Questions tagged [levy-processes]
Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.
335
questions
0
votes
0
answers
20
views
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 ...
- 37
0
votes
0
answers
18
views
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 ...
0
votes
1
answer
56
views
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 ...
- 1,124
2
votes
1
answer
27
views
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{...
- 422
2
votes
0
answers
18
views
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 ...
- 73
3
votes
1
answer
38
views
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 $\psi(\xi)$ ...
- 69
1
vote
0
answers
28
views
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 $\...
- 71
0
votes
0
answers
15
views
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 ...
- 1,597
0
votes
0
answers
22
views
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.
- 1
-1
votes
1
answer
36
views
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 ...
- 12.8k
1
vote
1
answer
20
views
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 $$\int\left|\...
- 12.8k
0
votes
0
answers
20
views
Lévy characterization of Hilbert-space-valued Wiener process
Let $H$ be a $\mathbb R$-Hilbert space (assume $H=\mathbb R^d$ for some $d\in\mathbb N$, if this is helpful for you to understand the following) and $(X_t)_{t\ge0}$ be an $H$-valued continuous Lévy ...
- 12.8k
1
vote
1
answer
62
views
Show that every continuous Lévy process is a Wiener process
Let $H$ be a $\mathbb R$-Hilbert space (assume $H=\mathbb R^d$ for some $d\in\mathbb N$, if this is helpful for you to understand the following) and $(X_t)_{t\ge0}$ be an $H$-valued continuous Lévy ...
- 12.8k
0
votes
1
answer
25
views
If $X$ is a Lévy process, then $\operatorname{Var}[X_n]=n\operatorname{Var}[X_1]$
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $H$ be a $\mathbb R$-Hilbert space, $(X_t)_{t\ge0}$ be an $H$-valued Lévy process on $(\Omega,\mathcal A,\operatorname P)$ and $\mu_t:=...
- 12.8k
1
vote
0
answers
19
views
Is the jump measure of a space homogeneous Markov process translation invariant?
Let $E$ be a normed $\mathbb R$-vector space and $(X_t)_{t\ge0}$ be an $E$-valued Lévy process. $X$ is a space- and time-homogeneous Markov process with transition semigroup $$\kappa_t(x,B):=\...
- 12.8k
2
votes
0
answers
68
views
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 ...
- 12.8k
1
vote
1
answer
64
views
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 $E$...
- 12.8k
2
votes
0
answers
22
views
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 $\...
- 12.8k
2
votes
0
answers
31
views
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. ...
- 21
3
votes
1
answer
53
views
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 $E$...
- 12.8k
1
vote
0
answers
16
views
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)\...
- 12.8k
1
vote
0
answers
17
views
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 $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$, $B\in\mathcal B(E)$ ...
- 12.8k
1
vote
0
answers
85
views
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 $$\Delta x(t):=x(t+)-x(t-)\;\;\;\text{for ...
- 12.8k
1
vote
0
answers
32
views
Is the reciprocal of Lévy-Khintchine theorem for Levy process true?
We know that if $X = [X_t , t \in \mathbb{Z}]$ is a Levy process, then any marginal distribution has the charasteristic function - ch. f. of $X_t$ - given by $\varphi_t(r)=e^{t \phi(r)}$ whith:
$$\...
- 637
2
votes
0
answers
36
views
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 ...
- 12.8k
2
votes
0
answers
59
views
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:
...
- 21
1
vote
1
answer
41
views
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 $$\pi_\omega(B):=\sum_{\substack{s\:\ge\:...
- 12.8k
0
votes
0
answers
39
views
An application of Ito lemma for Ito-Levy processes
For an Ito-Levy process $X_t$ with the dynamics given by
$$
\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}(\...
- 91
1
vote
0
answers
20
views
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 ...
- 381
1
vote
0
answers
63
views
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 ...
- 51
2
votes
0
answers
30
views
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 ...
- 61
2
votes
0
answers
40
views
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 ...
- 637
0
votes
0
answers
42
views
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 ...
- 51
0
votes
1
answer
50
views
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
1
answer
59
views
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:
...
- 367
2
votes
1
answer
94
views
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$^1$ $$N(t,A):=\left|\left\{s\in(0,t]...
- 12.8k
0
votes
0
answers
37
views
How the integration over $\mathbb{R}$ is transformed into summation over $\mathbb{R}_+$?
I'm not sure how this translation from integration (green box) to summation (blue box) occurred in the screenshot above. Furthermore, I'm also unsure about the change of $\mathit{v}_k$ into $\mathit{v}...
- 13
0
votes
0
answers
36
views
How to interpret the cartesian product of Borel sets, $\mathcal B[0,∞)\times \mathcal B(\Bbb R \setminus \{0\})$?
I am a self-taught learner, have no formal training in measure theory.
After plenty of readings, I'm assuming that the Borel field appears to contain almost every possible collection of intervals that ...
- 13
1
vote
0
answers
14
views
Solution of a special case of Lèvy flight
Consider the diffusion equation
$\partial_t p=D\frac{\partial}{\partial |x|}p$
where for simplicity we write p=p(x,t). By using the Fourier transform on both sides we get $\partial_t \tilde{p}=-D|k|\...
4
votes
1
answer
174
views
If $\tau$ is a stopping time with $\text P[\tau>s+t]=\text P[\tau>s]\text P[\tau>t]$, how do we determine the rate of the exponential distribution?
Let $X$ be a càdlàg Lévy process, $\pi$ denote the unique random measure with $$\pi_\omega([0,t]\times B)=\sum_{s\in[0,\:t]}1_B\left({\Delta X_s(\omega)}\right),$$ where $\Delta x(t):=x(t)-\lim_{s\to ...
- 12.8k
1
vote
1
answer
55
views
How do we see that the jump time of a Lévy process is exponentially distributed using the Markov property?
Let $E$ be a normed $\mathbb R$-vector space, $X$ be a càdlàg Lévy process on a probability space $(\Omega,\mathcal A,\operatorname P)$, $B\in\mathcal B(E)$ with $0\not\in\overline B$ and$^1$ $$\tau:=\...
- 12.8k
0
votes
1
answer
53
views
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-)\...
- 12.8k
2
votes
1
answer
95
views
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: \...
- 623
2
votes
0
answers
84
views
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,\:\...
- 12.8k
1
vote
1
answer
74
views
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\...
- 12.8k
0
votes
1
answer
65
views
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 ...
1
vote
1
answer
30
views
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 $$\Delta x(t):=x(t)-...
- 12.8k
1
vote
0
answers
34
views
Existence of jump intensity (random) measure of a Lévy process
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a normed $\mathbb R$-vector space and $(X_t)_{t\ge0}$ be an $E$-valued càdlàg Lèvy process on $(\Omega,\mathcal A,\operatorname ...
- 12.8k
1
vote
1
answer
62
views
How do we show that the jumping times $\inf\{t:\Delta X_t\in B\}$ of a Lévy process $X$ are stopping times?
We can show the following: If $E$ is a normed $\mathbb R$-vector space, $x:[0,\infty)\to E$ is càdlàg, $B\subseteq E\setminus\{0\}$, $\tau_0:=0$ and $$\tau_n:=\inf\underbrace{\{t>\tau_{n-1}:\Delta ...
- 12.8k
1
vote
2
answers
67
views
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 ...
- 623