Asymptotic properties of symmetric Lévy process expectations and spectral distributions of integro-differential operators with random potentials

1983 ◽  
pp. 497-506
Author(s):  
Hiroyuki Ôkura
Keyword(s):  
Lévy Process ◽  
Differential Operators ◽  
Asymptotic Properties ◽  
Levy Process ◽  
Random Potentials ◽  
Spectral Distributions ◽  
Symmetric Lévy Process

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 Estimation for Nonnegative Lévy-Driven Ornstein-Uhlenbeck Processes

2007 ◽  
Vol 44 (04) ◽  
pp. 977-989 ◽  
Author(s):  
Peter J. Brockwell ◽  
Richard A. Davis ◽  
Yu Yang
Keyword(s):  
Continuous Time ◽  
Lévy Process ◽  
Moving Average ◽  
Asymptotic Properties ◽  
Estimation Procedure ◽  
Financial Econometrics ◽  
Levy Process ◽  
Efficient Estimation ◽  
Autoregressive Moving Average ◽  
Continuous Time Processes

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a very general class of stationary, nonnegative continuous-time processes. In financial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Lévy process, was introduced by Barndorff-Nielsen and Shephard (2001) as a model for stochastic volatility to allow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes, we take advantage of the nonnegativity of the increments of the driving Lévy process to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0,h,…,Nh. We also show how to reconstruct the background driving Lévy process from a continuously observed realization of the process and use this result to estimate the increments of the Lévy process itself when h is small. Asymptotic properties of the coefficient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck process.

scholarly journals A geometric interpretation of the transition density of a symmetric Lévy process

2012 ◽  
Vol 55 (6) ◽  
pp. 1099-1126 ◽  
Author(s):  
Niels Jacob ◽  
Victorya Knopova ◽  
Sandra Landwehr ◽  
René L. Schilling
Keyword(s):  
Lévy Process ◽  
Transition Density ◽  
Geometric Interpretation ◽  
Levy Process ◽  
Symmetric Lévy Process

scholarly journals Estimation for Nonnegative Lévy-Driven Ornstein-Uhlenbeck Processes

2007 ◽  
Vol 44 (4) ◽  
pp. 977-989 ◽  
Author(s):  
Peter J. Brockwell ◽  
Richard A. Davis ◽  
Yu Yang
Keyword(s):  
Continuous Time ◽  
Lévy Process ◽  
Moving Average ◽  
Asymptotic Properties ◽  
Estimation Procedure ◽  
Financial Econometrics ◽  
Levy Process ◽  
Efficient Estimation ◽  
Autoregressive Moving Average ◽  
Continuous Time Processes

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a very general class of stationary, nonnegative continuous-time processes. In financial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Lévy process, was introduced by Barndorff-Nielsen and Shephard (2001) as a model for stochastic volatility to allow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes, we take advantage of the nonnegativity of the increments of the driving Lévy process to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0,h,…,Nh. We also show how to reconstruct the background driving Lévy process from a continuously observed realization of the process and use this result to estimate the increments of the Lévy process itself when h is small. Asymptotic properties of the coefficient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck 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.

Linear MMSE Estimation of Large-Magnitude Symmetric Levy-Process Phase-Noise

2010 ◽  
Vol 58 (12) ◽  
pp. 3369-3374 ◽  
Author(s):  
Yeong-Tzay Su ◽  
Kainam Thomas Wong ◽  
Keang-Po Ho
Keyword(s):  
Phase Noise ◽  
Lévy Process ◽  
Levy Process ◽  
Large Magnitude ◽  
Mmse Estimation ◽  
Symmetric Lévy Process

scholarly journals Threshold Estimation for a Spectrally Negative Lévy Process

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Honglong You ◽  
Chuncun Yin
Keyword(s):  
High Frequency ◽  
Lévy Process ◽  
Asymptotic Properties ◽  
Levy Process ◽  
Simulation Studies ◽  
Finite Sample ◽  
Threshold Estimation ◽  
Lévy Measure ◽  
Spectrally Negative Lévy Process ◽  
Unknown Diffusion Coefficient

Consider a spectrally negative Lévy process with unknown diffusion coefficient and Lévy measure and suppose that the high frequency trading data is given. We use the techniques of threshold estimation and regularized Laplace inversion to obtain the estimator of survival probability for a spectrally negative Lévy process. The asymptotic properties are given for the proposed estimator. Simulation studies are also given to show the finite sample performance of our estimator.

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