共查询到20条相似文献,搜索用时 15 毫秒
1.
In this paper, we investigate the long-range dependence of fractional Lévy processes on Gel’fand triple and construct stochastic
integral with respect to fractional Lévy processes for a class of deterministic integrands.
相似文献
2.
Young Shin Kim 《Applied Mathematical Finance》2016,23(4):309-322
This paper discusses the long-range dependence in the risk-neutral stock return process of the S&P 500 index option market. To observe the long-range dependence together with fat-tails, I define the parametric model of fractional Lévy process. Since the continuous time fractional Lévy process allows arbitrage, I use discrete time option pricing model based on the fractional Lévy process. By model calibration, we can capture the long-range dependence in the S&P 500 index option market. The paper finds that the long range dependence becomes stronger for the volatile market caused by the Lehman Brothers Collapse, comparing with other less volatility markets. 相似文献
3.
We review functional central limit theorems (FCLTs) for the queue-content process in a single-server queue with finite waiting
room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with
reflected Brownian motion (RBM) limit process that arise with heavy-tailed probability distributions and strong dependence.
Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem
with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume
that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite variance
or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable
process, whose steady-state distribution can be calculated by numerically inverting its Laplace transform. For a sequence
of models, the queue need not be in heavy traffic, and the limit can be a general reflected Lévy process. When the Lévy process
representing the net input has no negative jumps, the steady-state distribution of the reflected Lévy process again can be
calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queue-content process when the
input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence.
Then the limiting input process is a Gaussian process. When the limiting net-input process is also a Gaussian process and
there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian process can be conveniently
approximated.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
4.
We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein–Uhlenbeck processes driven by Lévy motion and their finite and infinite superpositions. We construct the multifractal, such as log-gamma, log-tempered stable, or log-normal tempered stable scenarios. 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.
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
7.
Vladimir Panov 《Methodology and Computing in Applied Probability》2017,19(1):97-119
In this paper, we analyze a Lévy model based on two popular concepts - subordination and Lévy copulas. More precisely, we consider a two-dimensional Lévy process such that each component is a time-changed (subordinated) Brownian motion and the dependence between subordinators is described via some Lévy copula. The main result of this paper is the series representation for our model, which can be efficiently used for simulation purposes. 相似文献
8.
It has recently been shown that in the heavy traffic limit, the stationary distribution of the scaled queue length process
of a Generalized Jackson Network converges to the stationary distribution of its corresponding Reflected Brownian Motion limit.
In this paper, we show that this “interchange of limits” is valid for Stochastic Fluid Networks with Lévy inputs. Furthermore,
under additional assumptions, we extend the result to show that the interchange is valid for moments of the stationary distribution
and for state-dependent routing. The results are obtained using monotonicity and sample-path arguments. 相似文献
9.
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
10.
This paper considers queues with server vacations, but departs from the traditional setting in two ways: (i) the queueing
model is driven by Lévy processes rather than just compound Poisson processes; (ii) the vacation lengths depend on the length
of the server’s preceding busy period. Regarding the former point: the Lévy process active during the busy period is assumed
to have no negative jumps, whereas the Lévy process active during the vacation is a subordinator. Regarding the latter point:
where in a previous study (Boxma et al. in Probab. Eng. Inf. Sci. 22:537–555, 2008) the durations of the vacations were positively correlated with the length of the preceding busy period, we now introduce
a dependence structure that may give rise to both positive and negative correlations. We analyze the steady-state workload of the resulting queueing (or: storage) system, by first considering
the queue at embedded epochs (viz. the beginnings of busy periods). We show that this embedded process does not always have
a proper stationary distribution, due to the fact that there may occur an infinite number of busy-vacation cycles in a finite
time interval; we specify conditions under which the embedded process is recurrent. Fortunately, irrespective of whether the
embedded process has a stationary distribution, the steady-state workload of the continuous-time storage process can be determined. In addition, a number of ramifications are presented. The theory is illustrated by several examples. 相似文献
11.
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
12.
In this paper, we analyse processes of Ornstein-Uhlenbeck (OU) type, driven by Lévy processes. This class is designed to capture
mean reverting behaviour if it exists; but the data may in fact be adequately described by a pure Lévy process with no OU
(autoregressive) effect. For an appropriate discretised version of the model, we utilise likelihood methods to test for such
a reduction of the OU process to Lévy motion, deriving the distribution of the relevant pseudo-log-likelihood ratio statistics,
asymptotically, both for a refining sequence of partitions on a fixed time interval with mesh size tending to zero, and as
the length of the observation window grows large. These analyses are non-standard in that the mean reversion parameter vanishes
under the null of a pure Lévy process for the data. Despite this we are able to give a very general analysis with no technical
restrictions on the underlying processes or parameter sets, other than a finite variance assumption for the Lévy process.
As a special case, for Brownian motion as driving process, we deduce the limiting distribution in a quite explicit way, finding
results which generalise the well-known Dickey-Fuller (‘unit-root’) theory.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
13.
Vicky Fasen 《Queueing Systems》2010,66(4):313-350
We consider a cluster Poisson model with heavy-tailed interarrival times and cluster sizes as a generalization of an infinite
source Poisson model where the file sizes have a regularly varying tail distribution function or a finite second moment. One
result is that this model reflects long-range dependence of teletraffic data. We show that depending on the heaviness of the
file sizes, the interarrival times and the cluster sizes we have to distinguish different growths rates for the time scale
of the cumulative traffic. The mean corrected cumulative input process converges to a fractional Brownian motion in the fast
growth case. However, in the intermediate and the slow growth case we can have convergence to a stable Lévy motion or a fractional
Brownian motion as well depending on the heaviness of the underlying distributions. These results are contrary to the idea
that cumulative broadband network traffic converges in the slow growth case to a stable process. Furthermore, we derive the
asymptotic behavior of the cluster Poisson point process which models the arrival times of data packets and the individual
input process itself. 相似文献
14.
Horst Osswald 《Journal of Theoretical Probability》2009,22(2):441-473
An approach to Malliavin calculus for Lévy processes, discrete in time and smooth in chance, is presented. Each Lévy triple
can be satisfied by a Lévy process living on a fixed sample space Ω, which is, in a certain sense, a finite dimensional Euclidean
space. The probability measures on Ω characterize the Lévy processes. We compare these measures with the associated Lévy measures,
and present several examples. Using chaos expansions for Lévy functionals, even for those having no moments, we can represent
all these functionals by polynomials in several variables. There exists an effective method to compute the kernels of the
chaos decomposition. Finally, we point out several applications, which are postponed to a succession of papers.
Dedicated to Helmut Schwichtenberg. 相似文献
15.
Fractional Brownian motion can be represented as an integral of a deterministic kernel w.r.t. an ordinary Brownian motion either on infinite or compact interval. In previous literature fractional Lévy processes are defined by integrating the infinite interval kernel w.r.t. a general Lévy process. In this article we define fractional Lévy processes using the com pact interval representation. We prove that the fractional Lévy processes presented via different integral transformations have the same finite dimensional distributions if and only if they are fractional Brownian motions. Also, we present relations between different fractional Lévy processes and analyze the properties of such processes. A financial example is introduced as well. 相似文献
16.
We deal with the least squares estimator for the drift parameters of an Ornstein-Uhlenbeck process with periodic mean function driven by fractional Lévy process. For this estimator, we obtain consistency and the asymptotic distribution. Compared with fractional Ornstein-Uhlenbeck and Ornstein-Uhlenbeck driven by Lévy process, they can be regarded both as a Lévy generalization of fractional Brownian motion and a fractional generaliza- tion of Lévy process. 相似文献
17.
WANG DingCheng 《中国科学 数学(英文版)》2013,56(11):2353-2366
In the paper,using Lvy processes subordinated by‘asymptotically self-similar activity time’processes with long-range dependence,we set up new asset pricing models.Using the diferent construction for gamma(Γ)based‘asymptotically self-similar activity time’processes with long-range dependence from Finlay and Seneta(2006)we extend the constructions for inverse-gamma and gamma based‘asymptotically selfsimilar activity time’processes with integer-valued parameters and long-range dependence in Heyde and Leonenko(2005)and Finlay and Seneta(2006)to noninteger-valued parameters. 相似文献
18.
Michele Nguyen Almut E. D. Veraart 《Stochastics An International Journal of Probability and Stochastic Processes》2018,90(7):1023-1052
While short-range dependence is widely assumed in the literature for its simplicity, long-range dependence is a feature that has been observed in data from finance, hydrology, geophysics and economics. In this paper, we extend a Lévy-driven spatio-temporal Ornstein–Uhlenbeck process by randomly varying its rate parameter to model both short-range and long-range dependence. This particular set-up allows for non-separable spatio-temporal correlations which are desirable for real applications, as well as flexible spatial covariances which arise from the shapes of influence regions. Theoretical properties such as spatio-temporal stationarity and second-order moments are established. An isotropic g-class is also used to illustrate how the memory of the process is related to the probability distribution of the rate parameter. We develop a simulation algorithm for the compound Poisson case which can be used to approximate other Lévy bases. The generalized method of moments is used for inference and simulation experiments are conducted with a view towards asymptotic properties. 相似文献
19.
We construct random locally compact real trees called Lévy trees that are the genealogical trees associated with continuous-state
branching processes. More precisely, we define a growing family of discrete Galton–Watson trees with i.i.d. exponential branch
lengths that is consistent under Bernoulli percolation on leaves; we define the Lévy tree as the limit of this growing family
with respect to the Gromov–Hausdorff topology on metric spaces. This elementary approach notably includes supercritical trees
and does not make use of the height process introduced by Le Gall and Le Jan to code the genealogy of (sub)critical continuous-state
branching processes. We construct the mass measure of Lévy trees and we give a decomposition along the ancestral subtree of
a Poisson sampling directed by the mass measure.
T. Duquesne is supported by NSF Grants DMS-0203066 and DMS-0405779. M. Winkel is supported by Aon and the Institute of Actuaries,
EPSRC Grant GR/T26368/01, le département de mathématique de l’Université d’Orsay and NSF Grant DMS-0405779. 相似文献
20.
A class of generalized Lévy Laplacians which contain as a special case the ordinary Lévy Laplacian are considered. Topics
such as limit average of the second order functional derivative with respect to a certain equally dense (uniformly bounded)
orthonormal base, the relations with Kuo’s Fourier transform and other infinite dimensional Laplacians are studied. 相似文献