Least squares estimator for stochastic differential equations driven by small fractional Lévy noises from discrete observations

2021 ◽  
pp. 2150047
Author(s):  
Qian Yu ◽  
Guangjun Shen ◽  
Wentao Xu
Keyword(s):  
Asymptotic Behavior ◽  
Parameter Estimation ◽  
Differential Equations ◽  
Least Squares ◽  
Stochastic Differential Equations ◽  
Dispersion Coefficient ◽  
Least Squares Estimator ◽  
Regularity Conditions ◽  
Discrete Observations ◽  
Small Dispersion

In this paper, we consider the problem of parameter estimation for stochastic differential equations with small fractional Lévy noises, based on discrete observations. Under certain regularity conditions on drift function, the consistency of least squares estimation has been established as a small dispersion coefficient [Formula: see text] and the number of discrete points [Formula: see text] simultaneously. We also obtain the asymptotic behavior of the estimator.

scholarly journals Statistical Inference for Stochastic Differential Equations with Small Noises

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Liang Shen ◽  
Qingsong Xu
Keyword(s):  
Differential Equations ◽  
Normal Distribution ◽  
Least Squares ◽  
Stochastic Differential Equations ◽  
Asymptotic Distribution ◽  
Asymptotic Property ◽  
Stable Distribution ◽  
Least Squares Method ◽  
Regularity Conditions ◽  
Drift Parameter

This paper proposes the least squares method to estimate the drift parameter for the stochastic differential equations driven by small noises, which is more general than pure jumpα-stable noises. The asymptotic property of this least squares estimator is studied under some regularity conditions. The asymptotic distribution of the estimator is shown to be the convolution of a stable distribution and a normal distribution, which is completely different from the classical cases.

On effects of discretization on estimators of drift parameters for diffusion processes

1996 ◽  
Vol 33 (04) ◽  
pp. 1061-1076 ◽  
Author(s):  
P. E. Kloeden ◽  
E. Platen ◽  
H. Schurz ◽  
M. Sørensen
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Diffusion Processes ◽  
Maximum Likelihood Estimators ◽  
Statistical Properties ◽  
Stochastic Integrals

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

Continuous Autoregressive Moving Average Models

2021 ◽  
pp. 118-141
Author(s):  
Yakup Ari
Keyword(s):  
Time Series ◽  
Parameter Estimation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Moving Average ◽  
Likelihood Estimation ◽  
Real Data ◽  
Autoregressive Moving Average ◽  
The Difference

The financial time series have a high frequency and the difference between their observations is not regular. Therefore, continuous models can be used instead of discrete-time series models. The purpose of this chapter is to define Lévy-driven continuous autoregressive moving average (CARMA) models and their applications. The CARMA model is an explicit solution to stochastic differential equations, and also, it is analogue to the discrete ARMA models. In order to form a basis for CARMA processes, the structures of discrete-time processes models are examined. Then stochastic differential equations, Lévy processes, compound Poisson processes, and variance gamma processes are defined. Finally, the parameter estimation of CARMA(2,1) is discussed as an example. The most common method for the parameter estimation of the CARMA process is the pseudo maximum likelihood estimation (PMLE) method by mapping the ARMA coefficients to the corresponding estimates of the CARMA coefficients. Furthermore, a simulation study and a real data application are given as examples.

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