共查询到20条相似文献,搜索用时 15 毫秒
1.
B. Grigelionis 《Lithuanian Mathematical Journal》2008,48(1):61-69
Lévy processes with marginal relativistic α-stable distributions are described. Strictly stationary Ornstein-Uhlenbeck type processes with one-dimentional relativistic
α-stable distributions are constructed. The exponential family as Esscher transforms of distributions on D
[0,∞)(R
d
) of relativistic α-stable Lévy processes is obtained and the corresponding mixed exponential processes are characterized. 相似文献
2.
Ante Mimica 《Potential Analysis》2010,32(3):275-303
In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝ
d
. We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Lévy process
in ℝ
d
with a Lévy density given by $c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}$c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}, where 0 ≤ j(r) ≤ cr
− d − α
, ∀ r > 1, for some constant c. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion
with subordinator with Laplace exponent ϕ(λ) = λ
α/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2). 相似文献
3.
Nicole Bäuerle Anja Blatter Alfred Müller 《Mathematical Methods of Operations Research》2008,67(1):161-186
In this paper we investigate dependence properties and comparison results for multidimensional Lévy processes. In particular
we address the questions, whether or not dependence properties and orderings of the copulas of the distributions of a Lévy
process can be characterized by corresponding properties of the Lévy copula, a concept which has been introduced recently
in Cont and Tankov (Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton, 2004) and Kallsen and Tankov
(J Multivariate Anal 97:1551–1572, 2006). It turns out that association, positive orthant dependence and positive supermodular dependence of Lévy processes can be characterized in terms of the Lévy measure as well as in terms of the Lévy copula. As
far as comparisons of Lévy processes are concerned we consider the supermodular and the concordance order and characterize
them by orders of the Lévy measures and by orders of the Lévy copulas, respectively. An example is given that the Lévy copula
does not determine dependence concepts like multivariate total positivity of order 2 or conditionally increasing in sequence. Besides these general results we specialize our findings for subfamilies of Lévy processes. The last section contains some
applications in finance and insurance like comparison statements for ruin times, ruin probabilities and option prices which
extends the current literature.
Anja Blatter was supported by the Deutsche Forschungsgemeinschaft (DFG). 相似文献
4.
Joshua Rushton 《Journal of Theoretical Probability》2007,20(3):397-427
We establish a functional LIL for the maximal process M(t) :=sup 0≤s≤t
‖X(s)‖ of an ℝ
d
-valued α-stable Lévy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing
a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum
process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition
necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Lévy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X.
Supported in part by NSF Grant DMS 02-05034. 相似文献
5.
Brice Franke 《Journal of Theoretical Probability》2007,20(4):1087-1100
We prove a functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Lévy-processes
with stability index α>1. The limit process turns out to be an α-stable Lévy process with an averaged jump-measure. Unlike in the situation where the diffusion is driven by Brownian motion,
there is no drift related enhancement of diffusivity. 相似文献
6.
A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov
processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably
time-changing the exponential of the Lévy process. In this paper we generalise Lamperti's result to processes in n dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of
the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher
dimensions, referred to as multi-self-similarity in the paper.
The special case where the Lévy process ξ is standard Brownian motion in n dimensions is studied in detail. There are also specific comments on the case where ξ is an n-dimensional compound Poisson process with drift.
Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond
to n-dimensional subordinators.
Received: 22 August 2002 / Revised version: 10 February 2003
Published online: 15 April 2003
RID="*"
ID="*" MaPhySto – Centre for Mathematical Physics and Stochastics, funded by a grant from the Danish National Research Foundation
Mathematics Subject Classification (2000): 60G18, 60G51, 60J25, 60J60, 60J75
Key words or phrases: Lévy process – Self-similarity – Time-change – Exponential functional – Brownian motion – Bessel process – Piecewise deterministic
Markov process – Moment sequence 相似文献
7.
M. Zähle 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2009,44(2):117-145
Potential spaces and Dirichlet forms associated with Lévy processes subordinate to Brownian motion in ℝ
n
with generator f(−Δ) are investigated. Estimates for the related Rieszand Bessel-type kernels of order s are derived which include the classical
case f(r) = r
α/2 with 0 < α < 2 corresponding to α-stable Lévy processes. For general (tame) Bernstein functions f potential representations of the trace spaces, the trace Dirichlet forms, and the trace processes on fractal h-sets are derived. Here we suppose the trace condition ∫01
r
−(n+1)
f(r
−2)−1
h(r) dr < ∞ on f and the gauge function h.
Dedicated to the 80th birthday of Klaus Krickeberg 相似文献
8.
Paulius Miškinis 《Applied Categorical Structures》2008,16(1-2):213-221
In the case of the quantum generalization of stable Lévy processes, expressions for the Hermitian operator of momentum and
its eigenfunctions are proposed. The normalization constant has been determined and its relation to the translation operator
is shown. The interrelation between the momentum and the wave number has been generalized for the processes with a non-integer
dimensionality α. The simplest nonlocal superalgebra is introduced.
相似文献
9.
The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place
for half-spaces of codimension one for a diffusion in ℝ
n + 1, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this
amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent
Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space or of a strip for the operator (I − Δ)
α/2, 0 < α < 2. In the case of a half-space, this result was recently found, by different methods, in Byczkowski et al. (Trans Am Math Soc 361:4871–4900, 2009). As an application of our method we also compute various formulas for first hitting places for the isotropic stable Lévy process. 相似文献
10.
Stéphane Jaffard 《Probability Theory and Related Fields》1999,114(2):207-227
We show that the sample paths of most Lévy processes are multifractal functions and we determine their spectrum of singularities.
Received: 21 February 1997 / Revised version: 27 July 1998 相似文献
11.
Tom Lindstrøm 《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(6):517-548
A hyperfinite Lévy process is an infinitesimal random walk (in the sense of nonstandard analysis) which with probability one is finite for all finite times. We develop the basic theory for hyperfinite Lévy processes and find a characterization in terms of transition probabilities. The standard part of a hyperfinite Lévy process is a (standard) Lévy process, and we show that given a generating triplet (γ, C, μ) for standard Lévy processes, we can construct hyperfinite Lévy processes whose standard parts correspond to this triplet. Hence all Lévy laws can be obtained from hyperfinite Lévy processes. The paper ends with a brief look at Malliavin calculus for hyperfinite Lévy processes including a version of the Clark-Haussmann-Ocone formula. 相似文献
12.
Steven N. Evans 《Probability Theory and Related Fields》2000,118(1):37-48
If X is a symmetric Lévy process on the line, then there exists a non-decreasing, càdlàg process H such that X(H(x)) = x for all x≥ 0 if and only if X is recurrent and has a non-trivial Gaussian component. The minimal such H is a subordinator K. The law of K is identified and shown to be the same as that of a linear time change of the inverse local time at 0 of X. When X is Brownian motion, K is just the usual ladder times process and this result extends the classical result of Lévy that the maximum process has
the same law as the local time at 0. Write G
t
for last point in the range of K prior to t. In a parallel with classical fluctuation theory, the process Z := (X
t
−X
Gt
)
t
≥0 is Markov with local time at 0 given by (X
Gt
)
t
≥0. The transition kernel and excursion measure of Z are identified. A similar programme is outlined for Lévy processes on the circle. This leads to the construction of a stopping
time such that the stopped local times constitute a stationary process indexed by the circle.
Received: 7 September 1999 / Revised version: 9 November 1999 / Published online: 8 August 2000 相似文献
13.
Endre Csáki Miklós Csörgő Antónia Földes Zhan Shi 《Probability Theory and Related Fields》2000,117(4):515-531
Let W be a standard Brownian motion, and define Y(t)= ∫0
t
ds/W(s) as Cauchy's principal value related to local time. We determine: (a) the modulus of continuity of Y in the sense of P. Lévy; (b) the large increments of Y.
Received: 1 April 1999 / Revised version: 27 September 1999 / Published online: 14 June 2000 相似文献
14.
Mladen Savov 《Journal of Theoretical Probability》2010,23(1):209-236
We specify a function b
0(t) in terms of the Lévy triplet such that lim sup
t→0
X
t
/b
0(t)∈[1,1.8] a.s. iff
ò01[` \varPi ](+)(b0(t)) dt < ¥\int_{0}^{1}\overline{ \varPi }^{(+)}(b_{0}(t))\,dt<\infty
for any Lévy process X with unbounded variation and a Brownian component σ=0. We show with an example that there are cases where lim sup
t→0
X
t
/b(t)=1 a.s. but b(t) is not asymptotically equivalent to b
0(t) as t tends to 0. We achieve this by introducing an integral criterion which checks whether lim sup
t→0
X
t
/b(t) is 0, infinity, or a finite positive value for b(t) satisfying very mild conditions and any Lévy process. 相似文献
15.
Liming Wu 《Probability Theory and Related Fields》2000,118(3):427-438
By means of the martingale representation, we establish a new modified logarithmic Sobolev inequality, which covers the previous
modified logarithmic Sobolev inequalities of Bobkov-Ledoux and the L
1-logarithmic Sobolev inequality obtained in our previous work. From it we derive several sharp deviation inequalities of Talagrand's
type, by following the powerful Herbst method developed recently by Ledoux and al. Moreover this new modified logarithmic
Sobolev inequality is transported on the discontinuous path space with respect to the law of a Lévy process.
Received: 16 June 1999 / Revised version: 13 March 2000 / Published online: 12 October 2000 相似文献
16.
Michael G. Akritas 《Annals of the Institute of Statistical Mathematics》1982,34(1):259-280
Summary We consider consistency and asymptotic normality of maximum likelihood estimators (MLE) for parameters of a Lévy process of
the discontinuous type. The MLE are based on a single realization of the process on a given interval [0,t]. Depending on properties of the Lévy measure we either consider the MLE corresponding to jumps of size greater than ε and,
keepingt fixed, we let ε tend to 0, or we consider the MLE corresponding to the complete information of the realization of the process
on [0,t] and lett tend to ∞. The results of this paper improve in both generality and rigor previous asymptotic estimation results for such
processes. 相似文献
17.
For any α∈(0,2), a truncated symmetric α-stable process in ℝ
d
is a symmetric Lévy process in ℝ
d
with no diffusion part and with a Lévy density given by c|x|−d−α
1{|x|<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, [2007]) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary
Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that
the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However,
for truncated symmetric stable processes, the boundary Harnack principle is not valid in non-convex domains. In this paper,
we show that, for a large class of not necessarily convex bounded open sets in ℝ
d
called bounded roughly connected κ-fat open sets (including bounded non-convex κ-fat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean
boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly
connected κ-fat open sets.
The research of P. Kim is supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic
Research Promotion Fund) (KRF-2007-331-C00037).
The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167. 相似文献
18.
Franziska Kühn 《Mathematische Nachrichten》2019,292(2):358-376
We present an existence result for Lévy‐type processes which requires only weak regularity assumptions on the symbol with respect to the space variable x. Applications range from existence and uniqueness results for Lévy‐driven SDEs with Hölder continuous coefficients to existence results for stable‐like processes and Lévy‐type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy‐type processes. The paper includes an extensive list of Lévy(‐type) processes satisfying the assumptions of our results. 相似文献
19.
Maria E. Vares 《Bulletin of the Brazilian Mathematical Society》1981,12(2):33-55
Summary In this article we consider some problems on local growth of 2-parameter Lévy processes, i.e., processes with independent
and stationary increments, indexed by [0,+∞)×[0,+∞). The results are for the upper growth of these processes, at a fixed “time”z
0. 相似文献
20.
Serguei Foss Takis Konstantopoulos Stan Zachary 《Journal of Theoretical Probability》2007,20(3):581-612
We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to −∞ and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural
conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the
process S in continuous time reduces to a Lévy process with heavy-tailed Lévy measure. A central point of the paper is that we make
full use of the so-called “principle of a single big jump” in order to obtain both upper and lower bounds. Thus, the proofs
are entirely probabilistic. The paper is motivated by queueing and Lévy stochastic networks. 相似文献