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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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Holomorphic extensions of characteristic functions

Let $\chi(\xi) := \mathbb{E}e^{i \xi X}$, $\xi \in \mathbb{R}^d$, be the characteristic function of a random variable $X$. It is widely known that $\chi$ admits a holomorphic extension to the strip $\{...
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172 views

Inequality for Lévy SDE

Let $X_{s}^{t,x}$ denote the solution at time $s$ of an Ito SDE whose coefficients are Lipschitz continuous with initial condition $X_t=x$. Let $t\leq s\leq T<\infty$. The inequality $$ \mathbb{E}\...
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363 views

Proof of strong Markov Property of double sided Levy Process

Let $(X_t)_{t\in\mathbb{R}}$ be a Levy Process, i.e. $X_0 = 0$ a.s., $X$ has independent and stationary increments, and almost all paths $t\mapsto X_t(\omega)$ are right continuous with left hand ...
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148 views

Potential measure of the product of (independent $\alpha$-stable) subordinators

For a nondecreasing Levy process $\mathbf{X}$ with values in $[0,\infty)$ (i.e. a subordinator) Jean Bertoin defines the potential measure of $\mathbf{X}$ in his book "Levy processes" as follows (p. ...
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How to Show RV Related to Poisson Random Measure is a.s. Finite?

I'm really new to this area of random measures, and I'm a bit confused on how to get started on this problem. Let $\mu$ be a measure on $\mathbb{R}$ with $\mu(\left\{0\right\}) = 0$ and $\int_\...
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145 views

Central Limit Theorem for a Lévy Process (mild assumptions)

I want to discuss the Central Limit Theorem (CLT) of a Lévy process under some assumptions. The answer of the previous post Central Limit Theorem for Lévy Process required a wide base of knowledge, ...
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148 views

What is the Lévy measure of the Student's $t$-distribution?

It is known since the 1970's that the Student's $t$-distribution is infinitely divisible. We can therefore apply the Lévy-Khintchine representation to it, and define the Lévy measure associated to a ...
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118 views

Bounded L2 increments for an Ornstein Uhlenbeck type process

Let $Z$ be an increasing Levy process (i.e. a subordinator). Let $\lambda>0$ and consider the Ornstein Uhlenbeck type SDE $$ d V_t = - \lambda V_t dt + d Z_{\lambda t } $$ where the integral can e....
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135 views

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let $\...
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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) = \...
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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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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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62 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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49 views

The Lévy-Khinchine and the Kolmogorov canonical representation

I am currently reading a book by Lukacs on characteristic functions. He states that the Kolmogorov canonical representation, given by $$\log \phi(\omega)=i\omega c+\int_{-\infty}^{\infty}\Big(\exp(i\...
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76 views

Modulus of continuity of Levy process as jump size tends to zero

When reading Kallenberg's "Foundations of Modern Probability Theory", 2nd edition, I have a question regarding an argument in the proof of Lemma 15.19. Let $X_n(t)$ be a sequence of Levy processes. ...
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40 views

Stopping times of Lévy-type processes

I'm trying to find conditions on the symbol $q(x,\xi)$ of a Lévy-type process $X$ (d-dimensional) such that the following stopping time $\tau_u^{\alpha}$ is a.s. finite; $$ \tau^\alpha _u=\inf\left\{t\...
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Differentiability of $E[1_{\tau > T} \mid X_t = x]$ where $X_t$ is a Lévy process

Let $X$ be a finite-variation Lévy process which starts at $X_0>0$, has positive drift, and has only downward jumps. Also define a stopping time $\tau := \inf(0\leq t \leq T: X_t<0)$, the first ...
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51 views

If two Brownian motion starts and end at the same points, can we say something about there difference?

Let $X$ and $Y$ be two standard Brownian motions with mean $0$ and variance $1$, both started at zero. If we know that \begin{align} X_n &= Y_n, \end{align} for some $n>0$, can we say ...
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122 views

How to apply Ito formula on non-gaussian Ornstein Uhlenbeck process

I have an non-Gaussian Ornstein Uhlenbeck process in the framework of Barndorff-Nielsen and Shephard (2001, 2003) $d\sigma^2(t)=-\lambda\sigma^2(t)dt+dz(\lambda t)$ where $z(t)$ is a Levy jump ...
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153 views

Higher order expectation of Lévy process using Teugels martingales

I am new about stochastic calculus but I would like to know if the following procedure for computing $E\left(L^{2}_{t}\right)$ and $E\left(L^{3}_{t}\right)$ if $L_{t}$ is a Lévy pure jump process is ...
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559 views

Characteristic function of compound Poisson process

It is widely known that the characteristic function of a compound Poisson process is $$ \phi_X(u) = \exp \left(t\lambda \int_{\mathbb{R}} (e^{iux}-1) F(dx) \right). $$ But if I try to derive it via ...
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70 views

Change of Measures for Lévy-Processes

If $X$ is a Lévy-Process on a filtered probability space $(\Omega,\mathcal{F}_t, \mathbf P)$ and $Q$ an equivalent probability measure. Under which circumstances is $X$ also a Lévy-Process under $\...
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76 views

A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian motion,...
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394 views

Lévy measure of a pure jump process

Let ($\xi_t)_{t \geq 0}$ an infinititely divisible cadlag process on $[0,\infty)$ and denote by $p$ its jump measure. Define a measure $p_t$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ by $$p_t(B) := p((...
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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 $$ X_t=\sigma (X_t) B_t + \sum_{n=1}^{N_t}\eta(X_t) $$ a Levy process? Here $B_t$ is a standard Brownian motion and $N_t$ ...
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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}\...
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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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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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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 $T_{B}^{'}=...
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$(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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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^{...
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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 ...
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59 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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46 views

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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139 views

Prove the following distribution is infinitely divisible

Consider the following characteristic function and show that the distribution is infinitely divisible $$\phi(u)=exp(imu-\sigma\vert u\vert[1+i\beta\frac{2}{\pi}\text{sgn}(u)\text{log}\vert u\vert])$$ ...
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Square root of a stationary OU process

We know that an OU process $\sigma_t^2=\int_{-\infty}^t e^{-\delta (t-x)}dV_x$, where $\delta$ is a positive constant and $V_t$ is a subordinator, is stationary, i.e. for all $t$ we have $\mathbb{E}[\...
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200 views

Sum of independent Lévy processes is a Lévy process

I'm currently reading Cont & Tankov's "Financial Modelling With Jump Processes" and they state the sum of independent Lévy processes is itself a Lévy process but only provide a working example to ...
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Questions about Distribution Seen in a Paper

I'm reading the following paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2543719 . On page 15, the author states that in a jump-diffusion process, the log jump sizes, $Y_k$, follow a ...
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37 views

Independent increment of Lévy process

I'm a beginner on Lévy process and I got a question as follows. Let $X=(X_t)_{t\geqslant 0}$ be a zero mean Lévy process having form $X_t=\sigma B_t + J_t$, where $B$ and $J$ are the Brownian part and ...
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Heuristics on subordinated Brownian motions and time-changed Levy processes

Studying time-changed Levy processes, I ended up writing the following equation: $$W\left(\int_0^t v(s) \, ds\right) = \int_0^t \sqrt{v(s)} \, d W(s)$$ where $v(s)$ can be either a deterministic ...
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67 views

Augmented Kalman-Bucy filter with exponential distribution.

Suppose that we have the regular system used in Kalman-Bucy filtering as $$dX(t)=F(t)X(t)dt+G(t)dB^1_t $$ $$ dZ(t)=H(t)X(t)dt+V(t)dB^2_t $$ The only difference is that $(H(t),V(t)) $ are randomly ...
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702 views

What is levy walk?

I am trying to understand Levy walk. I went through the wiki page and there was a line saying: A Lévy flight is a random walk in which the steps are defined in terms of the step-lengths, which ...
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46 views

Generalized OU process and Poisson random measure

If I consider $X_t$ is a generalized Ornstein uhlenbeck process, how I can prove that $$\text{sign}\left(X_t\right). N(dt,du)$$ is a Poisson measure? Same question if I consider the compensated ...
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148 views

Differences between additive processes and Lévy processes

A real valued stochastic process $\left\{ X_{t}:\ t\in\mathbb{R}^{+}\right\} $ is termed additive if $\forall n\in\mathbb{N}$, $0\leq t_{0}<t_{1}<...<t_{n}<+\infty$, $X_{t_{0}},X_{t_{1}}-...
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49 views

Role of alpha-stability for subordinators

A Lévy process $\left\{ X_{t}\right\} $with values in $\mathbb{R}^{+}$ is termed a subordinator if it is a.s. increasing as a function of $t$, i.e. the map $t\mapsto X_{t}(\omega)$ is increasing for ...
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194 views

Brownian Motion maximum process intuition

I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. ...
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62 views

book reference of Ito integral for jump processes

I would like your help in this: Which book would you recommend as a beginners book into Ito's integral with diffusion and jump processes (specially the latter). A for dummies sort of book. I have ...
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62 views

Conditional density and Lévy processes (with application to option pricing)

Given a Brownian motion $X_t$, we know (by definition) that the density is $f(x,t)=\frac{1}{\sqrt{2 \pi t}} e^{-\frac{x^2}{2 t}}$ Now, if our Brownian motion starts from a certain point $y$ instead ...
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49 views

Fractional moments of stochastic integrals

I want to bound the moments of stochastic integrals as $$E\left|\int_0^1 f(s)d L_s\right|^\alpha,\alpha\in[0,1],$$ where $(L_s)_{s\ge0}$ is a Lévy process with Gaussian part $\sigma^2$ and Lévy ...
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637 views

Compensated Poisson Random Measures and Levy Processes

This is a proposition from my lecture notes (no proof is given -- it's left as an exercise). The section on Poisson random measures starts on page 43 of these lecture notes. At this point in the ...