scholarly journals Limit Theory for High Frequency Sampled MCARMA Models

2014 ◽  
Vol 46 (03) ◽  
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 {h n , 2h n ,…, nh n }, where h n ↓ 0 and nh n → ∞ as n → ∞, or at a constant time grid where h n = 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 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 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.

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 Recovering Brownian and jump parts from high-frequency observations of a Lévy process

Bernoulli ◽  
2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Jorge González Cázares ◽  
Jevgenijs Ivanovs
Keyword(s):  
High Frequency ◽  
Lévy Process ◽  
Levy 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 FINANCIAL MODELING AND OPTION THEORY WITH THE TRUNCATED LEVY PROCESS

2000 ◽  
Vol 03 (01) ◽  
pp. 143-160 ◽  
Author(s):  
ANDREW MATACZ
Keyword(s):  
High Frequency ◽  
Lévy Process ◽  
Levy Process ◽  
Hedging Strategy ◽  
Financial Modeling ◽  
Option Hedging ◽  
Market Index ◽  
Delta Hedging ◽  
Option Theory ◽  
Finite Moments

In recent studies the truncated Levy process (TLP) has been shown to be very promising for the modeling of financial dynamics. In contrast to the Levy process, the TLP has finite moments and can account for both the previously observed excess kurtosis at short timescales, along with the slow convergence to Gaussian at longer timescales. In this paper I further test the truncated Levy paradigm using high frequency data from the Australian All Ordinaries share market index. I then consider an optimal option hedging strategy which is appropriate for the early Levy dominated regime. This is compared with the usual delta hedging approach and found to differ significantly.

ASYMPTOTIC BEHAVIOR OF TRIMMED SUMS

2012 ◽  
Vol 12 (01) ◽  
pp. 1150002 ◽  
Author(s):  
ISTVÁN BERKES ◽  
LAJOS HORVÁTH ◽  
JOHANNES SCHAUER
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Stable Law ◽  
Limit Theory ◽  
Trimmed Sums ◽  
The Central Limit Theorem

Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators. It is also a powerful tool in understanding deeper properties of partial sums of independent random variables. In this paper we review some basic results of the theory and discuss new results in the central limit theory of trimmed sums. In particular, we show that for random variables in the domain of attraction of a stable law with parameter 0 < α < 2, the asymptotic behavior of modulus trimmed sums depends sensitively on the number of elements eliminated from the sample. We also show that under moderate trimming, the central limit theorem always holds if we allow random centering factors. Finally, we give an application to change point problems.

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.

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