Questions tagged [levy-processes]

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

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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 \...
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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 ...
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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}\...
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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 $\beta_s$, $\...
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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 ? ...
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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(\...
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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
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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 (...
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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 ...
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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 ...
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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 $X_V - X_U$ ...
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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 ...
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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 $...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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^{\...
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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 ...
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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 ...
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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_{...
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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(...
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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 ...
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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) ...
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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 ...
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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) = \...
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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}...
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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 ...
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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 ...
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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}\...
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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,...
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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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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)=...
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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)....
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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 ...
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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. ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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\...
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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\...
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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 $\...
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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 ...

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