Sub-fractional Brownian Motion Research Articles

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.

The sub-fractional CEV model

The sub-fractional Brownian motion (sfBm) is a stochastic process, characterized by non-stationarity in their increments and long-range dependence, considered as an intermediate step between the standard Brownian motion (Bm) and the fractional Brownian motion (fBm). The mixed process, a linear combination between a Bm and an independent sfBm, called mixed sub-fractional Brownian motion (msfBm), keeps the features of the sfBm adding the semi-martingale property for H>3∕4, is a suitable candidate to use in price fluctuation modeling, in particular for option pricing. In this note, we arrive at the European Call price under the Constant Elasticity of Variance (CEV) model driven by a mixed sub-fractional Brownian motion. Empirical tests show the capacity of the proposed model to capture the temporal structure of option prices across different maturities.

Approximation of stochastic differential equations driven by subfractional Brownian motion at discrete time observation

In this paper, we consider discrete time approximations for stochastic differential equations with the form: where is a continuous function with locally bounded variation, are measurable functions, and the integral with respect to is the pathwise Riemann-Stieltjes integral, SH is a subfractional Brownian motion with σ is a deterministic (possibly discontinuous) function.

Pricing geometric asian power options in the sub-fractional brownian motion environment

This paper aims of obtaining the closed form expressions for the prices of the geometric Asian options and power options when the payoff function is a power function. After discussing the option pricing in the sub-fractional Brownian motion environment, by the fractional Ito^ formula which is based on the theory of stochastic differential equation, the sub-fractional Ito^ formula is derived. Furthermore, the solution of the stochastic differential equation satisfied by stock prices is obtained. The stock price process is modeled well with the driving force as the sub-fractional Brownian motion. The empirical results show that the fitting effect is better than the Brownian motion.

Parameter Estimation for an Ornstein-Uhlenbeck Process Driven by a General Gaussian Noise

In this paper, we consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process (Gt)t≥0. The second order mixed partial derivative of the covariance function $$R(t,s) = \mathbb{E}\left[ {{G_t}{G_s}} \right]$$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by (ts)β−1 up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or to have stationary increments; some examples of this include the subfractional Brownian motion and the bi-fractional Brownian motion. Under this assumption, we study the parameter estimation for a drift parameter in the Ornstein-Uhlenbeck process driven by the Gaussian noise (Gt)t≥0. For the least squares estimator and the second moment estimator constructed from the continuous observations, we prove the strong consistency and the asympotic normality, and obtain the Berry-Esseen bounds. The proof is based on the inner product’s representation of the Hilbert space $$\mathfrak{h}$$ associated with the Gaussian noise (Gt)t≥0, and the estimation of the inner product based on the results of the Hilbert space associated with the fractional Brownian motion.

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

<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 > 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>

Asymptotics of the Cross-Variation of Young Integrals with Respect to a General Self-Similar Gaussian Process

We show in this work that the limit in law of the cross-variation of processes having the form of Young integral with respect to a general self-similar centered Gaussian process of order β ∈ (1/2, 3/4] is normal according to the values of β. We apply our results to two self-similar Gaussian processes: the subfractional Brownian motion and the bifractional Brownian motion.

CLT for quadratic variation of Gaussian processes and its application to the estimation of the Orey index

We give a two-dimensional central limit theorem (CLT) for the second-order quadratic variation of the centered Gaussian processes on [0,T]. Though the approach we use is well known in the literature, the conditions under which the CLT holds are usually based on differentiability of the corresponding covariance function. In our case, we replace differentiability conditions by the convergence of the scaled sums of the second-order moments. To illustrate the usefulness and easiness of use of the approach, we apply the obtained CLT to proving the asymptotic normality of the estimator of the Orey index of a subfractional Brownian motion.

Nonparametric estimation of trend for stochastic differential equations driven by sub-fractional Brownian motion

AbstractWe discuss nonparametric estimation of a trend coefficient in models governed by a stochastic differential equation driven by a sub-fractional Brownian motion with small noise.

Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

Large deviations for functionals of some self-similar Gaussian processes

ABSTRACT We prove large deviation principles for , where X is a d-dimensional self-similar Gaussian process and takes the form of the Dirac delta function , with , or with . In particular, large deviations are obtained for the functionals of d-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.

More on maximal inequalities for sub-fractional Brownian motion

We derive some maximal inequalities for the sub-fractional Brownian motion using comparison theorems for Gaussian processes.

Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

Approximation to two independent Gaussian processes from a unique Lévy process and applications

In this article, we construct two families of processes, from a unique Lévy process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As applications of this result, we obtain families of processes that converge in law towards fractional Brownian motion, sub-fractional Brownian motion and bifractional Brownian motion, respectively.

Nonparametric estimation of linear multiplier for processes driven by subfractional Brownian motion

We study the problem of nonparametric estimation of linear multiplier function for processes satisfying stochastic differential equations of the type where is a subfractional Brownian motion with known Hurst index H and study the asymptotic behavior of the estimator as

Stochastic partial functional integrodifferential equations driven by a sub-fractional Brownian motion, existence and asymptotic behavior

Abstract This paper deals with the existence and uniqueness of mild solutions to stochastic partial functional integro-differential equations driven by a sub-fractional Brownian motion S Q H ⁢ ( t ) {S_{Q}^{H}(t)} , with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} . By the theory of resolvent operator developed by R. Grimmer (1982) to establish the existence of mild solutions, we give sufficient conditions ensuring the existence, uniqueness and the asymptotic behavior of the mild solutions. An example is provided to illustrate the theory.

Limit theorems for functionals of two independent Gaussian processes

Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. A new and interesting phenomenon is that, in comparison with the results for fractional Brownian motion, extra randomness appears in the limiting distributions for Gaussian processes with nonstationary increments, say sub-fractional Brownian motion and bi-fractional Brownian. The results are obtained based on the method of moments, in which Fourier analysis, the chaining argument introduced in [11] and a pairing technique are employed.

Statistical inference for Vasicek-type model driven by self-similar Gaussian processes

In this paper, we consider the drift parameters estimation problem for the Vasicek-type model defined aswhere a < 0 and are considered as unknown drift parameters and Gt is a self-similar Gaussian process with index . We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distributions of our estimators of a and of b based on the observation as . Our approach extend the result of Xiao and Yu (2017) for the case when G is a fractional Brownian motion with Hurst parameter . We also discuss the cases of sub-fractional Browian motion and bi-fractional Brownian motion. The conclusion can also be extended to more general self-similarity processes, such as Hermite processes.

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

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.

On the Generalized Fractional Brownian Motion

The generalized fractional Brownion motion (gfBm) is a new extension of both fractional and sub-fractional Brownian motions, introduced very recently. We show that this process could serve to obtain new models, better than those constructed from fractional and sub-fractional Brownian motions, permitting to take the level of correlation between the increments of the studied phenomenon into account. We also expand explicitly this process, we study the rate of convergence of the obtained expansion and, we apply our result to get a computer generation of some gfBm sample paths. In particular we present some sample paths of the even and odd parts of the fractional Brownian motion.