Sub-fractional Brownian Motion Research Articles

On the nth-order subfractional Brownian motion

In the present work, we introduce the nth-Order subfractional Brownian motion (S_H^n (t), t ≥ 0) with Hurst index H ∈ (n − 1,n) and order n ≥ 1; then we examine some of its basic properties: self-similarity, long-range dependence, non Markovian nature and semimartingale property. A local law of iterated logarithm for S_H^n (t) is also established.

European Option Pricing under Sub-Fractional Brownian Motion Regime in Discrete Time

In this paper, the approximate stationarity of the second-order moment increments of the sub-fractional Brownian motion is given. Based on this, the pricing model for European options under the sub-fractional Brownian regime in discrete time is established. Pricing formulas for European options are given under the delta and mixed hedging strategies, respectively. Furthermore, European call option pricing under delta hedging is shown to be larger than under mixed hedging. The hedging error ratio of mixed hedging is shown to be smaller than that of delta hedging via numerical experiments.

The estimations of drift parameters for the Gaussian Vasicek process with time-varying volatility

This article mainly considers the parameter estimation problem for the Gaussian Vasicek process defined as d X t = ( α − β X t ) d t + σ t d G t , t≥0 with unknown parameters α∈ℝ and β < 0, where G t is a continuous centered Gaussian process. The volatility parameter σ t is allowed to vary with time t. Based on the time-continuous observations and least squares method, the Least squares estimations of drift parameters for the non ergodic Gaussian Vasicek model with time-varying volatility are obtained. The strong consistency and asymptotic distributions of proposed estimations are verified under some sufficient conditions. Finally, we verify that the results are still valid under different forms of the volatility function when G t is fractional Brownian motion, sub-fractional Brownian motion, and bi-fractional Brownian motion, respectively. In some sense, our results extend the findings of El Machkouri, Es-Sebaiy, and Ouknine (2016) and Es-Sebaiy and Sebaiy (2021).

Stability Analysis of Neutral Fractional Stochastic Differential Equations Driven by Mixed Brownian Motion and Subfractional Brownian Motion With Applications in the I-Cub Robot

Exponential stability is often used to barricade stochastic disturbance in visual trajectory tracking of a robotic system. In this work, a nonlinear mathematical model to deal with the exponential stability of visual trajectory in the I-cub robot is developed. The main contributions of this article are: 1) exponential stability is studied for the first time in the literature for the neutral fractional stochastic differential equations (NFSDEs) driven by mixed Brownian motion (Bm) and subfractional Bm (sub-fBm); 2) existence and stability results are derived based on the Banach contraction principle, semi-group theory, and fractional calculus in stochastic settings; 3) stability results of mixed Bm and sub-fBm are established and applied to avoid dense environment stochastic disturbance in the visual trajectory of the I-cub robot; and 4) stability of sub-fBm is entrusted in the dense environment even in small particles from the shorter length. A weak solution of the proposed result ensures a sufficiently smooth solution of the considered robotic system. The obtained results are new and innovative in the sense that the proposed algorithms have several advantages for gaze shift framed to the I-cub robot, including helping to maintain the fixation of eye movement in the microscopic laboratory. Furthermore, the results are used to enhance the performance of decreasing rendering gaze shift to achieve high-quality visual tracking.

An approximate approach to fuzzy stochastic differential equations under sub-fractional Brownian motion

In this paper, we introduce fuzzy stochastic differential equations (FSDEs) driven by sub-fractional Brownian motion (SFBM) which are applied to describe phenomena subjected to randomness and fuzziness simultaneously. The SFBM is an extension of the Brownian motion that retains many properties of fractional Brownian motion (FBM), but not the stationary increments. This property makes SFBM a possible candidate for models that include long-range dependence, self-similarity, and non-stationary increments which is suitable for the construction of stochastic models in finance and non-stationary queueing systems. We apply an approximation method to stochastic integrals, and a decomposition of the SFBM to find the existence and uniqueness of the solutions.

Berry-Esséen bounds and almost sure CLT for the quadratic variation of a class of Gaussian process

We propose a condition which is valid for a class of continuous Gaussian processes that may fail to be self-similar or have stationary increments. Some examples include the sub-fractional Brownian motion and the bi-fractional Brownian motion and the sub-bifractional Brownian motion. Under this assumption, we show an upper bound for the difference between the inner product associated with a class of Gaussian process and that associated with the fractional Brownian motion. This inequality relates a class of Gaussian processes to the well studied fractional Brownian motion, which characterizes their relationship quantitatively. As an application, we obtain the optimal Berry-Esséen bounds for the quadratic variation when H ∈ ( 0 , 2 3 ] and the upper Berry-Esséen bounds when H ∈ ( 2 3 , 3 4 ] . As a by-product, we also show the almost sure central limit theorem (ASCLT) for the quadratic variation when H ∈ ( 0 , 3 4 ] . The results in the present paper extend and improve those in the literature.

Exponential stability of second-order fractional stochastic integro-differential equations

In this paper studies the exponential stability result is derived for the second-order fractional stochastic integro-differential equations (FSIDEs) driven by sub-fractional Brownian motion (sub-fBm). By constructing a successive approximation method, we present pth moment exponential stability result of second-order FSIDEs using stochastic analysis techniques and fractional calculus (FC). At last, an example is demonstrated to illustrate the obtained theoretical result.

Pendekatan Analisis Derau Putih untuk Arus Stokastik dari Gerak Brown Subfraksional

The subfractional Brownian motion is a Gaussian generalization of the Brownian motion whose increments are not stationary. In this paper we study the stochastic current of the one dimensional subfractional Brownian motion. For this purpose we use the method from white noise analysis by representing the subfractional Brownian motion as stochastic functionals of white noise. As the main result we prove that the stochastic current of the one dimensional subfractional Brownian motion is a generalized function in the space of Hida distributions.

Estimating Drift Parameters in a Sub-Fractional Vasicek-Type Process.

This study deals with drift parameters estimation problems in the sub-fractional Vasicek process given by , with , being unknown and ; here, represents a sub-fractional Brownian motion (sfBm). We introduce new estimators for and for based on discrete time observations and use techniques from Nordin–Peccati analysis. For the proposed estimators and , strong consistency and the asymptotic normality were established by employing the properties of . Moreover, we provide numerical simulations for sfBm and related Vasicek-type process with different values of the Hurst index H.

Mixed Sub-fractional Brownian Motion and Drift Estimation of Related Ornstein–Uhlenbeck Process

In this paper, we will first give the numerical simulation of the sub-fractional Brownian motion through the relation of fractional Brownian motion instead of its representation of random walk. In order to verify the rationality of this simulation, we propose a practical estimator associated with the LSE of the drift parameter of mixed sub-fractional Ornstein–Uhlenbeck process, and illustrate the asymptotical properties according to our method of simulation when the Hurst parameter \(H>1/2\).

Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion II: Self-Attracting Case

In this study, as a continuation to the studies of the self-interaction diffusion driven by subfractional Brownian motion SH, we analyze the asymptotic behavior of the linear self-attracting diffusion:dXtH=dStH−θ∫0t(XtH−XsH)dsdt+νdt,X0H=0,where θ &amp;gt; 0 and ν∈R are two parameters. When θ &amp;lt; 0, the solution of this equation is called self-repelling. Our main aim is to show the solution XH converges to a normal random variable X∞H with mean zero as t tends to infinity and obtain the speed at which the process XH converges to X∞H as t tends to infinity.

Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion

The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions {BtH,t≥0} and sub-fractional Brownian motions {ξtH,t≥0} with Hurst parameter H∈(12,1). We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.

Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case

Let SH be a sub-fractional Brownian motion with index 12&lt;H&lt;1. In this paper, we consider the linear self-interacting diffusion driven by SH, which is the solution to the equationdXtH=dStH−θ(∫0tXtH−XsHds)dt+νdt,X0H=0,where θ &amp;lt; 0 and ν∈R are two parameters. Such process XH is called self-repelling and it is an analogue of the linear self-attracting diffusion [Cranston and Le Jan, Math. Ann. 303 (1995), 87–93]. Our main aim is to study the large time behaviors. We show the solution XH diverges to infinity, as t tends to infinity, and obtain the speed at which the process XH diverges to infinity as t tends to infinity.

Prognostics of State-dependent Fractional Degradation Processes with Stochastic Disturbance

Predicting the remaining useful life (RUL) of in-service commercial plants is closely linked with safety production and maintenance cost optimization. It is noteworthy that real-life degradation processes could be affected by their past states coupled with unknown disturbances. To make a comprehensive consideration of multisource dependencies and uncertainties, we develop a generalized non-stationary, nonlinear, and non-Markovian degradation model consisting of a Gaussian disturbed drift term and a sub-fractional Brownian motion (sub-FBM) based diffusion term. Both parts are state-dependent, and reflect two different types of memory effects. The main parameters and the approximate probability density function (PDF) of RUL can be solved on foundation of the Markovian transformation theories. A case study finally illustrates the effectiveness of the proposed scheme.

Asymptotics of Karhunen–Loève Eigenvalues for Sub-Fractional Brownian Motion and Its Application

In the present paper, the Karhunen–Loève eigenvalues for a sub-fractional Brownian motion are considered. Rigorous large n asymptotics for those eigenvalues are shown, based on the functional analysis method. By virtue of these asymptotics, along with some standard large deviations results, asymptotical estimates for the small L2-ball probabilities for a sub-fractional Brownian motion are derived. Asymptotic analysis on the Karhunen–Loève eigenvalues for the corresponding “derivative” process is also established.

Maximum likelihood estimation for sub-fractional Vasicek model

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.

A New Algorithm for Approximate Maximum Likelihood Estimation in Sub-fractional Chan-Karolyi-Longstaff-Sanders Model

The paper introduces several approximate maximum likelihood estimators of the parameters of the sub-fractional Chan-Karolyi-Longstaff-Sanders (CKLS) interest rate model and obtains their rates of convergence. A new algorithm inspired by Newton-Cotes formula is presented to improve the accuracy of estimation. The estimators are useful for simulation of interest rates. The proposed new algorithm could be useful for other stochastic computation. It also proposes a generalization of the CKLS interest rate model with sub-fractional Brownian motion drivers which preserves medium range memory.

Impulsive fractional stochastic differential inclusions driven by sub-Fractional Brownian motion with infinite delay and sectorial operators

Impulsive fractional stochastic differential inclusions driven by sub-Fractional Brownian motion with infinite delay and sectorial operators

Stochastic Integral for Non-Adapted Processes Related to Sub-Fractional Brownian Motion when $H&gt;1/2$

Stochastic Integral for Non-Adapted Processes Related to Sub-Fractional Brownian Motion when $H&gt;1/2$

High-dimensional regimes of non-stationary Gaussian correlated Wishart matrices

We study the high-dimensional asymptotic regimes of correlated Wishart matrices [Formula: see text], where [Formula: see text] is a [Formula: see text] Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both [Formula: see text] and [Formula: see text] grow to infinity. The first regime is the one of central convergence, where the law of the properly renormalized Wishart matrices becomes close in Wasserstein distance to that of a Gaussian orthogonal ensemble matrix. In the second regime, a non-central convergence happens, and the law of the normalized Wishart matrices becomes close in Wasserstein distance to that of the so-called Rosenblatt–Wishart matrix recently introduced by Nourdin and Zheng. We then proceed to show that the convergences stated above also hold in a functional setting, namely as weak convergence in [Formula: see text]. As an application of our main result (in the central convergence regime), we show that it can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. Our findings complement and extend a rich collection of results on the study of the fluctuations of Gaussian Wishart matrices, and we provide explicit examples based on Gaussian entries given by normalized increments of a bi-fractional or a sub-fractional Brownian motion.