geq 0}$ be an $\mathbb{R}^d$-valued Lévy process and consider its associated jump measure $N_t: \Omega \times \mathbb{B}(\mathbb{R}^d \setminus \{0\}) \to \bar{\mathbb{N}}_0$ given by
\...
1
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0answers
19 views
Why does the infinitesimal of a poisson process behave as it does in the Ito multiplication table
In more informal derivations of Ito's formula for jump processes, the multiplication table for $dt, dW_t$ and $dN_t$give that $dN_tdN_t=dN_t$. Why is this? I have tried deriving this from the ...
0
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0answers
47 views
Levy distribution, cuckoo search
In cuckoo search, in some point you have to generate new solution using formula $$x_k(t+1) = x_k(t) + \alpha L(s, \lambda), $$ where
$$L(s,\lambda)=\frac{\lambda\Gamma(\lambda)\sin(\pi\lambda/2)}{\pi}\...
0
votes
0answers
14 views
solution of linear SDE (driven by Levy Process)
Let $Y$ be a levy process, and $V$ an independent stable process. For the following SDE,
$U_t= \int_0^t \beta_s U_{s-} dY_s - \int_0^t \gamma_{s-} dV_s$.
with stochastic coefficients $\beta_s$, $\...
0
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0answers
19 views
Path of $\alpha$ stable process
I'm studying Lévy processes and more particulary $\alpha$-stable processes. I was wondering: if I watch the path of a Lévy process, are there some particularities if it is an $\alpha$-stable process ? ...
0
votes
1answer
34 views
Strong Markov Property for Lévy processes
I do not understand one step in the proof of Theorem 32 in Chapter 1.
How can we conclude that $Y_t = X_{T + t} - X_{T}$ is independent from $\mathcal{F}_T$ using the identity
$$\mathbb{E}\left(\...
0
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0answers
8 views
Q: Levy process as a combination of wiener process poisson procerss and determenistic process
Is it true that any Levy process can be decomposed to a 3 processes:
1. Poisson process
2. Wiener process
3. determenistic process
If so, what is the proof/intuitaion for that?
thank you very much
0
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0answers
11 views
Path of Excursion Process
I am learning the excursion theory for Markov Process and have some difficulties to visualize the excursion process $\epsilon_t$ of a Markov Process $X_t$. Based on my understanding of the concept (...
0
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0answers
36 views
Proof Levy Khintchine
I am supposed to proof this theorem
Theorem 1.16
For a semimartingale levy process X, there is a continuous non decreasing function $\beta$ with $\beta_0 = 0$, a measure $\kappa$ satisfying the ...
0
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0answers
12 views
Two questions about densities of a Levy process
I have the following two questions about densities of a Levy process $(X_t)_{t\ge 0}$.
Does $X_t$ have a density wrt. Lebesque measure? If not, under what conditions does this hold?
If it does have ...
1
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0answers
10 views
Independence of increments with stopping times in Levy processes
Let $X$ be a Levy Process and $S<T<U<V$ be stopping times. Let $F^X$ be the natural filtration of $X$. How can one show that
$X_V - X_U$ and $X_T - X_S$
are independent and
$X_V - X_U$ ...
0
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0answers
23 views
subordinator, which process has this levy measure
which subordinator has this levy measure
$v_Z(dx)={1}_{x>0}\gamma x^{-a-1}e^{-\lambda x},dx(a\in [0,1),\gamma,\lambda>0)$
with positive drift $\sigma_Z$. Can I find this in any book? I suppose ...
0
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0answers
11 views
Simulation of optimal transport coupling between compound Poisson and normal distributions
Consider a compound Poisson random variable $X=\sum_{n=1}^{N}Y_n$,
where $N$ is a Poisson random variable with mean $\lambda$ and
$(Y_n)_{n\in\mathbb{N}}$ are iid bounded random variables.
Suppose $...
0
votes
0answers
15 views
Left limit of Levy Process at time t almost surely equal to Levy Process at time t
I am a little bit confused about the fact that for a cadlag Levy process and a fixed $t>0$ it is true that $X_t=X_{t-}$ almost-surely but we only have stochastic continuity of the Levy process.
...
1
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0answers
38 views
What is the Levy measure of $\int_0^\cdot \int_{0<|x|\le r} x \tilde N(dx,dt) + \int_0^\cdot \int_{|x|>r} x N(dx,dt)$?
Let $N$ be a Poisson random measure on $(\mathbf R^d\setminus\{0\})\times \mathbf R_+$ with intensity measure $\nu\otimes dt$. Here $dt$ is the Lebesgue measure on $\mathbf R_+$. Let $\tilde N$ be the ...
1
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0answers
75 views
Quadratic variation of a Lévy process is finite and itself a Lévy process
I want to show that given a Lévy measure $l(dx)$ such that
$$\int_\mathbb R x^2\land 1l(x)<\infty$$
Then for the quadratic variation of the Lévy process $X$ almost surely
$$\sum_{s\leq t} \Delta ...
1
vote
0answers
16 views
How to calculate the mean correcting argument for Lèvy processes?
To calculate stock path of Lévy process, I need to calculate the mean-correcting equivalent martingale measure: $\omega= log (E[e^{X_t}| y(0)])$.
I found in a thesis that $\omega$ can be calculated ...
2
votes
0answers
47 views
Infinitesimal generator of stochastic integral of a diffusion process with deterministic jumps
I have little knowledge on jump processes, so I am trying to make an analogous case of a normal stochastic process here but I my intuition says that I am doing something wrong
Suppose you have a ...
0
votes
0answers
31 views
Intuitive reason that stationary increments create nonstationarity?
In an earlier question I asked Why are processes with stationary independent increments nonstationary?, i.e. why are Lévy processes nonstationary, and saz gave a nice proof using characteristic ...
2
votes
1answer
217 views
Why are processes with stationary independent increments nonstationary?
This answer states that any process with stationary independent increments is nonstationary.
Why? Specifically:
Let $X(t-s) = N(t) - N(s)$ have distribution $F(t-s)$ for all $s\leq t$ [increments ...
5
votes
1answer
89 views
What is the true definition of a Lévy process?
What is the “true” definition of a Lévy process?
I notice that definitions vary in non-equivalent ways:
1) Wikipedia states that a Lévy process is one that satisfies four particular properties, but ...
1
vote
1answer
73 views
Are nonnegative Lévy processes almost always nondecreasing?
Let $X(t)$ be a Lévy process defined on $t\geq 0$. Suppose that, with probability 1, $X(t)\geq 0$ for all $t \geq 0$. That is:
$$ P\left[\left(\inf_{t\geq 0} X(t)\right) \geq 0\right] =1$$
Does that ...
2
votes
1answer
67 views
Wald Martingale for Lévy Processes
in the book "ruin probabilities" by Asmussen and Albrecher, in Chapter II Thm. 1.2, the following statement is made:
Let $\{X_t\}$ be a Lévy process and $\alpha\in\mathbb{R}$. If $\mathbb{E}[e^{\...
3
votes
1answer
54 views
Renewal for Levy Processes
Suppose $X(t)$ is a Levy process with almost surely positive increments (for all $t_1 < t_2$ $P(X(t_1) < X(t_2)) = 1$)
Define
$$\nu X(t) := \sup \{\tau \in \mathbb{R_+}| X(\tau) < t\}$$
It ...
1
vote
0answers
30 views
Lévy processes, Brownian motion and Lie groups
We know that Brownian motion describes any (non-deterministic) Lévy process with continuous sample paths.
The above statement is true in Euclidean space. My question is: does it stand for other ...
1
vote
1answer
89 views
Independent increment vs independent sigma-algebras
Suppose we want to define a Lévy process $\{ X_t \vert \ t \geq 0\} $. Is it equivalent to demand independent increments i.e. $$ \forall n \geq 1, \forall t_n \geq t_{n-1} \geq ...\geq t_1 \geq 0: X_{...
3
votes
1answer
108 views
Independent increments and stationary increments, Lévy process
Prop:Let $(X_t)$ be a $\mathbb{R}^d$ valued stochastic process with transition probability $P_t(x,dy)$. We assume there exsist probabilities on $\mathbb{R}^d$ $\{m_t\}_{t\geq 0}$ s.t., $P_t (x,B)=m_t(...
1
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0answers
62 views
Do jumps in Levy processes need to be independent of the process itself?
I have a very basic question about Levy processes. Is the process of the form
$$ X_t=\sigma B_t + \sum_{i=1}^{N_t}\eta_i(X_t) $$ a Levy process? Here $B_t$ is a standard Brownian motion and $N_t$ is a ...
1
vote
1answer
53 views
Convergence of the Laplace Exponent of a Compound Poisson Process, Lèvy Fluctuation Theory
The question is about spectrally positive Lévy processes.
For certain $d, \sigma^{2} \geq 0$ and measure $\Pi_{\varphi}(\cdot)$ such that $\int_{(0,\infty)} \min \{1, x^2 \} \Pi_{\varphi}(\cdot) (dx) ...
0
votes
1answer
39 views
Does partial averaging allow moving increments in and out of an expectation?
Given a Levy process $X$ at different points in time $s$ and $t$, and if I have an expression like this:
$$\mathbb{E}[X_t \cdot \mathbb{E}[X_s]]$$
I want to know if I can use partial averaging to ...
2
votes
0answers
93 views
How to calculate expected value of integral?
How to calculate
$E \big[(\int ^{t} _{-\infty} e^{\lambda u} d\tilde{L_\alpha}(u))^A (\int ^{t+h} _{-\infty} e^{\lambda u} d\tilde{L_\alpha}(u))^B\big]$,
where
\begin{align}
\tilde{L}_\alpha (t) = \...
1
vote
1answer
81 views
Lévy process + scaling property $\implies$ Brownian motion
How can I show that if $\xi_t$ is a Lévy process distributed as $\xi_{t+s}- \xi_s$ for all $t,s \in [0,\infty)$ and has independence of increments, and also is distributed as $\lambda\xi_{\lambda^{-2}...
1
vote
1answer
78 views
Stochastic integral with Poisson random measure
The following is what I read in paper and I am confused by some parts.
We consider a one-dimensional Itô semimartingale $X$ which is defined
on some probability space $(\Omega,\mathcal F,\{\mathcal ...
2
votes
1answer
120 views
The Levy measure of a multivariate alpha-stable Levy motion
I am having difficulties to understand the form of the Levy measure of the multivariate Levy-stable motion. Let me start by defining the one dimensional motion in order to clarify my question.
The ...
1
vote
0answers
118 views
Levy construction of Brownian motion by Haar function and Schauder function
For every $t \in [0,1]$, we set $h_0(t) = 1$, and then, for every integer $n \geq 0$ and every $k \in \{0,1,2,...,2^n-1\},$
$$h^n_k(t) = 2^{n/2}\mathbb{1}_{[(2k)2^{-n-1},(2k+1)2^{-n-1})}(t) - 2^{n/2}\...
3
votes
2answers
149 views
How to compute the levy path integral with zero potential?
In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as
$$K(x_b,t_b|x_a,t_a)=\int_{x_{t_a}=x_a,...
1
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0answers
57 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 $\...
2
votes
0answers
45 views
Construction of semimartingales
Let $(B,C,\nu)$ be the characteristics of a semimartingale $\{X_t\}_t$ on $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$.
If
\begin{equation}
B_t(\omega)=bt, \ C_t(\omega)=ct, \ \nu(\omega,dt,dx)=...
1
vote
1answer
112 views
Characteristic function of a Levy process
When I was reading Protter's textbook "Stochastic Integration and Differential Equations", it is stated without proof that (page 20)
If we take the Fourier transform of of each $X_t$ (a levy process)....
1
vote
1answer
199 views
Book on stochastic differential equations
I'm applying for a job at a sports forecaster on the mathematical modeling side. My interview ended with the handing out of a test for which I have a week.
With only a MSc thesis on Sasaki-Einstein ...
3
votes
1answer
115 views
Definition of Lévy process
I know that Lévy process $\{X_t\}_{t\geq 0}$ is a stochastic process that satisfies few conditions:
$\mathbb{P}(X_0 = 0) = 1.$
$X_t$ has stationary increments and $X_t$ has independent increments.
...
0
votes
0answers
160 views
Proof that the Expected Value of a Levy Distribution diverges?
If this question has already been answered, please link me because I could not find anything online.
I have been using exponential Brownian motion in my models of stochastic population dynamics. The ...
1
vote
1answer
195 views
Scaling property for $\alpha-$stable subordinators
I'm trying to prove that, for a $\alpha-$stable subordinator, with $\alpha\in(0,1)$ the next equality holds:$X_{t}\overset{d}{=}t^{1/\alpha}X_{1}.$
The definition of $\alpha-$stable subordinator is ...
0
votes
1answer
59 views
Lévy's triplet of $Y_{t}=aB_{t}+Z_{t}+rt$
I'm trying to find the triplet of the next Lévy process:
$$Y_{t}=aB_{t}+Z_{t}+rt,$$ where $\{B_{t}\}_{t\geq 0}$ is a standard Brownian Motion, $\{Z_{t}\}_{t\geq 0}$ is a Compound Poisson Process and $...
0
votes
0answers
331 views
Laplace exponent of a standard $\alpha-$stable subordinator
I'm trying to calculate the Laplace exponent of a standar $\alpha-$stable subordinator.
An $\alpha-$stable subordinator has Lévy measure $\frac{c}{x^{1+\alpha}}dx,$ where $\alpha\in(0,1)$ and $c$ is ...
1
vote
1answer
49 views
Does the Levy process stay in any open ball at any fixed time with positive probability?
Question: Let $X$ be a $d$-dimensional Levy process. Then for every $t>0$ and $a>0$,
\begin{equation}\tag{1}
\mathbf P\{|X_t|<a\}>0\ ?
\end{equation}
The question comes from the proof ...
0
votes
1answer
47 views
Recurrent Lévy process implies $\limsup_{t\rightarrow\infty}X_{t}=\infty$ and $\liminf_{t\rightarrow\infty}X_{t}=-\infty$
Let $\{X_{t}\}_{t\geq 0}$ be a non-zero recurrent Lévy process on $\mathbb{R}.$ Then
$$\space\displaystyle\limsup_{t\rightarrow\infty}X_{t}=\infty\space\text{a.s.}\space\text{and}\space\displaystyle\...
2
votes
1answer
63 views
Asymptotic behavior of Lévy processes
I'm trying to prove the next proposition:
Let $\{X_{t}\}_{t\geq 0}$ be a non-zero Lévy process on $\mathbb{R}.$ Then it satisfies one of the following three conditions:
$$i)\space\displaystyle\lim_{t\...
0
votes
2answers
37 views
Measurability of inferior limit and limit of a Lévy Process
I'm reading about Lévy Processes.
As a definition of a recurrence and transient Lévy process we have:
Def. For a Lévy process $\{X_{t}\}_{t\geq 0}$ in $\mathbb{R}^{d}$ is called recurrent if $\...
0
votes
0answers
43 views
Support of Potential Measure for Lévy Processes
I'm reading about Support of Potential measures for Lévy Processes. $\Sigma$ denotes the support of $U(0,\cdot),$ where $U(x,B)=\int_{0}^{\infty}P_{x}(X_{t}\in B)\,\mathrm dt$ is the potential measure ...
