Questions tagged [levy-processes]
Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.
1
vote
0answers
40 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
26 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
32 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
75 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) = \...
0
votes
0answers
32 views
Inverting a Laplace transform (for a Lévy process)
Let $\psi(\theta) = c\theta + \frac{\sigma^{2}}{2}\theta^{2} - \frac{\lambda\theta}{\alpha + \theta}.$
For those who are wondering where this function comes from, $\psi$ is the Laplace exponent for a ...
0
votes
0answers
11 views
Simulating a Cauchy process
My question is how to I simulate sample paths from a Cauchy process? I know this can be done using two Brownian motions, but I am trying to do it from the basics.
It's known that if we have a Levy ...
1
vote
1answer
43 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
37 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
66 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 ...
0
votes
0answers
15 views
generate random numbers of Lévy distribution
I am going to generate a random step length which is drawn from a Lévy distribution
$$
\textrm{Lévy } \sim u= t^{-\lambda},\;\; 1<λ ≤3
$$
How to (in computer) generate $u$ of such probability?
...
1
vote
0answers
57 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 $k \in \{0,1,2,...,2^n-1\},$
$$h^n_k(t) = 2^{n/2}\mathbb{1}_{[(2k)2^{-n-1},(2k+1)2^{-n-1})}(t) - 2^{n/2}\...
3
votes
2answers
94 views
How to compute the levy path integral with zero potential?
In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as
$$K(x_b,t_b|x_a,t_a)=\int_{x_{t_a}=x_a,...
1
vote
0answers
52 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
40 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
65 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
57 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 ...
1
vote
1answer
56 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
100 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
56 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
41 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
152 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
46 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
37 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
$$\space\displaystyle\limsup_{t\rightarrow\infty}X_{t}=\infty\space\text{a.s.}\space\text{and}\space\displaystyle\...
2
votes
1answer
58 views
Asymptotic behavior of Lévy processes
I'm trying to prove the next proposition:
Let $\{X_{t}\}_{t\geq 0}$ be a non-zero Lévy process on $\mathbb{R}.$ Then it satisfies one of the following three conditions:
$$i)\space\displaystyle\lim_{t\...
0
votes
2answers
33 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
20 views
Doubts in equivalence between recurrence and Potential measure for Lévy processes
I'm reading about Potential measures of Lévy processes. There is a theorem which proof is not totally clear to me. The teorem is the next:
Suposse that $H=\mathbb{R}^{d},$ where $H$ is the spanned of ...
0
votes
0answers
41 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 ...
0
votes
1answer
32 views
Resolvent operators describe distribution of Lévy Process evaluated at independent exponential times
I'm reading about resolvent operators of Lévy processes. The definition is the next:
$$U^{q}f(x)=\int_{0}^{\infty}e^{-qt}P_{t}f(x)dt,$$ where $(P_{t})_{t\geq 0}$ is the semigroup, $f$ is non negative ...
0
votes
0answers
25 views
Doubt in the proof of existence and uniqueness of Laplace exponent for subordinators
I was reading a proof of the existence and uniqueness of Laplace exponent of subordinators:
If $\Phi$ is the Laplace exponent of a subordinator,then there exist a unique pair $(k,d)$ of nonnegative ...
0
votes
0answers
44 views
Lévy measure as a limit
I'm reading about Lévy processes. Durig this, I've found with the next proposition without proof:
For every fixed $a>0,$ the measure $\frac{1}{\epsilon}P_{0}(X_{\epsilon}\in dx)$ converges vaguely ...
0
votes
0answers
20 views
Independent components of a $d-$dimesional Lévy process
I'm tryig to prove the next proposition but I'm lost:
Let $X=(X^{1},\ldots,X^{d})$ be a $d-$dimensional Lévy process with Gaussian coefficient $Q$ and Lévy measure $\Pi.$ Then the Levy processes $X^1,...
1
vote
0answers
20 views
Comparing two sum of fractal moments for heavy-tail distribution
Assume a heavy tailed distribution whose tail can be approximated as
$$P(X\geq x)\sim x^{-\alpha}$$
Consider some fractal moment of iid $X_i$, we have
$$\frac{1}{n}\sum_{i=1}^nX_i^{\theta}\sim O(n^{\...
1
vote
0answers
33 views
First passage time $T_{B}^{'}=\inf\{t\geq0:X_{t}\in B\}$ is stopping time when $B$ is open or closed set
I'm reading a proof of the following proposition:
Let $(X_{t})_{t\geq 0}$ be a Lévy process on $\mathbb{R}^{d}$ and $B\subset\mathbb{R}^{d}.$ We deine $T_{B}=\inf\{t>0:X_{t}\in B\}$ and $T_{B}^{'}=...
2
votes
1answer
33 views
$f(X_{T})=f(X_{T^{-}})$ a.s. for all $f \in C_0$ implies $X_{T}=X_{T^{-}}$ a.s.
During a proof of quasi-left continuity of Lévy processes, there is the next step:
$$E(f(X_{T})g(X_{T}))=E(f(X_{T^{-}})g(X_{T}))$$ for all $f,g\in C_{0},$ where $C_{0}$ is the set of all continuous ...
1
vote
0answers
26 views
$(X_{z+t}-X_{z})_{t\geq 0}$ satisfies “Strong Markov Property” where $X$ is càdlag process and $z$ discrete stopping time.
I'm reading about strong Markov property. In the text there is the next proposition which I'm, trying to prove:
If $X$ is a càdlag process and $z$ is a discrete stopping time, then $(X_{z+t}-X_{z})_{...
2
votes
0answers
41 views
Convergence of stopping times and limit of a right continuous process
I'm trying to prove the next:
If $X$ is cád (right continuous) and adapted process, then $\displaystyle\lim_{n\rightarrow\infty}X_{Z_{n}}=X_{T}$ and $X_{T}$ is random variable.
Here $T$ is a ...
2
votes
2answers
97 views
Levy measure of borel sets away from $0$
The following is of Philip Protter at page 26 of the book Stochastic integration and Differential equations that I have not been able to proved yet.
Let $X$ a Levy process, and $\Lambda$ a borel set ...
2
votes
1answer
63 views
convergence towards infinity of jumping times of Levy processes
The following is a remark of Philip Protter at page 26 of the book Stochastic integration and Differential equation that I have not been able to proved yet.
Let $\Lambda$ be a borel set in $\mathbb{...
0
votes
1answer
54 views
Show that $ (e^{\alpha X_t} \int^ t_ 0 e ^{-\alpha X_u}du, t \geq 0) $ is a Markov process
I want to show that $ (e^{\alpha X_t} \int^ t_
0 e ^{-\alpha X_u}du, t \geq 0) $ is a Markov process
whereas $(\int^ t_
0 e ^{-\alpha X_u}du, t\geq 0)$ is not. Here $X_t$ is a Levy Process and ...
1
vote
0answers
22 views
Fraction of the largest element of a sum of $N$ i.i.d. random variates sampled from power law distribution
For a probability distribution
$$ p(x) \propto x^{-(\mu + 1)} \qquad 0 < \mu < 1$$
both the sum of $N$ i.i.d. samples $S_N$ and the largest element of those samples $x_{\text{max}}$ scale as $N^{...
1
vote
1answer
98 views
Give canonical decomposition of semimartingales $Z_t$ and $W_t$ based on $\mathscr{A}_t$ Levy's area
Let $(X_t, Y_t)$ be a two-dimensional $(\mathscr{F}_t)$-Brownian motion started from 0. We set, for every $t \geq 0$
$$\mathscr{A}_t = \int_0^t X_s dY_s - \int_0^t Y_s dX_s$$
(Levy's area).
...
0
votes
2answers
94 views
Levy Process and characteristic function
Let $X$ be a Levy process. Let $$f(t):=E[e^{i u^{tr}X_t}]$$ I want to show that $f(t+h)=f(t)f(h)$ holds and I know that $M_t:=e^{i u^{tr}X_t}/f(t)$ is a martingale.
I tried to replace $X_{t+h}=X_t+...
1
vote
0answers
35 views
Independence of subordinator and inverse subordinator. [closed]
Let $\{D(t)\}_{t\geq0}$ be a subordinator, that is, a one dimensional L\'evy process with increasing sample paths. Also, the inverse subordinator $\{E(t)\}_{t\geq0}$ is defined as $$E(t):=\inf\{x\geq0:...
1
vote
1answer
105 views
Generalizing a proof for the density of stopped subordinators
Is it possible to generalize the proof of the statement below to general Lévy kernels $\rho(r)dr$ or even Lévy-type kernels $\rho(x,r)dr$ (in the sense of Lévy Matters III by Böttcher et al.)?
...
1
vote
1answer
21 views
(In)dependence of solutions to certain SDEs
Consider a Levy process $L$ in $\mathbb R^d$ written in the Levy-Kchinchine decomosition as $$L(t)=bt + W(t) + Z(t),$$ where $bt$ is the drift part, $W(t)$ is the Wiener part and $Z(t)$ is the jump ...
1
vote
1answer
78 views
Stochastic exponential and strong Markov property of Levy process
Let $X$ be a (cadlag) Levy process with a triplet $(\gamma, \sigma, \nu)$ and it stochastic exponential $\mathcal E$, which is the (cadlag) solution of SDE $d\mathcal E_t=\mathcal E_{t-}dX_t$, $\...
1
vote
1answer
105 views
Construction of Stochastic integration with respect to levy processes
I am asking if anyone could recommand me a book containing the rigorous construction of stochastic integral with respect to levy processes(as integrators). Many thanks!
1
vote
0answers
43 views
Complete randomness imply Poisson process
In this paper, the authors claim that
Theorem 1. A random point process $\Pi$ on a regular measure space is a Poisson process if and only if $N_\Pi$ defined by $N_\Pi(A) = \#\{\Pi \cap A\}$ is a ...
1
vote
0answers
60 views
Lévy–Khintchine representation for distributions
I have read that the Lévy–Khintchine representation exists for any infinitely divisible distribution. However, all the references I could find on Lévy–Khintchine representations are for Lévy processes....
2
votes
1answer
91 views
Exponential martingales and changes of measure
Suppose $X$ is a subordinator (an increasing Levy process) with Laplace exponent $\Phi$, i.e.
$$
\exp(-\Phi(\lambda)) = E(\exp(-\lambda X_1)).
$$
Let $\mathcal{F} = (\mathcal{F}_t)$ denote the ...
