Sub-fractional Brownian Motion Research Articles

A Simple Proof of Berry–Esséen Bounds for the Quadratic Variation of the Subfractional Brownian Motion

We give a simple technic to derive the Berry-Esseen bounds for the quadratic variation of the subfractional Brownian motion (subfBm). Our approach has two main ingredients: (i) bounding from above the covariance of quadratic variation of subfBm by the covariance of the quadratic variation of fractional Brownian motion (fBm); and (ii) using the existing results on fBm in [1, 3, 2]. As a result, we obtain simple and direct proof to derive the rate of convergence of quadratic variation of subfBm. In addition, we also improve this rate of convergence to meet the one of fractional Brownian motion in [2].

Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes

The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein–Uhlenbeck process defined as dXt=θXtdt+dGt,t≥0 with an unknown parameter θ>0, where G is a Gaussian process. We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distribution of our estimator θ˜t of θ based on the observation {Xs,s∈[0,t]} as t→∞. Our approach offers an elementary, unifying proof of Belfadli (2011), and it allows to extend the result of Belfadli (2011) to the case when G is a fractional Brownian motion with Hurst parameter H∈(0,1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.

Parameter estimations for the sub-fractional Brownian motion with drift at discrete observation

In this paper, we investigate the $L^{2}$-consistency and the strong consistency of the maximum likelihood estimators (MLE) of the mean and variance of the sub-fractional Brownian motion with drift at discrete observation. By combining the Stein’s method with Malliavin calculus, we obtain the central limit theorem and the Berry–Esséen bounds for these estimators.

Occupation densities for certain processes related to subfractional Brownian motion

Occupation densities for certain processes related to subfractional Brownian motion

Weighted power variation of integrals with respect to a Gaussian process

We consider a stochastic process $Y$ defined by an integral in quadratic mean of a deterministic function $f$ with respect to a Gaussian process $X$, which need not have stationary increments. For a class of Gaussian processes $X$, it is proved that sums of properly weighted powers of increments of $Y$ over a sequence of partitions of a time interval converge almost surely. The conditions of this result are expressed in terms of the $p$-variation of the covariance function of $X$. In particular, the result holds when $X$ is a fractional Brownian motion, a subfractional Brownian motion and a bifractional Brownian motion.

On the sub-mixed fractional Brownian motion

Let {S , t ≥ 0} be a linear combination of a Brownian motion and an independent sub-fractional Brownian motion with Hurst index 0 < H < 1. Its main properties are studied. They suggest that S H lies between the sub-fractional Brownian motion and the mixed fractional Brownian motion. We also determine the values of H for which S H is not a semi-martingale.

Stochastic delay evolution equations driven by sub-fractional Brownian motion

In this paper, we investigate the existence, uniqueness and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a sub-fractional Brownian motion \(S^{H}_{Q}(t)\): \(dX(t)=(AX(t)+f(t,X_{t}))\, dt+g(t)\, dS^{H}_{Q}(t)\) with index \(H\in(1/2,1)\).

Variations and estimators for self-similarity parameter of sub-fractional Brownian motion via Malliavin calculus

ABSTRACTUsing multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of the adjusted quadratic variation for a sub-fractional Brownian motion. We apply our results to construct strongly consistent statistical estimators for the self-similarity of sub-fractional Brownian motion.

Strong Local Non-Determinism of Sub-Fractional Brownian Motion

Let be a subfractional Brownian motion in . We prove that is strongly locally nondeterministic.

Asymptotic behavior of weighted cubic variation of sub-fractional brownian motion

ABSTRACTIn this article, we investigate the convergence of renormalized weighted cubic variation of a sub-fractional Brownian motion SH with Hurst index H. When , we prove by means of Malliavin calculus that the convergence holds in L2 toward an explicit limit which only depends on SH. We also numerically simulate the sample paths of such a type of sub-fractional Brownian motion regulated by different Hurst index H.

INTERSECTION LOCAL TIME OF SUBFRACTIONAL ORNSTEIN-UHLENBECK PROCESSES

In this paper, we consider Ornstein-Uhlenbeck process dX t = −X t dt+ vdS t , X 0 = x, driven by a subfractional Brownian motion S . We prove that the subfractional Ornstein-Uhlenbeck process X is local nondeterministic and give some properties of this process. As an application, assume d ≥ 2, we prove that the intersection local time of two independent, d−dimensional subfractional Ornstein-Uhlenbeck process,X and X , exists in L if and only if Hd < 2. 2000 AMS Classification: 60G15, 60G18, 60H07.

Mixed sub-fractional Brownian motion

Abstract A new extension of the sub-fractional Brownian motion, and thus of the Brownian motion, is introduced. It is a linear combination of a finite number of sub-fractional Brownian motions, that we have chosen to call the mixed sub-fractional Brownian motion. In this paper, we study some basic properties of this process, its non-Markovian and non-stationarity characteristics, the conditions under which it is a semimartingale, and the main features of its sample paths. We also show that this process could serve to get a good model of certain phenomena, taking not only the sign (like in the case of the sub-fractional Brownian motion), but also the strength of dependence between the increments of this phenomena into account.

An Approximation of Subfractional Brownian Motion

In this article, we obtain an approximation theorem for subfractional Brownian motion with H > 1/2, using martingale differences. The proof involves the tightness and identification of finite dimensional distributions.

Estimators for the Drift of Subfractional Brownian Motion

In this paper, we consider, using technique based on Girsanov theorem, the problem of efficient estimation for the drift of subfractional Brownian motion SH ≔ (SHt)t ∈ [0, T]. We also construct a class of biased estimators of James-Stein type which dominate, under the usual quadratic risk, the natural maximum likelihood estimator.

A self-similar Gaussian process

Abstract. In this paper we introduce and study a self-similar Gaussian process denoted by S H , K ${{S^{H,\,K}}}$ with parameters H ∈ ( 0 , 1 ) ${{H\in (0,1)}}$ and K ∈ [ 0 , 1 ] ${{K\in [0,1]}}$ . This process generalizes the well-known fractional Brownian motion introduced by Mandelbrot and Van Ness [SIAM Rev. 10 (1968), no. 4, 422–437], the sub-fractional Brownian motion introduced by Bojdecki, Gorostiza and Talarczyket [Statist. Probab. Lett. 69 (2004), 405–419] and the Gaussian process introduced by Lei and Nualart [Statist. Probab. Lett. 79 (2009), 619–624] in order to obtain a decomposition in law of the bifractional Brownian motion.

Nonparametric Regression with Subfractional Brownian Motion via Malliavin Calculus

We study the asymptotic behavior of the sequenceSn=∑i=0n-1K(nαSiH1)(Si+1H2-SiH2),asntends to infinity, whereSH1andSH2are two independent subfractional Brownian motions with indicesH1andH2, respectively.Kis a kernel function and the bandwidth parameterαsatisfies some hypotheses in terms ofH1andH2. Its limiting distribution is a mixed normal law involving the local time of the sub-fractional Brownian motionSH1. We mainly use the techniques of Malliavin calculus with respect to sub-fractional Brownian motion.

Maximum likelihood estimator for the sub-fractional Brownian motion approximated by a random walk

We estimate the drift parameter in a simple linear model driven by sub-fractional Brownian motion. We construct a maximum likelihood estimator (MLE) for the drift parameter by using a random walk approximation of the sub-fractional Brownian motion and study the asymptotic behaviors of the estimator. Simulations confirm the theoretical results and indicate superiority of the new proposed estimator.

Power Variation of Subfractional Brownian Motion and Application

In this paper, we consider the power variation of subfractional Brownian motion. As an application, we introduce a class of estimators for the index of a subfractional Brownian motion and show that they are strongly consistent.

An extension of sub-fractional Brownian motion

In this paper, firstly, we introduce and study a self-similar Gaussian process with parameters H ∈ (0; 1) and K ∈ (0; 1] that is an extension of the well known sub-fractional Brownian motion introduced by Bojdecki et al. [4]. Secondly, by using a decomposition in law of this process, we prove the existence and the joint continuity of its local time.

The generalized Bouleau-Yor identity for a sub-fractional Brownian motion

Let S H be a sub-fractional Brownian motion with index 0 < H < 1/2. In this paper we study the existence of the generalized quadratic covariation [f(S H), S H](W) defined by $[f(S^H ),S^H ]_t^{(W)} = \mathop {\lim }\limits_{\varepsilon \downarrow 0} \tfrac{1} {{\varepsilon ^{2H} }}\int_0^t {\{ f(S_{s + \varepsilon }^H ) - f(S_s^H )\} (S_{s + \varepsilon }^H - S_s^H )ds^{2H} } , $ provided the limit exists in probability, where x ↦ f(x) is a measurable function. We construct a Banach space ℋ of measurable functions such that the generalized quadratic covariation exists in L 2 provided f ∈ ℋ. Moreover, the generalized Bouleau-Yor identity takes the form for all f ∈ ℋ, where ℒH(x, t) is the weighted local time of S H. This allows us to write the generalized Itô’s formula for absolutely continuous functions with derivative belonging to ℋ.