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.

Maximum likelihood estimation for sub-fractional Vasicek model

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
B. L. S. Prakasa Rao
Keyword(s):  
Brownian Motion ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Fractional Brownian Motion ◽  
Asymptotic Properties ◽  
Maximum Likelihood Estimators ◽  
Likelihood Estimation ◽  
Vasicek Model ◽  
Model Driven

Abstract We investigate the asymptotic properties of maximum likelihood estimators of the drift parameters for the fractional Vasicek model driven by a sub-fractional Brownian motion.

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→∞.

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.

scholarly journals Solving Arbitrage Problem on the Financial Market Under the Mixed Fractional Brownian Motion With Hurst Parameter H ∈]1/2,3/4[

2019 ◽  
Vol 11 (1) ◽  
pp. 76
Author(s):  
Eric Djeutcha ◽  
Didier Alain Njamen Njomen ◽  
Louis-Aimé Fono
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Financial Market ◽  
Existence And Uniqueness ◽  
Hurst Parameter ◽  
Existence And Uniqueness Theorem ◽  
Martingale Approximation ◽  
Arbitrage Opportunities ◽  
Mixed Fractional Brownian Motion ◽  
Geometric Fractional Brownian Motion

This study deals with the arbitrage problem on the financial market when the underlying asset follows a mixed fractional Brownian motion. We prove the existence and uniqueness theorem for the mixed geometric fractional Brownian motion equation. The semi-martingale approximation approach to mixed fractional Brownian motion is used to eliminate the arbitrage opportunities.

Harnack inequalities and applications for functional SDEs driven by fractional Brownian motion

2014 ◽  
Vol 22 (4) ◽  
Author(s):  
Zhi Li ◽  
Jiaowan Luo
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Functional Differential Equations ◽  
Hurst Parameter ◽  
Stochastic Functional Differential Equations ◽  
Harnack Inequalities ◽  
Functional Differential ◽  
Stochastic Functional

AbstractIn this paper, Harnack inequalities are established for stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter

scholarly journals ASYMPTOTIC THEORY FOR ESTIMATING DRIFT PARAMETERS IN THE FRACTIONAL VASICEK MODEL

2018 ◽  
Vol 35 (1) ◽  
pp. 198-231 ◽  
Author(s):  
Weilin Xiao ◽  
Jun Yu
Keyword(s):  
Strong Consistency ◽  
Asymptotic Theory ◽  
Least Squares Method ◽  
Estimation Method ◽  
Hurst Parameter ◽  
Stationary Case ◽  
Vasicek Model ◽  
Continuous Record ◽  
Recurrent Case ◽  
Two Parameters

This article develops an asymptotic theory for estimators of two parameters in the drift function in the fractional Vasicek model when a continuous record of observations is available. The fractional Vasicek model with long-range dependence is assumed to be driven by a fractional Brownian motion with the Hurst parameter greater than or equal to one half. It is shown that, when the Hurst parameter is known, the asymptotic theory for the persistence parameter depends critically on its sign, corresponding asymptotically to the stationary case, the explosive case, and the null recurrent case. In all three cases, the least squares method is considered, and strong consistency and the asymptotic distribution are obtained. When the persistence parameter is positive, the estimation method of Hu and Nualart (2010) is also considered.

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