# 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 $\psi(\xi)$ ...
1 vote
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1 vote
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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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1 vote
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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 $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$, $B\in\mathcal B(E)$ ...
1 vote
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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: ...
1 vote
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1 vote
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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 ...
1 vote
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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.
1 vote
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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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### 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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### 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 ...