scholarly journals On Drift Parameter Estimation in Models with Fractional Brownian Motion by Discrete Observations

2014 ◽  
Vol 43 (3) ◽  
pp. 218-228 ◽  
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Maximum Likelihood ◽  
Fractional Brownian Motion ◽  
Strong Consistency ◽  
Discrete Observations ◽  
Likelihood Estimator ◽  
True Value ◽  
Differen Tial Equation ◽  
Drift Parameter

We study a problem of an unknown drift parameter estimation in a stochastic differen- tial equation driven by fractional Brownian motion. We represent the likelihood ratio as a function of the observable process. The form of this representation is in general rather complicated. However, in the simplest case it can be simplified and we can discretize it to establish the a. s. convergence of the discretized version of maximum likelihood estimator to the true value of parameter. We also investigate a non-standard estimator of the drift parameter showing further its strong consistency. 

Large deviations and Berry–Esseen inequalities for estimators in nonhomogeneous diffusion driven by fractional Brownian motion

2020 ◽  
Vol 28 (3) ◽  
pp. 183-196
Author(s):  
Kouacou Tanoh ◽  
Modeste N’zi ◽  
Armel Fabrice Yodé
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Maximum Likelihood ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Maximum Likelihood Estimator ◽  
Bayes Estimator ◽  
Likelihood Estimator

AbstractWe are interested in bounds on the large deviations probability and Berry–Esseen type inequalities for maximum likelihood estimator and Bayes estimator of the parameter appearing linearly in the drift of nonhomogeneous stochastic differential equation driven by fractional Brownian motion.

Drift Parameter Estimation for a Reflected Fractional Brownian Motion Based on its Local Time

2013 ◽  
Vol 50 (02) ◽  
pp. 592-597 ◽  
Author(s):  
Yaozhong Hu ◽  
Chihoon Lee
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Service Rate ◽  
Time Process ◽  
State Process ◽  
Drift Parameter

We consider a drift parameter estimation problem when the state process is a reflected fractional Brownian motion (RFBM) with a nonzero drift parameter and the observation is the associated local time process. The RFBM process arises as the key approximating process for queueing systems with long-range dependent and self-similar input processes, where the drift parameter carries the physical meaning of the surplus service rate and plays a central role in the heavy-traffic approximation theory for queueing systems. We study a statistical estimator based on the cumulative local time process and establish its strong consistency and asymptotic normality.

scholarly journals Drift Parameter Estimation for a Reflected Fractional Brownian Motion Based on its Local Time

2013 ◽  
Vol 50 (2) ◽  
pp. 592-597 ◽  
Author(s):  
Yaozhong Hu ◽  
Chihoon Lee
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Service Rate ◽  
Time Process ◽  
State Process ◽  
Drift Parameter

We consider a drift parameter estimation problem when the state process is a reflected fractional Brownian motion (RFBM) with a nonzero drift parameter and the observation is the associated local time process. The RFBM process arises as the key approximating process for queueing systems with long-range dependent and self-similar input processes, where the drift parameter carries the physical meaning of the surplus service rate and plays a central role in the heavy-traffic approximation theory for queueing systems. We study a statistical estimator based on the cumulative local time process and establish its strong consistency and asymptotic normality.

scholarly journals Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments

2017 ◽  
Vol 46 (3-4) ◽  
pp. 67-78 ◽  
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Sergiy Shklyar
Keyword(s):  
Brownian Motion ◽  
Maximum Likelihood ◽  
Gaussian Process ◽  
Unknown Parameter ◽  
Strong Consistency ◽  
Likelihood Function ◽  
Likelihood Estimator ◽  
Stationary Increments ◽  
Drift Estimation ◽  
Mixed Fractional Brownian Motion

The paper deals with the regression model X_t = \theta t + B_t , t\in[0, T ],where B=\{B_t, t\geq 0\} is a centered Gaussian process with stationary increments.We study the estimation of the unknown parameter $\theta$ and establish the formula for the likelihood function in terms of a solution to an integral equation.Then we find the maximum likelihood estimator and prove its strong consistency. The results obtained generalize the known results for fractional and mixed fractional Brownian motion.

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