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Questions tagged [levy-processes]

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

3
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1answer
156 views

Lévy Process existence of the expectation of the supremum of the past process.

Given a Lévy Process $X_{t}$ in $\mathbb{R}^{d}$, with $X_{t}^{*}:=\sup_{s\in[0,t]}|X_{s}|$. I want to show, that for $t>0$ with $E[|X_{t}|]<\infty$ for $t>0$, then $E[X_{t}^{*}]<\...
6
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1answer
201 views

Central Limit Theorem for Lévy Process

I am reading a book, which uses the Central Limit Theorem of Lévy Processes $X_t$ without mentioning the exact theorem. Due to the infinite divisible property I can write $X_t$ as a sum of $N$ iid ...
2
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2answers
130 views

Small time behavior of Lévy processes

Let $(X_t)_{t \geq 0}$ a Lévy process and $\varepsilon>0$. Is there anything known about the asymptotics of the probability $$\mathbb{P}(|X_t| > \varepsilon)$$ as $t \to 0$? Obviously, by the ...
0
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1answer
209 views

Finiteness of the number of big jumps of a Lévy process on a finite interval

Let $Y(t)$, $0 \leq t \leq 1$, be a Lévy process. Denote by $\{ \Delta Y_i \}$ the jumps of the process. I would like to show that the set $\{i: |\Delta Y_i| > r\}$ is finite almost surely. However,...
2
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1answer
358 views

Etemadi's inequality

In another post an inequality referred to as "Etemadi's Inequality" is mentioned twice - in the original post as well as in the answer. However, the contexts of usage are such as to raise the question ...
2
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1answer
280 views

Fixed-time Jumps of a Lévy process

If one defines a Levy process as a stochastic process $(X_t)_{t\geq0}$ that has independent and stationary increments with (a.s.) cadlag paths (hence a def. withouth stochastic continuity). How can I ...
2
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1answer
378 views

Equation involving expectations of Levy processes

I have the following equation : $$ E[e^{(\theta +1) X_T} | \mathcal{F}_t] = e^{X_t} E[e^{\theta X_T} | \mathcal{F}_t] $$ where $X_t$ is a Levy process, $\mathcal{F_t}$ the filtration at time $t$, $T &...
1
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1answer
349 views

Conditional law of the arrival times of a Poisson process

The following is Exercise 2.1.(i) from the book "Introductory Lectures on Fluctuations of Levy Processes with Applications". I use the book for self study and couldn't prove this exercise. I would be ...
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1answer
97 views

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). ...
5
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1answer
169 views

A proof by René Schilling that a continuous Lévy process is integrable

In his treatise "An Introduction to Lévy and Feller Processes" (arXiv link), Prof. Dr. René Schilling gives a short and seemingly straightforward proof for the claim that a continuous Lévy process is ...
5
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2answers
1k views

Good book that contains stochastic integration, martingales and Lévy-processes?

Does anyone know about any good and easy interoductory books which contins information about martingales, sotchastic integration and Lévy-processes? I have tried reading: http://www.cambridge.org/us/...
3
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3answers
404 views

Jumps of independent Lévy processes

Suppose I have two independent Lévy processes $(X_t)_t$ and $(Y_t)_t$, both not continuous. Is anyone familiar and can refer me to a result (or a counterexample) which states that ${\displaystyle \...
1
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1answer
90 views

If $X$ is a Lévy process w.r.t. the natural filtration $\mathcal{F}$, is it so w.r.t. the right-continuous modified filtration $\mathcal{F}^+$?

Given a probability space $S=(\Omega, \mathcal{A}, P)$, a filtration $\mathcal{F}=(\mathcal{F}_t)_{t\in[0,\infty)}$ on $\mathcal{A}$, and a separable normed space $H$, whose induced topology shall be ...
1
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1answer
107 views

Every zero-mean Lévy process has linear variance (wrt $t$)

I'd like to show that every Lévy process with $\mathbb{E}X_t=0, \:\forall t\ge0$ has linear variance, namely $t\mapsto\mathbb{E}X^2_t$ is linear. I showed that indeed the additivity holds, i.e. $t+s\...
0
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1answer
434 views

How do I find the Levy triplet of a Levy process

I know the levy triplet of a Poisson process $N_t$- $(0,0,λδ_1(y))$ and its characteristic function is $exp[-t\Bigl(\intop_{0}^{\infty}(1-e^{iuy}+iuy1_{\{\mathbf{|}\mathbf{y}|<1\}})\delta_{1}(y)\...
2
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1answer
91 views

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 ...
2
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0answers
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 ...
1
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1answer
355 views

Stationary Markov process properties

Let $X$ be a right-continuous process with values in $(E,\mathcal{E})$, defined on $(\Omega, \mathcal{F}_t,P)$. Suppose that $X$ has stationary, independent increments. I now want to show the ...
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0answers
78 views

Doubt in the proof of right continuity of the completed Levy Filtration

I was self-studying the book on Stochastic Integration by Protter for my Phd seminar in statistics and I am stuck on theorem 31 where the author proves that the filtration of the Levy Process(...
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0answers
157 views

Proving that the Poisson process has a.s. jumps of value 1.

A poisson process $N(t)$ is a Levy process, where where $N(t)-N(s), t-s=h$ has a discrete distribution $(\lambda h)^k e^{-\lambda h}/k!$. I want to prove that the jumps of this process has value 1 a. ...