scholarly journals Parametric Estimation in the Vasicek-Type Model Driven by Sub-Fractional Brownian Motion

Algorithms ◽  
2018 ◽  
Vol 11 (12) ◽  
pp. 197
Author(s):  
Shengfeng Li ◽  
Yi Dong
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Asymptotic Distributions ◽  
Least Square ◽  
Parametric Estimation ◽  
Type Model ◽  
Unknown Parameters ◽  
Model Driven ◽  
Least Squares Estimators ◽  
Least Square Estimators

In the paper, we tackle the least squares estimators of the Vasicek-type model driven by sub-fractional Brownian motion: d X t = ( μ + θ X t ) d t + d S t H , t ≥ 0 with X 0 = 0 , where S H is a sub-fractional Brownian motion whose Hurst index H is greater than 1 2 , and μ ∈ R , θ ∈ R + are two unknown parameters. Based on the so-called continuous observations, we suggest the least square estimators of μ and θ and discuss the consistency and asymptotic distributions of the two estimators.

scholarly journals Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 716 ◽  
Author(s):  
Pavel Kříž ◽  
Leszek Szała
Keyword(s):  
Brownian Motion ◽  
Least Squares ◽  
Fractional Brownian Motion ◽  
Numerical Experiments ◽  
Covariance Structure ◽  
Mesh Size ◽  
Uhlenbeck Process ◽  
Least Squares Estimators ◽  
Discrete Time Observations ◽  
Drift Parameter

We introduce three new estimators of the drift parameter of a fractional Ornstein–Uhlenbeck process. These estimators are based on modifications of the least-squares procedure utilizing the explicit formula for the process and covariance structure of a fractional Brownian motion. We demonstrate their advantageous properties in the setting of discrete-time observations with fixed mesh size, where they outperform the existing estimators. Numerical experiments by Monte Carlo simulations are conducted to confirm and illustrate theoretical findings. New estimation techniques can improve calibration of models in the form of linear stochastic differential equations driven by a fractional Brownian motion, which are used in diverse fields such as biology, neuroscience, finance and many others.

ARBITRAGE IN FRACTAL MODULATED BLACK–SCHOLES MODELS WHEN THE VOLATILITY IS STOCHASTIC

2005 ◽  
Vol 08 (03) ◽  
pp. 283-300 ◽  
Author(s):  
ERHAN BAYRAKTAR ◽  
H. VINCENT POOR
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stock Prices ◽  
Stock Price ◽  
Time Change ◽  
Quadratic Variation ◽  
Model Driven ◽  
Black Scholes ◽  
Arbitrage Strategy ◽  
Adapted Process

In this paper an arbitrage strategy is constructed for the modified Black–Scholes model driven by fractional Brownian motion or by a time changed fractional Brownian motion, when the volatility is stochastic. This latter property allows the heavy tailedness of the log returns of the stock prices to be also accounted for in addition to the long range dependence introduced by the fractional Brownian motion. Work has been done previously on this problem for the case with constant "volatility" and without a time change; here these results are extended to the case of stochastic volatility models when the modulator is fractional Brownian motion or a time change of it. (Volatility in fractional Black–Scholes models does not carry the same meaning as in the classic Black–Scholes framework, which is made clear in the text.) Since fractional Brownian motion is not a semi-martingale, the Black–Scholes differential equation is not well-defined sense for arbitrary predictable volatility processes. However, it is shown here that any almost surely continuous and adapted process having zero quadratic variation can act as an integrator over functions of the integrator and over the family of continuous adapted semi-martingales. Moreover it is shown that the integral also has zero quadratic variation, and therefore that the integral itself can be an integrator. This property of the integral is crucial in developing the arbitrage strategy. Since fractional Brownian motion and a time change of fractional Brownian motion have zero quadratic variation, these results are applicable to these cases in particular. The appropriateness of fractional Brownian motion as a means of modeling stock price returns is discussed as well.

scholarly journals Tail distribution estimates of the mixed-fractional CEV model

2021 ◽  
Vol 5 (1) ◽  
pp. 371-379
Author(s):  
Nguyen Thu Hang ◽  
◽  
Pham Thi Phuong Thuy ◽  
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Malliavin Calculus ◽  
Explicit Estimate ◽  
Cev Model ◽  
Model Driven ◽  
Tail Distribution ◽  
Tail Distributions

The aim of this paper is to study the tail distribution of the CEV model driven by Brownian motion and fractional Brownian motion. Based on the techniques of Malliavin calculus and a result established recently in [<a href="#1">1</a>], we obtain an explicit estimate for tail distributions.

Parameter estimation for Gaussian mean-reverting Ornstein–Uhlenbeck processes of the second kind: Non-ergodic case

2019 ◽  
Vol 20 (02) ◽  
pp. 2050011 ◽  
Author(s):  
Fares Alazemi ◽  
Abdulaziz Alsenafi ◽  
Khalifa Es-Sebaiy
Keyword(s):  
Brownian Motion ◽  
Continuous Time ◽  
Least Square ◽  
Hurst Parameter ◽  
Unknown Parameters ◽  
Uhlenbeck Process ◽  
Type Method ◽  
Bifractional Brownian Motion ◽  
Subfractional Brownian Motion ◽  
The Mean

We consider a least square-type method to estimate the drift parameters for the mean-reverting Ornstein–Uhlenbeck process of the second kind [Formula: see text] defined as [Formula: see text], with unknown parameters [Formula: see text] and [Formula: see text], where [Formula: see text] with [Formula: see text], and [Formula: see text] is a Gaussian process. In order to establish the consistency and the asymptotic distribution of least square-type estimators of [Formula: see text] and [Formula: see text] based on the continuous-time observations [Formula: see text] as [Formula: see text], we impose some technical conditions on the process [Formula: see text], which are satisfied, for instance, if [Formula: see text] is a fractional Brownian motion with Hurst parameter [Formula: see text], [Formula: see text] is a subfractional Brownian motion with Hurst parameter [Formula: see text] or [Formula: see text] is a bifractional Brownian motion with Hurst parameters [Formula: see text]. Our method is based on pathwise properties of [Formula: see text] and [Formula: see text] proved in the sequel.

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