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1.
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
2.
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
3.
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
4.
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
5.
A martingale measure is constructed by using a mean correcting transform for the geometric Lévy processes model. It is shown that this measure is the mean correcting martingale measure if and only if, in the Lévy process, there exists a continuous Gaussian part. Although this measure cannot be equivalent to a physical probability for a pure jump Lévy process, we show that a European call option price under this measure is still arbitrage free. 相似文献
6.
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
7.
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
8.
A common way to inject long-range dependence in a stochastic traffic model possessing a weak regenerative structure is to
make the variance of the underlying period infinite (while keeping the mean finite). This method is supported both by physical
reasoning and by experimental evidence. We exhibit the long-range dependence of such a process and, by studying its second-order
properties, we asymptotically match its correlation structure to that of a fractional Brownian motion. By studying a certain
distributional limit theorem associated with such a process, we explain the emergence of an extremely skewed stable Lévy motion
as a macroscopic model for the aforementioned traffic. Surprisingly, long-range dependence vanishes in the limit, being “replaced”
by independent increments and highly varying marginals. The marginal distribution is computed and is shown to match the one
empirically obtained in practice. Results on performance of queueing systems with Lévy inputs of the aforementioned type are
also reported in this paper: they are shown to be in agreement with pre-limiting models, without violating experimental queueing
analysis.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
9.
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
10.
AbstractIn this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor. 相似文献
11.
Paweł J. Szabłowski 《随机分析与应用》2017,35(5):852-872
We study a class of stationary Markov processes with marginal distributions identifiable by moments such that every conditional moment of degree say m is a polynomial of degree at most m. We show that then under some additional, natural technical assumption there exists a family of orthogonal polynomial martingales. More precisely we show that such a family of processes is completely characterized by the sequence {(αn, pn)}n ? 0 where α′ns are some positive reals while p′ns are some monic orthogonal polynomials. Bakry and Mazet (Séminaire de Probabilit?s, vol. 37, 2003) showed that under some additional mild technical conditions each such sequence generates some stationary Markov process with polynomial regression.We single out two important subclasses of the considered class of Markov processes. The class of harnesses that we characterize completely. The second one constitutes of the processes that have independent regression property and are stationary. Processes with independent regression property so to say generalize ordinary Ornstein–Uhlenbeck (OU) processes or can also be understood as time scale transformations of Lévy processes. We list several properties of these processes. In particular we show that if these process are time scale transforms of Lévy processes then they are not stationary unless we deal with classical OU process. Conversely, time scale transformations of stationary processes with independent regression property are not Lévy unless we deal with classical OU process. 相似文献
12.
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
13.
We give a ??small time?? functional version of Chung??s ??other?? law of the iterated logarithm for Lévy processes with non-vanishing Brownian component. This is an analogue of the ??other?? law of the iterated logarithm at ??large times?? for Lévy processes and random walks with finite variance, as extended to a functional version by Wichura. As one of many possible applications, we mention a functional law for a two-sided passage time process. 相似文献
14.
A semi-Lévy process is an additive process with periodically stationary increments. In particular, it is a generalization of a Lévy process. The dichotomy of recurrence and transience of Lévy processes is well known, but this is not necessarily true for general additive processes. In this paper, we prove the recurrence and transience dichotomy of semi-Lévy processes. For the proof, we introduce a concept of semi-random walk and discuss its recurrence and transience properties. An example of semi-Lévy process constructed from two independent Lévy processes is investigated. Finally, we prove the laws of large numbers for semi-Lévy processes. 相似文献
15.
Jamison Wolf 《Journal of Theoretical Probability》2010,23(4):1182-1203
Several indices, such as the Blumenthal–Getoor indices, have been defined to help describe various sample path properties
for Lévy processes. These indices can be used to obtain bounds on the Hausdorff dimension of the range, graph, and zero set
for a special subclass of Lévy processes. However, there has yet to be found an index that precisely determines the dimension
of the graph for a general Lévy process. While surveying many of these results with a focus on general Lévy processes, some
of the results are generalized or improved. The culmination of this synthesis is a new index that specifies the dimension
of the graph of a general multidimensional Lévy process. 相似文献
16.
We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga, Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg, The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910]. 相似文献
17.
We consider a Lévy stochastic network as a regulated multidimensional Lévy process. The reflection direction is constant on each boundary of the positive orthant and the corresponding reflection matrix corresponds to a single-class network. We use the representation of the Lévy process and Itô's formula to arrive at some equations for the steady-state process; the latter is shown to exist, under natural stability conditions. We specialize first to the class of Lévy processes with non-negative jumps and then add the assumption of self-similarity. We show that the stationary distribution of the network corresponding the the latter process does not has product form (except in trivial cases). Finally, we derive asymptotic bounds for two-dimensional Lévy stochastic network. 相似文献
18.
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
19.
This paper extends the model and analysis in that of Vandaele and Vanmaele [Insurance: Mathematics and Economics, 2008, 42:
1128–1137]. We assume that parameters of the Lévy process which models the dynamic of risky asset in the financial market
depend on a finite state Markov chain. The state of the Markov chain can be interpreted as the state of the economy. Under
the regime switching Lévy model, we obtain the locally risk-minimizing hedging strategies for some unit-linked life insurance
products, including both the pure endowment policy and the term insurance contract. 相似文献
20.
Nikola Sandrić 《Stochastics An International Journal of Probability and Stochastic Processes》2016,88(7):1012-1040
In this paper, we study weak and strong transience of a class of Feller processes associated with pseudo-differential operators, the so-called Lévy-type processes. As a main result, we derive Chung-Fuchs type conditions (in terms of the symbol of the corresponding pseudo-differential operator) for these properties, which are sharp for Lévy processes. Also, as a consequence, we discuss the weak and strong transience with respect to the dimension of the state space and Pruitt indices, thus generalizing some well-known results related to elliptic diffusion and stable Lévy processes. Finally, in the case when the symbol is radial (in the co-variable) we provide conditions for the weak and strong transience in terms of the Lévy measures. 相似文献