Sub-fractional Brownian Motion Research Articles

Oscillatory Fractional Brownian Motion

We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the discrete and ultrametric structure of the hierarchical group, and they become slower as time tends to infinity and faster as time approaches zero. We also give other results to provide an overall picture of the behavior of this kind of systems, emphasizing the new phenomena that are caused by the ultrametric structure as compared with results for analogous models on Euclidean space.

Mutual information for stochastic differential equation with subfractional noises

Motivated by the Gaussian channel models, we calculate the mutual information for processes described by multidimensional stochastic differential equations driven by sub-fractional Brownian motion.

The 𝒮-Transform of Sub-fBm and an Application to a Class of Linear Subfractional BSDEs

Let SH be a subfractional Brownian motion with index 0<H<1. Based on the 𝒮-transform in white noise analysis we study the stochastic integral with respect to SH, and we also prove a Girsanov theorem and derive an Itô formula. As an application we study the solutions of backward stochastic differential equations driven by SH of the form -dYt=f(t,Yt,Zt)dt-ZtdStH, t∈[0,T],YT=ξ, where the stochastic integral used in the above equation is Pettis integral. We obtain the explicit solutions of this class of equations under suitable assumptions.

Limit theorems and strong approximation for occupation times problem of sub-fractional Brownian motion in a class of Besov spaces

In this paper, we use the tightness criterion in Besov spaces, proved by Ait Ouahra, Boufoussi and Lakhel [Stochastics Stochastics Rep. 74 (2002), 411–427] and Boufoussi, Lakhel and Dozzi [Stoch. Anal. Appl. 23 (2005), no. 4, 665–685], to give some limit theorems for occupation times problem of a sub-fractional Brownian motion. Finally, we give a strong approximation version of our limit theorems, more precisely Lp-estimate version.

Parametric estimation for sub-fractional Ornstein–Uhlenbeck process

We consider the parameter estimation problem for the sub-fractional Ornstein–Uhlenbeck process defined as X0=0,dXt=θXtdt+dStH, t⩾0, with parameter θ>0, where SH is a sub-fractional Brownian motion with index H>12. We study the consistency and the asymptotic distribution of the least squares estimator θ^t of θ based on the observation {Xs,s∈[0,t]} as t→∞.

Singularity of Subfractional Brownian Motions with Different Hurst Indices

We prove that the probability measures generated by two subfractional Brownian motions with different Hurst indices are singular with respect to each other.

Particle picture interpretation of some Gaussian processes related to fractional Brownian motion

We construct fractional Brownian motion, sub-fractional Brownian motion and negative sub-fractional Brownian motion by means of limiting procedures applied to some particle systems. These processes are obtained for full ranges of Hurst parameter.We employ the so-called white noise approach. Our construction is quite general, permitting to obtain also some other Gaussian processes, as well as multidimensional random fields. In particular, we generalize and presumably simplify some results by Hambly and Jones (2007). We also obtain a new class of S′-valued density processes, containing as a particular case the density process of Martin-Löf (1976).

On the convergence to the multiple subfractional Wiener–Itô integral

In this paper, we construct a family of continuous stochastic processes that converges in law to the multiple Wiener–Itô integrals with respect to the subfractional Brownian motion with H>12 for the integrand f in a rather general class of functions. We mainly use Donsker and Stroock approximations and the techniques of the multiple Wiener–Itô integral with respect to the Wiener process.

On the Self‐Intersection Local Time of Subfractional Brownian Motion

We study the problem of self‐intersection local time of d‐dimensional subfractional Brownian motion based on the property of chaotic representation and the white noise analysis.

Remarks on Confidence Intervals for Self‐Similarity Parameter of a Subfractional Brownian Motion

We first present two convergence results about the second‐order quadratic variations of the subfractional Brownian motion: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we combine these results and concentration inequalities to build confidence intervals for the self‐similarity parameter associated with one‐dimensional subfractional Brownian motion.

Limit theorems for a quadratic variation of Gaussian processes

In the paper a weighted quadratic variation based on a sequence of partitions for a class of Gaussian processes is considered. Conditions on the sequence of partitions and the process are established for the quadratic variation to converge almost surely and for a central limit theorem to be true. Also applications to bifractional and sub-fractional Brownian motion and the estimation of their parameters are provided.

Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions

Let SH and S ̃ H be two independent d-dimensional sub-fractional Brownian motions with indices H ∈ (0, 1). Assume d ≥ 2, we investigate the intersection local time of subfractional Brownian motions ℓ T = ∫ 0 T ∫ 0 T δ S t H - S ̃ s H d s d t , T > 0 , where δ denotes the Dirac delta function at zero. By elementary inequalities, we show that ℓ T exists in L2 if and only if Hd < 2 and it is smooth in the sense of the Meyer-Watanabe if and only if H < 2 d + 2 . As a related problem, we give also the regularity of the intersection local time process.2010 Mathematics Subject Classification: 60G15; 60F25; 60G18; 60J55.

Stochastic integration with respect to the sub-fractional Brownian motion with [formula omitted

We define a stochastic integral with respect to sub-fractional Brownian motion S H with index H ∈ ( 0 , 1 2 ) that extends the divergence integral from Malliavin calculus. For this extended divergence integral, we establish versions of the formulas of Itô and Tanaka that hold for all H ∈ ( 0 , 1 2 ) .

Remarks on asymptotic behavior of weighted quadratic variation of subfractional Brownian motion

The present note is devoted to prove, by means of Malliavin calculus, the convergence in L2 of some properly renormalized weighted quadratic variation of sub-fractional Brownian motion SH with parameter H<14.

Remarks on sub-fractional Bessel processes

Let S = {( S 1 t ,…, S d t )} t≥0 denote a d-dimensional sub-fractional Brownian motion with index H≥1/2. In this paper we study some properties of the process X of the form X t : = ∑ i = 1 d ∫ 0 t S s i R s d S s i , d ≥ 1 , where R t = ( S t 1 ) 2 + … + ( S t d ) 2 is the sub-fractional Bessel process.

Itô's formula for a sub-fractional Brownian motion

In this paper we consider stochastic calculus connected with sub- fractional Brownian motion S H with H 2 ( 1 ;1) and narrow the focus to obtain various versions of It^o's formula. We introduce the integral of deter- ministic functions f with respect to the local time L H (x;t) of S H and the weighted quadratic covariation (f(S H );S H ) (W) . We establish the Bouleau- Yor identity (2 2 2H 1 ) h f(S H );S H i (W) t = Z R

Remarks on an integral functional driven by sub-fractional Brownian motion

This paper studies the functionals A 1 ( t , x ) = ∫ 0 t 1 [ 0 , ∞ ) ( x − S s H ) d s , A 2 ( t , x ) = ∫ 0 t 1 [ 0 , ∞ ) ( x − S s H ) s 2 H − 1 d s , where ( S t H ) 0 ≤ t ≤ T is a one-dimension sub-fractional Brownian motion with index H ∈ ( 0 , 1 ) . It shows that there exists a constant p H ∈ ( 1 , 2 ) such that p -variation of the process A j ( t , S t H ) − ∫ 0 t ℒ j ( s , S s H ) d S s H ( j = 1 , 2 ) is equal to 0 if p > p H , where ℒ j , j = 1 , 2 , are the local time and weighted local time of S H , respectively. This extends the classical results for Brownian motion.

A complement to Gladyshev’s theorem

We prove the almost sure convergence of a weighted quadratic variation for a class of Gaussian processes. The result is applied to a bifractional Brownian motion and a subfractional Brownian motion.

Berry–Esséen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion

In this paper, we derive explicit bounds for the Kolmogorov distance in the CLT and we prove the almost sure CLT for the quadratic variation of the sub-fractional Brownian motion. We use recent results on the Stein method combined with the Malliavin calculus and an almost sure CLT for multiple integrals.

On the local time of sub-fractional Brownian motion

S H = {S H t , t ≥ 0} be a sub-fractional Brownian motion with H ∈ (0, 1). We establish the existence, the joint continuity and the Holder regularity of the local time L H of S H . We will also give Chung's form of the law of iterated logarithm for S H . This results are obtained with the decomposition of the sub-fractional Brownian motion into the sum of fractional Brownian motion plus a stochastic process with absolutely continuous trajectories. This decomposition is given by Ruiz de Chavez and Tudor [10].