Questions tagged [levy-processes]

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

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Law of $\sup_{s \geq 0} X_s$ for $X$ a Lévy process with derive

We know that the supremum of drifted brownian motion $\sup_{t \geq 0} B_t + t \mu$ is exponentially distributed. I want to know if $X$ is a Lévy process with derive (i.e. goes to $+\infty$) if we have ...
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Levy process "that has only jumps"

I post this today as I'm looking to some caractérisation of a Levy processes "that has only jumps" but I didn't found anything on the web neither on the classic books I know about the ...
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Why is $-cN/z^\alpha$ a good approximation for $\ln(1-F(z))^N$?

Bertin's Statistical Physics of Complex Systems, 3rd ed. p. 65 defines a "complementary cumulative distribution $\tilde{F}(z)$" equal to $\int_z^\infty p(x)\,dx$, where the density $p(z)$ is ...
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Equivalence of a Levy measure

We know that a measure $\nu$ is a Levy measure if: \begin{equation}\label{1}\tag{1} \nu(\{0\}),\quad \int_{|x|<1}|x|^{2} \nu (dx) <\infty, \quad \int_{|x| \geq 1} \nu (dx)<\infty. \end{...
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When are levy flights superdiffusive and why? [closed]

A stable distribution that is centered and symmetrical has a value $\alpha \leq 1$ the expected value of the distribution is undefined. I have also been told that if a Levy Flight has it's step ...
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Lévy process, characteristic function, $\frac{\psi(\xi)}{i \xi}$

Given a one dimensional Lévy process $X_t$ with characteristic exponent $\psi(\xi)$, so that \begin{align} \mathbf{E}[e^{i \xi X_t}]= e^{t \psi(\xi)} \end{align} for example we can find $\psi(\xi)$ ...
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From Donsker's theorem to subordinate Brownian motion

Denote $S_n := \sum_{k=1}^n X_i$, where the random variables $X_i$ are independent and identically distributed. Suppose $\mathbb{E}[X_i] = \mu$ and $\mathbb{V}[X_i] = \sigma^2$ are finite. Let $\...
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Suppose $\xi_t$ an infinitely divisible cadlag process with Lévy measure $\Lambda((-\infty,\,0])=0$. All jumps nonnegative?

Suppose $\xi_t$ defined on $[0,\,\infty)$ with $\xi_t(\omega)$ always nonnegative, is an infinitely divisible cadlag process whose Lévy measure $\Lambda$ concentrates on $(0,\,\infty)$. Is this ...
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What is the relation between \alpha stable and Levy Processes

I know that \alpha stable processes belongs to Levy Type Process class. If so, why is there a kind of process called alpha stable levy processes. Thank you for the answer in advance.
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Jumps of a compound Poisson process

If $(Z_n)_{n\in\mathbb N}$ is an i.i.d. process with values in a normed $\mathbb R$-vector space $E$, then $$W_n:=\sum_{i=1}^nZ_i$$ is called random walk with step distribution $\mathcal L(Z_1)$. Now ...
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Does $\int\min(1,\|x\|)<\infty$ imply $\int\left|e^{{\rm i}\langle x,\:x'\rangle}-1-{\rm i}\langle x,x'\rangle1_{\overline B_1(0)}(x)\right|<\infty$

Let $E$ be a normed $\mathbb R$-vector space and $\lambda$ be a measure on $\mathcal B(E)$ with $$\int\min(1,\|x\|_E)\:\lambda({\rm d}x)<\infty.\tag1$$ Are we able to conclude $$\int\left|\...
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Lévy characterization of Hilbert-space-valued Wiener process

Let $H$ be a $\mathbb R$-Hilbert space (assume $H=\mathbb R^d$ for some $d\in\mathbb N$, if this is helpful for you to understand the following) and $(X_t)_{t\ge0}$ be an $H$-valued continuous Lévy ...
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Show that every continuous Lévy process is a Wiener process

Let $H$ be a $\mathbb R$-Hilbert space (assume $H=\mathbb R^d$ for some $d\in\mathbb N$, if this is helpful for you to understand the following) and $(X_t)_{t\ge0}$ be an $H$-valued continuous Lévy ...
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If $X$ is a Lévy process, then $\operatorname{Var}[X_n]=n\operatorname{Var}[X_1]$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $H$ be a $\mathbb R$-Hilbert space, $(X_t)_{t\ge0}$ be an $H$-valued Lévy process on $(\Omega,\mathcal A,\operatorname P)$ and $\mu_t:=...
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Is the jump measure of a space homogeneous Markov process translation invariant?

Let $E$ be a normed $\mathbb R$-vector space and $(X_t)_{t\ge0}$ be an $E$-valued Lévy process. $X$ is a space- and time-homogeneous Markov process with transition semigroup $$\kappa_t(x,B):=\...
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Is the limit in probability of Lévy processes still a Lévy process?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ bea filtration on $(\Omega,\mathcal A)$ and $(X_t)_{t\ge0}$ be a process on $(\Omega,\mathcal A,\operatorname ...
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Bound for the probability of a Lévy process having a jump of a given size

Let $E$ be a normed $\mathbb R$-vector space; $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$; $(X_t)_{t\ge0}$ be an $E$...
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If $M,N$ are martingales, show that $\operatorname E\left[M_tN_t\right]=\operatorname E\left[\sum_{s\in(0,\:t]}\Delta M_s\Delta N_s\right]$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ and $M,N\in\mathcal V$ (see definition below$^2$) be càdlàg $\...
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Creeping for a Levy processes with infinite total mass

I am reading Theorem 7.11 from the book on Levy processes by Kyprianoy. I cannot intuitively grasp this: In this theorem it is said that if the process has a Gaussian component then it creeps upwards. ...
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If $\pi_t$ is the jump measure of a Lévy process and $B_1,\ldots,B_k$ are disjoint, then $\pi_t(B_1),\pi_t(B_k)$ are independent

Let $E$ be a normed $\mathbb R$-vector space; $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$; $(X_t)_{t\ge0}$ be an $E$...
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If $r:=\operatorname{dist}(0,\overline B)$, what's the relation between $\{t>τ^B_{n-1}:\Delta x(t)\in B\}$ and $\{s\in(0,t]:\|\Delta x(s)\|_E\ge r\}$

Let $E$ be a normed $\mathbb R$-vector space $x:[0,\infty)\to E$ be càdlàg $x(t-):=\lim_{s\to t-}x(s)$ and $\Delta x(t):=x(t)-x(t-)$ for $t\ge0$ $\tau^B_0:=0$, $I^B_n:=\{t>\tau^B_{n-1}:\Delta x(t)\...
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Show that the inter-jump times of a Lévy process are i.i.d.

Let $E$ be a normed $\mathbb R$-vector space, $(X_t)_{t\ge0}$ be a càdlàg Lévy process on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$, $B\in\mathcal B(E)$ ...
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Show that $\sum_{\substack{s\ge0\\\Delta X_s\ne0}}1_B(s,\Delta X_s)$ is measurable

Let $E$ be a normed $\mathbb R$-vector space and $x:[0,\infty)\to E$ be regular in the sense that $$x(t\pm):=\lim_{s\to t\pm}x(s)$$ exist for all $t\ge0$. Let $$\Delta x(t):=x(t+)-x(t-)\;\;\;\text{for ...
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Is the reciprocal of Lévy-Khintchine theorem for Levy process true?

We know that if $X = [X_t , t \in \mathbb{Z}]$ is a Levy process, then any marginal distribution has the charasteristic function - ch. f. of $X_t$ - given by $\varphi_t(r)=e^{t \phi(r)}$ whith: $$\...
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Distribution of first jump time

Let $(X_t)_{t\ge0}$ be a Lévy process, $\tau_0:=0$ and $$\tau_n:=\{t>\tau_{n-1}:\Delta X_t\in B\}\;\;\;\text{for }n\in\mathbb N$$ for some measurable set $B$ with $0\not\in B$. How can we show that ...
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Numerical simulation of SDE with Lévy noise in Python - Overflow issue

My goal is to simulate a SDE with alpha-stable noise in Python. This is my code: ...
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How can we show that the jump measure of a càdlàg process is a *counting* measure?

Let $E$ be a normed $\mathbb R$-vector space, $(X_t)_{t\ge0}$ be an $E$-valued càdlàg process on a probability space $(\Omega,\mathcal A,\operatorname P)$ and $$\pi_\omega(B):=\sum_{\substack{s\:\ge\:...
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An application of Ito lemma for Ito-Levy processes

For an Ito-Levy process $X_t$ with the dynamics given by $$ \mathrm{d}X_t = \sigma_t \mathrm{d}B_t + \int_{|z|<1}\gamma_t(z) \tilde{N}(\mathrm{d}t, \mathrm{d}z) + \int_{|z|\ge 1}\gamma_t(z) {N}(\...
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Intersection of random fractals; i.e. the range of $\alpha$-stable subordinators

I am thinking about the following: https://users.math.yale.edu/public_html/People/frame/Fractals/FracAndDim/DimAlg/intersection.html states that most placements of two fractals whose dimensions add to ...
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Numerical simulation of SDE's driven by Lévy processes (particularly stable processes)

I'm trying to learn how to numerically simulate SDE's of the form $$ dX(t) = f(t,X)dt + g(t,X)dZ(t) $$ Where Z(t) is a Lévy process with triplet $(a,0,\nu)$. My question: what is currently considered ...
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Theorem Sato 30.1 subordinated levy process

Referring to Theorem 30.1 of Sato's book "Levy processes and Infinitely divisible distributions". In the proof, page 200, when he shows that the subordinated levy process has independent ...
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A question about the Lévy-Khintchine formula

I am reading some references about the Lévy-Khintchine formula. I would like to understand the equivalence. The first reference is Takano's paper: The family of $p$-dimensional i.d. (infinitely ...
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how to generalize fractional laplacian to be interdimensionally correlated?

sorry if my question sounds weird. struggling to find the appropriate vernacular. the laplace operator Δ can be generalized to have "cross-diffusion" ∇·M∇ where ∇ is the gradient operator, M ...
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Can a stochastic process have stationary increments which are not independent increments?

I can see that increments could be independent but not stationary, but can't think of an example going in the other direction, i.e., stationary but dependent.
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Combining the derivative of two Brownian motions into one.

I am new to stochastic analysis. I am working on a project that uses splitting methods to numerically solve SDEs. I need to solve a 2D system - the catch however is I need both components in the form: ...
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2 votes
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Prove that $N(t,A):=\left|\left\{s\in(0,t]:\Delta X(s)\in A\right\}\right|$ has independent increments

Let $E$ be a normed $\mathbb R$-vector space, $(X(t))_{t\ge0}$ be an $E$-valued càdlàg Lévy process on a probability space $(\Omega,\mathcal A,\operatorname P)$ and$^1$ $$N(t,A):=\left|\left\{s\in(0,t]...
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How the integration over $\mathbb{R}$ is transformed into summation over $\mathbb{R}_+$?

I'm not sure how this translation from integration (green box) to summation (blue box) occurred in the screenshot above. Furthermore, I'm also unsure about the change of $\mathit{v}_k$ into $\mathit{v}...
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How to interpret the cartesian product of Borel sets, $\mathcal B[0,∞)\times \mathcal B(\Bbb R \setminus \{0\})$?

I am a self-taught learner, have no formal training in measure theory. After plenty of readings, I'm assuming that the Borel field appears to contain almost every possible collection of intervals that ...
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Solution of a special case of Lèvy flight

Consider the diffusion equation $\partial_t p=D\frac{\partial}{\partial |x|}p$ where for simplicity we write p=p(x,t). By using the Fourier transform on both sides we get $\partial_t \tilde{p}=-D|k|\...
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If $\tau$ is a stopping time with $\text P[\tau>s+t]=\text P[\tau>s]\text P[\tau>t]$, how do we determine the rate of the exponential distribution?

Let $X$ be a càdlàg Lévy process, $\pi$ denote the unique random measure with $$\pi_\omega([0,t]\times B)=\sum_{s\in[0,\:t]}1_B\left({\Delta X_s(\omega)}\right),$$ where $\Delta x(t):=x(t)-\lim_{s\to ...
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How do we see that the jump time of a Lévy process is exponentially distributed using the Markov property?

Let $E$ be a normed $\mathbb R$-vector space, $X$ be a càdlàg Lévy process on a probability space $(\Omega,\mathcal A,\operatorname P)$, $B\in\mathcal B(E)$ with $0\not\in\overline B$ and$^1$ $$\tau:=\...
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If a càdlàg function $x$ has a jump $\Delta x(t)$ of magnitude $\ge r$, then $\|x(s)-x(u)\|\ge r$ for all $t-\delta\le s\le t\le u<t+\delta$

Let $(E,d)$ be a metric space and $x\colon[0,\infty)\to E$ such that $x(0)=0$ and $$x(t\pm):=\lim_{s\to t\pm}x(s)$$ exists for all $t\ge0$ ($x(0-):=x(0)$). Let $t\ge0$ and $r>0$ with $d(x(t+),x(t-)\...
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2 votes
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Lemma 2.2 on Poisson Random Measures from Kyprianou's Introductory Lectures on Fluctuations of Levy Processes

I have several problems to understand the proof of the Lemma 2.2 (page 37) from Kyprianou's "Introductory Lectures On Fluctuations Levy Processes with Applications". We assume: Let $Z_i: \...
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2 votes
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universal Markov property of Lévy processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(E,\mathcal E)$ be a measurable space; $\delta_x$ denote the Dirac measure at $x\in E$; $\pi_I$ denote the projection from $E^{[0,\:\...
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If $X$ is a Lévy process, why is $t\mapsto\sum_{\substack{s\in[0,\:t]\\\Delta X_s(\omega)}}1_B(\Delta X_s(\omega))$ càdlàg?

Let $E$ be a normed $\mathbb R$-vector space, $(X_t)_{t\ge0}$ be an $E$-valued càdlàg Lévy process on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$, $B\in\...
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Distribution of a stochastic process at a stopping time.

Suppose I have a continuous-time stochastic process $\{X(t)\}$ defined on a filtered probability space $(\Omega, \mathcal{F},\{\mathcal{F}_t\},\mathbb{P})$ and that I know the distribution of a ...
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1 vote
1 answer
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If $x$ is right-continuous with left limits, does $\{t>0:x(t)\ne\lim_{s\to t-}x(s)\}$ admit a minimum?

Let $E$ be a normed $\mathbb R$-vector space and $x:[0,\infty)\to E$ be càdlàg with $x(0)=0$. Moreover, let $x(0-):=x(0)$, $$x(t-):=\lim_{s\to t-}x(s)\;\;\;\text{for }t>0$$ and $$\Delta x(t):=x(t)-...
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Existence of jump intensity (random) measure of a Lévy process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a normed $\mathbb R$-vector space and $(X_t)_{t\ge0}$ be an $E$-valued càdlàg Lèvy process on $(\Omega,\mathcal A,\operatorname ...
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1 vote
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How do we show that the jumping times $\inf\{t:\Delta X_t\in B\}$ of a Lévy process $X$ are stopping times?

We can show the following: If $E$ is a normed $\mathbb R$-vector space, $x:[0,\infty)\to E$ is càdlàg, $B\subseteq E\setminus\{0\}$, $\tau_0:=0$ and $$\tau_n:=\inf\underbrace{\{t>\tau_{n-1}:\Delta ...
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2 answers
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Probability measure $P_X$ on the space of paths of a Lévy process $(X_t)_{t \ge 0}$ determined by $P_{X_1}$

Let $X=(X_t)_{t \ge 0}$ a Lévy process for real valued $X_i: \Omega \to \mathbb{R}$. I heard about a slogan on Lévy processes that the "the probability measure on the space of paths $t \mapsto ...
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