scholarly journals Least Squares Estimation forα-Fractional Bridge with Discrete Observations

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Guangjun Shen ◽  
Xiuwei Yin
Keyword(s):  
Brownian Motion ◽  
Least Squares ◽  
Fractional Brownian Motion ◽  
Discrete Time ◽  
Unknown Parameter ◽  
Least Squares Estimation ◽  
Least Squares Estimator ◽  
Hurst Parameter ◽  
Discrete Observations ◽  
Observation Window

We consider a fractional bridge defined asdXt=-α(Xt/(T-t))dt+dBtH,  0≤t<T, whereBHis a fractional Brownian motion of Hurst parameterH>1/2and parameterα>0is unknown. We are interested in the problem of estimating the unknown parameterα>0. Assume that the process is observed at discrete timeti=iΔn,  i=0,…,n, andTn=nΔndenotes the length of the “observation window.” We construct a least squares estimatorα^nofαwhich is consistent; namely,α^nconverges toαin probability asn→∞.

scholarly journals Least squares type estimations for discretely observed nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind

2021 ◽  
Vol 7 (1) ◽  
pp. 1095-1114
Author(s):  
Huantian Xie ◽  
◽  
Nenghui Kuang ◽  
Keyword(s):  
Brownian Motion ◽  
Least Squares ◽  
Fractional Brownian Motion ◽  
Unknown Parameter ◽  
Discrete Observations ◽  
Bifractional Brownian Motion ◽  
Similar Index ◽  
Self Similar ◽  
Strongly Consistent ◽  
Theoretical Results

<abstract><p>We consider the nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind defined by $ dX_t = \theta X_tdt+dY_t^{(1)}, t\geq 0, X_0 = 0 $ with an unknown parameter $ \theta &gt; 0, $ where $ dY_t^{(1)} = e^{-t}dG_{a_{t}} $ and $ \{G_t, t\geq 0\} $ is a mean zero Gaussian process with the self-similar index $ \gamma\in (\frac{1}{2}, 1) $ and $ a_t = \gamma e^{\frac{t}{\gamma}} $. Based on the discrete observations $ \{X_{t_i}:t_i = i\Delta_n, i = 0, 1, \cdots, n\} $, two least squares type estimators $ \hat{\theta}_n $ and $ \tilde{\theta}_n $ of $ \theta $ are constructed and proved to be strongly consistent and rate consistent. We apply our results to the cases such as fractional Brownian motion, sub-fractional Brownian motion, bifractional Brownian motion and sub-bifractional Brownian motion. Moreover, the numerical simulations confirm the theoretical results.</p></abstract>

scholarly journals Asymptotic Properties of Parameter Estimators in Fractional Vasicek Model

2016 ◽  
Vol 55 (1) ◽  
pp. 102-111 ◽  
Author(s):  
Stanislav Lohvinenko ◽  
Kostiantyn Ralchenko ◽  
Olga Zhuchenko
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Unknown Parameter ◽  
Strong Consistency ◽  
Asymptotic Properties ◽  
Hurst Parameter ◽  
Vasicek Model ◽  
Parameter Estimators

We consider the fractional Vasicek model of the form dXt = (α-βXt)dt + γdBHt, driven by fractional Brownian motion BH with Hurst parameter H ∈ (0,1). We construct three estimators for an unknown parameter θ=(α,β) and prove their strong consistency.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

Malliavin calculus used to derive a stochastic maximum principle for system driven by fractional Brownian and standard Wiener motions with application

2020 ◽  
Vol 28 (4) ◽  
pp. 291-306
Author(s):  
Tayeb Bouaziz ◽  
Adel Chala
Keyword(s):  
Brownian Motion ◽  
Maximum Principle ◽  
Control Problem ◽  
Fractional Brownian Motion ◽  
Stochastic Control ◽  
Malliavin Calculus ◽  
Stochastic Maximum Principle ◽  
Hurst Parameter ◽  
Principle Of Optimality ◽  
Initial Cost

AbstractWe consider a stochastic control problem in the case where the set of the control domain is convex, and the system is governed by fractional Brownian motion with Hurst parameter {H\in(\frac{1}{2},1)} and standard Wiener motion. The criterion to be minimized is in the general form, with initial cost. We derive a stochastic maximum principle of optimality by using two famous approaches. The first one is the Doss–Sussmann transformation and the second one is the Malliavin derivative.

Sign in / Sign up

Export Citation Format

Share Document