Questions tagged [levy-processes]
Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.
269
questions
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68 views
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}...
3
votes
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 $\...
1
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0answers
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
$$
\lim_{n \to \infty} d^{Sk} (L,L^n) = ...
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0answers
40 views
On the Hypothesis and Form of the Levy-Khintchine / Ito Representation of IDDs
Background:
I should probably start by saying I am an autodidact in almost everything pure maths.
I have two questions regarding the Levy-Khintchine representation; from Sato [1]:
If $\mu$ is an ...
1
vote
1answer
30 views
How do you interpret the infinite activity property of variance gamma process?
I am a little bit confused with the infinite activity property of variance gamma(VG) process $X(t),$ where $$X(t)=\theta G(t) + \sigma W(G(t)),$$ for any finite interval, the VG process has infinitely ...
1
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0answers
15 views
Why is there no subordinator when alpha is 1?
I am trying to understand what happens if I take
$$ \frac{1}{n} \sum_{i=1}^{nt} \xi_i, $$
where $\xi_i$ are positive i.i.d.r.v. with tail being $\frac{l(x)}{x}$ for some slowly varying function $l(x)$....
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0answers
14 views
Finite intensity of Lévy measure implies compound Poisson process
Suppose $X$ is a Lévy process with triplet $(b,\sigma^2,\nu)$ and finite intensity, so $\nu(\mathbb R)<\infty$.
Why does it follow immediately that the jump part of $X$ can be described by a ...
1
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0answers
21 views
Definition of random measure (studying Lévy processes)
While studying Lévy processes and Poisson processes, I have encountered the jump random measure
$$\mu^{X}(\omega, \cdot)=\sum_{t \geq 0} \delta_{\left(t, \Delta X_{t}(\omega)\right)} .$$
My confusion ...
4
votes
1answer
78 views
Show that the “distinguished power” of a continuous complex-valued function is well-defined
Let $E$ be a normed $\mathbb R$-vector space$^1$ and $\ln$ denote the principal branch of the complex logarithm.
We can show the following result:
Theorem: Let $\varphi:E\to\mathbb C\setminus\{0\}$ ...
1
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0answers
20 views
Property of alpha-stable subordinator
Let $\alpha \in (0,2)$ and let $\eta(t,x)$ be the density of alpha stable subordinator, that is
$$ \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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0answers
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, ...
0
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0answers
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} + ... ...
2
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0answers
34 views
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 ...
3
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0answers
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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15 views
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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0answers
21 views
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)...
1
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1answer
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
$$ e^{2y} f(y) = C \frac{d}{dy}\int_y^\...
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0answers
25 views
Application of the Compensation Formula
I am having trouble understanding the proof of the next theorem:
T $5.6$: Suppose that $X$ is a killed subordinator. Then, for $u>0$ y $y\in [0,x]$:
$$P(X_{\tau_{x}^{+}} - x \in du, x - X_{\tau_{x}^...
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0answers
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-...
1
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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.
...
3
votes
1answer
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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0answers
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 '...
1
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1answer
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 $\...
1
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0answers
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 $E$-...
1
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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$$ denote the Laplace transform of $\mu$. By the Lévy-...
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0answers
16 views
Determine Lévy-Khintchine representation of $\frac c{\sqrt 2\pi}\int_{B\:\cap\:(0,\:\infty)}e^{-\frac{c^2}{2x}}x^{-\frac32}\:{\rm d}x$
Let $c>0$ and $$\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).$$
How can we determine the Lévy-...
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2answers
57 views
If $X$ and $Y$ are Lévy processes with $X_t\sim Y_t$ for all $t$, can we infer that $X\sim Y$?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$, $E$ be a $\mathbb R$-Banach space and $(X_t)_{t\ge0}$ be an $E$-...
2
votes
1answer
110 views
Randomly restarted Lévy process is again a Lévy process
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$;
$E$ be a $\mathbb R$-Banach space;
$(L_t)_{t\ge0}$ ...
1
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0answers
77 views
Poisson random measure corresponding to a compound Poisson process
Let $H$ be a $\mathbb R$-Hilbert space, $\mu$ be a finite measure on $\mathcal B(H)$ with $\mu(\{0\})=0$ and $(L_t)_{t\ge0}$ be a $H$-valued càdlàg Lévy process on a probability space $(\Omega,\...
0
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0answers
48 views
Properties preservation by stochastic integration: which integrands work?
Consider the stochastic integral
\begin{equation}\int_0^t H_s dX_s.\quad\quad\quad\quad\quad\quad\quad\quad(*) \end{equation}
For a general cadlag (not necessarily continuous) semimartingale $X$, I ...
3
votes
1answer
229 views
Convergence of infinitely divisible distributions
I want to show$$\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.$$
$M_n$ are ...
2
votes
0answers
41 views
Examples of stochastic integrals being a Lévy Process
Consider the stochastic integral:
$$X(t)=\int_0^tf(s-)dL(s)$$
with $L$ being a Lévy process (and $f$ properly chosen in order to let the SI exist).
Do there exists conditions on $f$, excluding the ...
3
votes
0answers
160 views
Expectation of product of two random variables.
If I have a compound possion process due to which I can generate a poison random measure $N$ then I can write my compound poisson process in the another form as
$$\sum_{i=1}^{P_t}Y_i = \int_{\mathbb{...
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0answers
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 $(M_t)_{t \geq 0}$ ...
1
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1answer
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 $...
0
votes
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\...
0
votes
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 ...
2
votes
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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0answers
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
$$X(t)=W(t)+\int_0^t\int_0^{1}(...
3
votes
1answer
147 views
Proof that the jump measure of a Lévy process is a Poisson random measure
Let $(X_t)_{t \geq 0}$ be an $\mathbb{R}^d$-valued Lévy process and consider its associated jump measure $N_t: \Omega \times \mathbb{B}(\mathbb{R}^d \setminus \{0\}) \to \bar{\mathbb{N}}_0$ given by
\...
2
votes
0answers
27 views
Why does the infinitesimal of a poisson process behave as it does in the Ito multiplication table
In more informal derivations of Ito's formula for jump processes, the multiplication table for $dt, dW_t$ and $dN_t$give that $dN_tdN_t=dN_t$. Why is this? I have tried deriving this from the ...
0
votes
1answer
48 views
Strong Markov Property for Lévy processes
I do not understand one step in the proof of Theorem 32 in Chapter 1.
How can we conclude that $Y_t = X_{T + t} - X_{T}$ is independent from $\mathcal{F}_T$ using the identity
$$\mathbb{E}\left(\...
1
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0answers
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
$X_V - X_U$ ...
1
vote
0answers
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 ...
1
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0answers
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 ...
1
vote
0answers
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 ...
2
votes
0answers
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 ...
