scholarly journals Invariance properties of the negative binomial Levy process and stochastic self-similarity

2007 ◽  
Vol 2 ◽  
pp. 1457-1468 ◽  
Author(s):  
T. J. Kozubowski ◽  
K. Podgorski
Keyword(s):  
Lévy Process ◽  
Negative Binomial ◽  
Levy Process ◽  
Self Similarity ◽  
Invariance Properties

scholarly journals A bivariate Lévy process with negative binomial and gamma marginals

2008 ◽  
Vol 99 (7) ◽  
pp. 1418-1437 ◽  
Author(s):  
Tomasz J. Kozubowski ◽  
Anna K. Panorska ◽  
Krzysztof Podgórski
Keyword(s):  
Lévy Process ◽  
Negative Binomial ◽  
Levy Process

scholarly journals Signal processing with Lévy information

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120433 ◽  
Author(s):  
Dorje C. Brody ◽  
Lane P. Hughston ◽  
Xun Yang
Keyword(s):  
Lévy Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Levy Process ◽  
Inverse Gaussian ◽  
Information Process ◽  
True Value ◽  
Information Processes ◽  
Noise Type ◽  
Exponential Moments

Lévy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Lévy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced with an independent random variable, the true value of which represents a ‘message’, then under the transformed measure the original Lévy process takes on the character of an ‘information process’. In this paper we develop a theory of such Lévy information processes. The underlying Lévy process, which we call the fiducial process, represents the ‘noise type’. Each such noise type is capable of carrying a message of a certain specification. A number of examples are worked out in detail, including information processes of the Brownian, Poisson, gamma, variance gamma, negative binomial, inverse Gaussian and normal inverse Gaussian type. Although in general there is no additive decomposition of information into signal and noise, one is led nevertheless for each noise type to a well-defined scheme for signal detection and enhancement relevant to a variety of practical situations.

scholarly journals LAN property for a simple Lévy process

2014 ◽  
Vol 352 (10) ◽  
pp. 859-864 ◽  
Author(s):  
Arturo Kohatsu-Higa ◽  
Eulalia Nualart ◽  
Ngoc Khue Tran
Keyword(s):  
Lévy Process ◽  
Levy Process

scholarly journals On the optimal dividend problem for a spectrally negative Lévy process

2007 ◽  
Vol 17 (1) ◽  
pp. 156-180 ◽  
Author(s):  
Florin Avram ◽  
Zbigniew Palmowski ◽  
Martijn R. Pistorius
Keyword(s):  
Lévy Process ◽  
Levy Process ◽  
Spectrally Negative Lévy Process ◽  
Optimal Dividend

scholarly journals Limit Theory for High Frequency Sampled MCARMA Models

2014 ◽  
Vol 46 (3) ◽  
pp. 846-877 ◽  
Author(s):  
Vicky Fasen
Keyword(s):  
Asymptotic Behavior ◽  
High Frequency ◽  
Lévy Process ◽  
Asymptotic Properties ◽  
Domain Of Attraction ◽  
Levy Process ◽  
Sample Variance ◽  
Limit Theory ◽  
Second Moments ◽  
Time Grid

We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {hn, 2hn,…, nhn}, where hn ↓ 0 and nhn → ∞ as n → ∞, or at a constant time grid where hn = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

scholarly journals Last Exit Before an Exponential Time for Spectrally Negative Lévy Processes

2009 ◽  
Vol 46 (02) ◽  
pp. 542-558 ◽  
Author(s):  
E. J. Baurdoux
Keyword(s):  
Laplace Transform ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Risk Theory ◽  
Exponential Time ◽  
Main Motivation ◽  
Spectrally Negative Lévy Process ◽  
The Laplace Transform ◽  
Spectrally Negative Lévy Processes

Chiu and Yin (2005) found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to ∞, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to ∞, is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in Doney (1991).

scholarly journals A factorization of a Lévy process over a phase-type horizon

2018 ◽  
Vol 34 (4) ◽  
pp. 397-408 ◽  
Author(s):  
Søren Asmussen ◽  
Jevgenijs Ivanovs
Keyword(s):  
Lévy Process ◽  
Levy Process ◽  
Phase Type
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