HARNACK TYPE INEQUALITIES FOR SDES DRIVEN BY FRACTIONAL BROWNIAN MOTION WITH MARKOVIAN SWITCHING

In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by fractional Brownian motion with Markovian switching. The Hurst parameter H is supposed to be in(1/2, 1). As a direct application, the strong Feller property is presented.

Dynamics of stochastic non-Newtonian fluids driven by fractional Brownian motion with Hurst parameter H∈(1/4,1/2)

A two-dimensional （2D） stochastic incompressible non-Newtonian fluid driven by the genuine cylindrical fractional Brownian motion （FBM） is studied with the Hurst parameter ∈ （1/4,1/2） under the Dirichlet boundary condition. The existence and regularity of the stochastic convolution corresponding to the stochastic non-Newtonian fluids are obtained by the estimate on the and the identity of the infinite double series spectrum of the spatial differential operator in the analytic number theory. The existence of the mild solution and the random attractor of a random dynamical system are then obtained for the stochastic non-Newtonian systems with ∈ （1/2,1） without any additional restriction on the parameter H.

THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT

Let B^(H) be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the equation(called the Ornstein-Uhlenbeck process with a linear self-repelling drift)dX_(t)^(H)=dB_(t)^(H)+σ X_(t)^(H)dt+vdt-θ(∫_(0)^(t)(X_(t)^(H)-X_(s)^(H))ds)dt,whereθ<0,σ,v∈ℝ.The process is an analogue of self-attracting diffusion(Cranston,Le Jan.Math Ann,1995,303:87–93).Our main aim is to study the large time behaviors of the process.We show that the solution X^(H)diverges to infinity as t tends to infinity,and obtain the speed at which the process X^(H)diverges to infinity.

EXACT MAXIMUM LIKELIHOOD ESTIMATOR FOR DRIFT FRACTIONAL BROWNIAN MOTION AT DISCRETE OBSERVATION

This paper deals with the problems of consistency and strong consistency of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants. Both the central limit theorem and the Berry-Ess′een bounds for these estimators are obtained by using the Stein’s method via Malliavin calculus.

LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED FRACTIONAL BROWNIAN MOTION

In this article, we study a least squares estimator （LSE） of θ for the Ornstein- Uhlenbeck process X0=0,dXt=θXtdt＋dBt^ab, t ≥ 0 driven by weighted fractional Brownian motion B^a,b with parameters a, b. We obtain the consistency and the asymptotic distribution of the LSE based on the observation {Xs, s∈[0,t]} as t tends to infinity.

Some It Formulas with Respect to Mixed Fractional Brownian Motion and Brownian Motion

Some It formulas with respect to mixed Fractional Brownian motion and Brownian motion were given in this paper.These extended the It formula for the fractional Wick It Skorohod integral with respect to Fractional Brownian motion,meanwhile extended the It formula for It Skorohod integral with respect to Brownian motion.Taylor's formula is applied to prove our conclusion in this article.

On the sub-mixed fractional Brownian motion

Let {S t H, 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 SH lies between the sub-fractional Brownian motion and the mixed fractional Brownian motion. We also determine the values of H for which SH is not a semi-martingale.

CONTROLLABILITY OF NEUTRAL STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION

In this paper,we investigate the controllability for neutral stochastic evolution equations driven by fractional Brownian motion with Hurst parameter H ∈（1/2,1） in a Hilbert space.We employ the α-norm in order to reflect the relationship between H and the fractional power α.Sufficient conditions are established by using stochastic analysis theory and operator theory.An example is provided to illustrate the effectiveness of the proposed result.

A LIMINF RESULT FOR HANSON-RUSSO TYPE INCREMENTS OF FRACTIONAL BROWNIAN MOTION

Let {X(t), t greater than or equal to 0} be a fractional Brownian motion of order 2 alpha with 0 < alpha < 1,beta > 0 be a real number, alpha(T) be a function of T and 0 < alpha(T), [GRAPHICS] (log T/alpha(T))/log T = r, (0 less than or equal to r less than or equal to infinity). In this paper, we proved that [GRAPHICS] where c(1), c(2) are two positive constants depending only on alpha,beta.

ERRATUM TO： LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED FRACTIONAL BROWNIAN MOTION （ACTA MATHEMATICA SCIENTIA 2016,36B （2） ：394-408）

We give a correction of Theorem 2.2 of Shen, Yin and Yan （2016）.

Strong Local Non-Determinism of Sub-Fractional Brownian Motion

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

Controllability of Fractional Order Stochastic Differential Inclusions with Fractional Brownian Motion in Finite Dimensional Space

In this paper,sufficient conditions are formulated for controllability of fractional order stochastic differential inclusions with fractional Brownian motion(f Bm) via fixed point theorems,namely the Bohnenblust-Karlin fixed point theorem for the convex case and the Covitz-Nadler fixed point theorem for the nonconvex case.The controllability Grammian matrix is defined by using Mittag-Leffler matrix function.Finally,a numerical example is presented to illustrate the efficiency of the obtained theoretical results.

Some limit results on supremum of Shepp statistics for fractional Brownian motion

Define the incremental fractional Brownian field ZH（τ, s） = BH（s＋τ） -BH（s）,where BH（s） is a standard fractional Brownian motion with Hurst parameter H ∈ （0, 1）. Inthis paper, we first derive an exact asymptotic of distribution of the maximum MH（Tu） =supτ∈[0,1],s∈[0,xτu] ZH（τ, s）, which holds uniformly for x ∈ [A, B] with A, B two positive con-stants. We apply the findings to analyse the tail asymptotic and limit theorem of MH （τ） witha random index τ. In the end, we also prove an almost sure limit theorem for the maximum M1/2（τ） with non-random index T.

IMAGE OBJECT DETECTION BASED ON FRACTIONAL BROWNIAN MOTION

Fractional Brownian motion, continuous everywhere and differentiable nowhere, offers a convenient modeling for irregular nonstationary stochastic processes with long-term dependencies and power law behavior of spectrum over wide ranges of frequencies. It shows high correlation at coarse scale and varies slightly at fine scale, which is suitable for and successful in describing and modeling natural scenes. On the other hand, man-made objects can be constructively well described by using a set of regular simple shape primitives such as line, cylinder, etc. and are free of fractal. Based on the difference, a method to discriminate man-made objects from natural scenes is provided. Experiments are used to demonstrate the good efficiency of developed technique.

SOLUTIONS TO BSDES DRIVEN BY BOTH FRACTIONAL BROWNIAN MOTIONS AND THE UNDERLYING STANDARD BROWNIAN MOTIONS

The local existence and uniqueness of the solutions to backward stochastic differential equations（BSDEs, in short） driven by both fractional Brownian motions with Hurst parameter H ∈ （1/2, 1） and the underlying standard Brownian motions are studied. The generalization of the It6 formula involving the fractional and standard Brownian motions is provided. By theory of Malliavin calculus and contraction mapping principle, the local existence and uniqueness of the solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian motions are obtained.

A limit theorem for the solutions of slow–fast systems with fractional Brownian motion

A limit theorem which can simplify slow–fast dynamical systems driven by fractional Brownian motion with the Hurst parameter H inside the（1/2, 1） interval has been proved. The slow variables of the original system can be approximated by the solution of the simplified equations in the sense of mean square. An example is presented to illustrate the applications of the limit theorem.

A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

Let B={B^H(t)}t≥0 be a d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1).Consider the functionals of k independent d-dimensional fractional Brownian motions 1/√n∫0^ent1⋯∫0^entk f(B^H,1(s1)+⋯+B^H,k(sk))ds1⋯dsk,where the Hurst index H=k/d.Using the method of moments,we prove the limit law and extending a result by Xu\cite{xu}of the case k=1.It can also be regarded as a fractional generalization of Biane\cite{biane}in the case of Brownian motion.

Stability of Linear Stochastic Differential Equations with Respect to Fractional Brownian Motion

This paper is concerned with the stochastically stability for the m-dimensional linear stochastic differential equations with respect to fractional Brownian motion(FBM)with Hurst parameter H∈(1/2,1).On the basis of the pioneering work of Duncan and Hu,a It's formula is given.An improved derivative operator to Lyapunov functions is constructed,and the sufficient conditions for the stochastically stability of linear stochastic differential equations driven by FBM are established.These extend the stochastic Lyapunov stability theories.

Pricing Perpetual American Put Option in theMixed Fractional Brownian Motion

Under the assumption of the underlying asset is driven by the mixed fractional Brownian motion, we obtain the mixed fractionalBlack-Scholes partial differential equation by fractional Ito formula, and the pricing formula of perpetual American put option bythis partial differential equation theory.

Pricing Formulae of Asian Options under the Fractional Brownian Motion

In this paper,the pricing formulae of the geometric average Asian call option with the fixed and floating strike price under the fractional Brownian motion(FBM)are given out by the method of partial differential equation(PDE).The call-put parity for the geometric average Asian options is given.The results are generalization of option pricing under standard Brownian motion.