Questions tagged [levy-processes]

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

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Given the characteristic function of Lèvy distribution, Find the density function.

In wireless communication, we characterize the total interference by the following characteristic function (CF): \begin{equation} \mathcal{F}_I(\omega) = \exp\left(-\lambda\pi \omega^{2/\alpha}...
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1answer
45 views

Poisson random measure and $\alpha-$stable processes

Let $N$ be a Poisson random measure in $(0,\infty)^2$ with intensity $\eta$ given by $$\eta(ds, dx) = \mathbb{I}_{\{x>0\}} \dfrac{C}{x^{\alpha + 1}} ds dx.$$ Find the values $\alpha$ for which $\...
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21 views

convergence of a Lévy process in Skorokhod implies convergence of jumps in Skorokhod

Assume we have a Lévy process $L=(L_t)_{t \geq 0}$ and an approximating sequence $(L^n)_{n \in \mathbb{N}} = ((L^n_t)_{t \geq 0})_{n \in \mathbb{N}}$ such that $$ \lim_{n \to \infty} d^{Sk} (L,L^n) = ...
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40 views

On the Hypothesis and Form of the Levy-Khintchine / Ito Representation of IDDs

Background: I should probably start by saying I am an autodidact in almost everything pure maths. I have two questions regarding the Levy-Khintchine representation; from Sato [1]: If $\mu$ is an ...
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30 views

How do you interpret the infinite activity property of variance gamma process?

I am a little bit confused with the infinite activity property of variance gamma(VG) process $X(t),$ where $$X(t)=\theta G(t) + \sigma W(G(t)),$$ for any finite interval, the VG process has infinitely ...
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15 views

Why is there no subordinator when alpha is 1?

I am trying to understand what happens if I take $$ \frac{1}{n} \sum_{i=1}^{nt} \xi_i, $$ where $\xi_i$ are positive i.i.d.r.v. with tail being $\frac{l(x)}{x}$ for some slowly varying function $l(x)$....
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14 views

Finite intensity of Lévy measure implies compound Poisson process

Suppose $X$ is a Lévy process with triplet $(b,\sigma^2,\nu)$ and finite intensity, so $\nu(\mathbb R)<\infty$. Why does it follow immediately that the jump part of $X$ can be described by a ...
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21 views

Definition of random measure (studying Lévy processes)

While studying Lévy processes and Poisson processes, I have encountered the jump random measure $$\mu^{X}(\omega, \cdot)=\sum_{t \geq 0} \delta_{\left(t, \Delta X_{t}(\omega)\right)} .$$ My confusion ...
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78 views

Show that the “distinguished power” of a continuous complex-valued function is well-defined

Let $E$ be a normed $\mathbb R$-vector space$^1$ and $\ln$ denote the principal branch of the complex logarithm. We can show the following result: Theorem: Let $\varphi:E\to\mathbb C\setminus\{0\}$ ...
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20 views

Property of alpha-stable subordinator

Let $\alpha \in (0,2)$ and let $\eta(t,x)$ be the density of alpha stable subordinator, that is $$ \int_0^{\infty}\eta(s,x)e^{-xt} dx = e^{-st^{\alpha/2}}, ~s\in(0,\infty).$$ I would like to calculate ...
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17 views

Potential measure of a Subordinator is Radon?

Is there a quick proof, that the potential measure of a Subordinator is radon? i.e. Let $(X_t)_{t\geq 0}$ be a Subordinator, that is a stochastic process with $X_0 = 0$ a.s. and stationary, ...
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18 views

Stationary process with independent increments and Laplace transform

Let $\sigma_ {t} $ be a process with independent and stationary increments, and increasing. It is known that the process is characterized by its Laplace transform $ \mathbb {E} (e ^ {-\lambda \sigma_ {...
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129 views

Two questions on stable Lévy processes : définition and uniqueness of maximum

Let me suppose that $X_1, ..., X_n$ are mutually independent and identically distributed random variables, the function $d P(X_i \leq x)$ is symmetric around $0$, $S_{0} = 0$ and $S_{k} = X_{1} + ... ...
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Formula for quadratic variation of a general Levy process

Let $X$ be a one-dimensional Levy process with generating triplet $(\gamma,\sigma^2,\nu)$. Is there a formula for the quadratic variation for this process without further restrictions? In the book ...
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30 views

Expectation of first Levy jump exceeding a given threshold

Given is a pure jump Lévy process $L$ with Lévy measure $\nu$. Let $m>0$ and set $J = \{x: ||x|| > m\}$. Moreover let $t_i$ be a strictly increasing sequence with $t_0=0$, $t_i \rightarrow \...
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What is an overshoot of a Levy process?

I am reading up about Levy processes and keep seeing the words overshoot and undershoot in the context of fluctuation theory and optimal stopping. Would anyone be able to clarify what these mean? Any ...
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Connections between Levy Processes and Exponential Families

Let $\{\mathbb{P}_{\theta}\}_{\theta}$ be an exponential family of probability measures on $\mathbb{R}^n$ with $$ \frac{d\mathbb{P}_{\theta}}{d\mathbb{P}} \propto \exp\left( \sum_{n=1}^N \eta_n(\theta)...
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38 views

Existence of a similar process to the Browinan motion

Let $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0},P)$ a filtered probability space satisfying the usual conditions. Is there a real-valued stochastic process $X=(X_t)_{t\geq0}$ satisfying the ...
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40 views

Asymptotic solution of an integro-differential equation

Motivated by a problem of Lévy-flights in a potential, I am looking for the asymptotics of a function $f(y)$, which obeys the the integro-differential equation $$ e^{2y} f(y) = C \frac{d}{dy}\int_y^\...
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25 views

Application of the Compensation Formula

I am having trouble understanding the proof of the next theorem: T $5.6$: Suppose that $X$ is a killed subordinator. Then, for $u>0$ y $y\in [0,x]$: $$P(X_{\tau_{x}^{+}} - x \in du, x - X_{\tau_{x}^...
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31 views

Compensation Formula

I am having troubles understanding the proof of the compensation formula. Here is the proof: Theorem $4.4$ Suppose $\phi:[0,\infty)\times \Re\times \Omega \rightarrow [0,\infty)$ is a random time-...
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1answer
43 views

Question on the proof of the uniqueness of Levy triplets in the Levy-Khintchine formula

I am reading the proof on the uniqueness of the Levy triplets in the Levy-Khintchine formula from Ken Iti Sato's Levy Processes. However, there are two questions I cannot answer from the proof below. ...
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51 views

Show that Wiener process with drift is a Levy process

Hey I have to check if process $X_t=\mu t+\sigma W_t$ is a Levy process where $W_t$ is a Wiener process and $\sigma>0,\mu\in\mathbb{R}$. First, $X_0=0$ so it is ok. Then, for $s<t$ we have $X_t-...
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18 views

Reference for stochastic integration w.r.t discontinuous martingales

I'm looking for a book/monograph which deals with stochastic integration w.r.t to general martingales, which in particular don't need to have continuous paths. One book I stumbled upon already is '...
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128 views

If $X$ is a Lévy process, can we show $\frac1tX_t\xrightarrow{t\to0+}\operatorname E\left[X_1\right]$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a $\mathbb R$-Banach space, $(X_t)_{t\ge0}$ be an $E$-valued Lévy$^1$ process on $(\Omega,\mathcal A,\operatorname P)$ and $\...
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55 views

Equivalent condition for a process to have independent increments

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$, $E$ be a $\mathbb R$-Banach space and $(X_t)_{t\ge0}$ be an $E$-...
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1answer
43 views

Determine the support of an infinite divisible probability measure on $[0,\infty)$

Let $\mu$ be a probability measure on $\mathbb R$ and $$\mathcal L_\mu(t):=\int e^{-tx}\:\mu({\rm d}x)\;\;\;\text{for }t\in\mathbb R$$ denote the Laplace transform of $\mu$. Assume $$\mu((-\infty,0))=...
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21 views

If $\mu$ is an infinite divisible distribution on $[0,\infty)$, then the deterministic part of $\mu$ is $\sup\left\{x\ge0:\mu([0,x))=0\right\}$

Let $\mu$ be an infinitely divisible probability measure on $[0,\infty)$ and $$\mathcal L_\mu(t):=\int\mu({\rm d}x)e^{-tx}\;\;\;\text{for }t\ge0$$ denote the Laplace transform of $\mu$. By the Lévy-...
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Determine Lévy-Khintchine representation of $\frac c{\sqrt 2\pi}\int_{B\:\cap\:(0,\:\infty)}e^{-\frac{c^2}{2x}}x^{-\frac32}\:{\rm d}x$

Let $c>0$ and $$\mu(B):=\frac c{\sqrt 2\pi}\int_{B\:\cap\:(0,\:\infty)}e^{-\frac{c^2}{2x}}x^{-\frac32}\:{\rm d}x\;\;\;\text{for }B\in\mathcal B(\mathbb R).$$ How can we determine the Lévy-...
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57 views

If $X$ and $Y$ are Lévy processes with $X_t\sim Y_t$ for all $t$, can we infer that $X\sim Y$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$, $E$ be a $\mathbb R$-Banach space and $(X_t)_{t\ge0}$ be an $E$-...
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1answer
110 views

Randomly restarted Lévy process is again a Lévy process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$; $E$ be a $\mathbb R$-Banach space; $(L_t)_{t\ge0}$ ...
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77 views

Poisson random measure corresponding to a compound Poisson process

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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48 views

Properties preservation by stochastic integration: which integrands work?

Consider the stochastic integral \begin{equation}\int_0^t H_s dX_s.\quad\quad\quad\quad\quad\quad\quad\quad(*) \end{equation} For a general cadlag (not necessarily continuous) semimartingale $X$, I ...
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229 views

Convergence of infinitely divisible distributions

I want to show$$\lim\limits_{t\to 0} \sup\limits_{n}\int (1-\cos tx)dM_n(x) = 0 \ \ \ \ \ \ \ \text{ implies } \ \ \ \ \ \ \ \sup\limits_{n}M_n\{ x : \varepsilon \leq |x| \} < \infty.$$ $M_n$ are ...
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41 views

Examples of stochastic integrals being a Lévy Process

Consider the stochastic integral: $$X(t)=\int_0^tf(s-)dL(s)$$ with $L$ being a Lévy process (and $f$ properly chosen in order to let the SI exist). Do there exists conditions on $f$, excluding the ...
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160 views

Expectation of product of two random variables.

If I have a compound possion process due to which I can generate a poison random measure $N$ then I can write my compound poisson process in the another form as $$\sum_{i=1}^{P_t}Y_i = \int_{\mathbb{...
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60 views

Complex-valued Martingale with Square Integrable Variation on Finite Intervals

Let $(X_t)_{t \geq 0}$ be an $\mathbb{R}^d$-valued compound Poisson process defined on a probability space ($\Omega, \mathbb{F}, P)$. Define the centered, complex-valued martingale $(M_t)_{t \geq 0}$ ...
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61 views

How to solve $ \frac{dS(t)}{S(t-)}=\mu dt+\sigma dW(t)+d\left( \sum_{i=1}^{N(t)}\left( V_{i}-1\right) \right) $

Hey I found following Ito formula for jump diffusion process. Let $$X_{t}=X_{0}+\int_{0}^{t}b_{s}ds+\int_{0}^{t}\sigma_{s}dW_{s}+\sum_{i=1}% ^{N_{t}}\Delta X_{i},$$ where $N_t$ is Poisson process and $...
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1answer
24 views

Expectation of a Levy process with exponiantially distributed time

I am reading about Levy processes and am having trouble to understand the following: Let $X$ be a subordinated Levy Process on ($\Omega$, $\mathcal{F}$, $\mathbb{P}$), and let $\tau$ = $\tau\left(q\...
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1answer
40 views

Subordinator - conditions for the Levy triple

different books write Levy Khinchnin's formula in different ways and thus I have a problem with understanding the conditions that Levy's triple must satisfy for the process to be a subordinator. In my ...
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2answers
72 views

Levy processes - infinitely divisible distribution

I am reading the following text but can't understand the last sentence (source: Andreas E. Kyprianou "Fluctuations of Levy Processes with Applications"): From the definition of a Levy ...
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65 views

The integrand of Levy-Ito decomposition

Suppose that $W(t)$ is a Brownian Motion, $N(ds,dz)$ is a Possion random measure with intensity $ds\mu(dz)$,$\tilde{N}(ds,dz)$ is the compensated measure. Shall we call $$X(t)=W(t)+\int_0^t\int_0^{1}(...
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147 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 \...
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27 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 ...
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1answer
48 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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12 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 $X_V - X_U$ ...
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64 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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124 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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17 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 ...
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74 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 ...

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