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Questions tagged [levy-processes]

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

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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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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
33 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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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? ...
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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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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
48 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
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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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45 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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76 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
39 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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33 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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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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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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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
57 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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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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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 ...
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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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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 ...
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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 ...
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38 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 ...
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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,...
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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^{\...
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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}^{'}=...
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Ordinary-Stochastic Levy Triplet Relation for Tempered Stable Process

I'm trying to verify a result in the paper of https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1700305 with regards to the tempered stable or CGMY process in option pricing literature, which has ...
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$f(X_{T})=f(X_{T^{-}})$ 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 ...
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$(X_{z+t}-X_{z})_{t\geq 0}$ 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})_{...
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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 ...
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2answers
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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 ...
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1answer
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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{...
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48 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 ...
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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^{...
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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). ...
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Step of the proof for a property of Levy Processes

Suppose $ν$ is a Levy measure. We want to prove $$∫(1 ∧|x^2|)ν(dx)<+∞ $$. In Financial Modelling with Jump Processes by Cont and Tankov, page 82, they arrive at a step when they showed: $$∫(1-...
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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+...
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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:...
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101 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.)? ...
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(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 ...
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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 $d\mathcal E_t=\mathcal E_{t-}dX_t$, $\...
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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!
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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 ...
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52 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....
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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 ...
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60 views

An intuitive property of Brownian motion

Let $B$ be a Brownian motion (in fact, I have in mind a Levy process). Let $I_s := [a_s,b_s]$ be an interval for all $s \in [0,2t]$. How do you prove the following statement rigorously, just using ...
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57 views

Application of Ito Formula

Hy everyone, My first question so please be gentle. I need to find the following PIDE for a Lévy market $$ -rf(x,t) +\partial_2 f(x,t)+(r-\frac{c}{2}) \partial_1 f(x,t)+\frac{c}{2} \partial_1^2 f(...
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Wiener-Hopf factorization of the characteristic function of a Levy process

Given $X_t$ a Levy process and $\Delta$ an interval of time, I have to compute the Wiener-Hopf factorization $\Phi_+$$\Phi_-$ of $$\Phi(u,q)=1-q\mathbf{E}[e^{iuX_{\Delta}}]=1-q\varphi(u)=\Phi_+(u,q)\...
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1answer
53 views

Characteristic function of maximum of Levy process

Nothing to add to the title, I'm looking for the characteristic function of the maximum of a Levy process, can someone help me out? Thanks