scholarly journals Parametric estimation of the driving Lévy process of multivariate CARMA processes from discrete observations

2013 ◽  
Vol 115 ◽  
pp. 217-251 ◽  
Author(s):  
Peter J. Brockwell ◽  
Eckhard Schlemm
Keyword(s):  
Lévy Process ◽  
Levy Process ◽  
Parametric Estimation ◽  
Discrete Observations

scholarly journals Parametric estimation of a bivariate stable Lévy process

2011 ◽  
Vol 102 (5) ◽  
pp. 918-930 ◽  
Author(s):  
Habib Esmaeili ◽  
Claudia Klüppelberg
Keyword(s):  
Lévy Process ◽  
Levy Process ◽  
Parametric Estimation ◽  
Stable Lévy Process

scholarly journals Nonparametric estimation of time-changed Lévy models under high-frequency data

2009 ◽  
Vol 41 (04) ◽  
pp. 1161-1188
Author(s):  
José E. Figueroa-López
Keyword(s):  
Lévy Process ◽  
Asymptotic Properties ◽  
Kernel Estimation ◽  
Uniform Boundedness ◽  
Building Blocks ◽  
Levy Process ◽  
Time Interval ◽  
Discrete Observations ◽  
Lévy Measure ◽  
Ergodic Diffusion

Let {Zt}t≥0be a Lévy process with Lévy measure ν, and let τ(t)=∫0tr(u)du, where {r(t)}t≥0is a positive ergodic diffusion independent fromZ. Based upon discrete observations of the time-changed Lévy processXt≔Zτtduring a time interval [0,T], we study the asymptotic properties of certain estimators of the parameters β(φ)≔∫φ(x)ν(dx), which in turn are well known to be the building blocks of several nonparametric methods such as sieve-based estimation and kernel estimation. Under uniform boundedness of the second moments ofrand conditions on φ necessary for the standard short-term ergodic property limt→ 0E φ(Zt)/t= β(φ) to hold, consistency and asymptotic normality of the proposed estimators are ensured when the time horizonTincreases in such a way that the sampling frequency is high enough relative toT.

scholarly journals Nonparametric estimation of time-changed Lévy models under high-frequency data

2009 ◽  
Vol 41 (4) ◽  
pp. 1161-1188 ◽  
Author(s):  
José E. Figueroa-López
Keyword(s):  
Lévy Process ◽  
Asymptotic Properties ◽  
Kernel Estimation ◽  
Uniform Boundedness ◽  
Building Blocks ◽  
Levy Process ◽  
Time Interval ◽  
Discrete Observations ◽  
Lévy Measure ◽  
Ergodic Diffusion

Let {Zt}t≥0 be a Lévy process with Lévy measure ν, and let τ(t)=∫0tr(u) d u, where {r(t)}t≥0 is a positive ergodic diffusion independent from Z. Based upon discrete observations of the time-changed Lévy process Xt≔Zτt during a time interval [0,T], we study the asymptotic properties of certain estimators of the parameters β(φ)≔∫φ(x)ν(d x), which in turn are well known to be the building blocks of several nonparametric methods such as sieve-based estimation and kernel estimation. Under uniform boundedness of the second moments of r and conditions on φ necessary for the standard short-term ergodic property limt→ 0 E φ(Zt)/t = β(φ) to hold, consistency and asymptotic normality of the proposed estimators are ensured when the time horizon T increases in such a way that the sampling frequency is high enough relative to T.

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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