A process of runs and its convergence to the brownian motion

Let X₁,X₂,… be i.i.d. random variables with a continuous distribution function. Let R₀=0, R_k=min{j>R_k?1, such that X_j>X_j+1}, k?1. We prove that all finite-dimensional distributions of a process $\text{W}{}^{\mathrm{(n)}}\text{(t)=}\text{(R}{}_{\mathrm{[nt]}}\text{?2[nt])}\text{2}\text{3}\text{n}\text{, t ? [0,1]}$, converge to those of the standard Brownian motion.     

Weak solutions to stochastic differential equations driven by fractional brownian motion

Existence of a weak solution to the n-dimensional system of stochastic differential equations driven by a fractional Brownian motion with the Hurst parameter H ∈ (0, 1) \ {1/2} is shown for a time-dependent but state-independent diffusion and a drift that may by split into a regular part and a singular one which, however, satisfies the hypotheses of the Girsanov Theorem. In particular, a stochastic nonlinear oscillator driven by a fractional noise is considered.     

Weak convergence to fractional Brownian motion in Brownian scenery

 Kesten and Spitzer have shown that certain random walks in random sceneries converge to stable processes in random sceneries. In this paper, we consider certain random walks in sceneries defined using stationary Gaussian sequence, and show their convergence towards a certain self-similar process that we call fractional Brownian motion in Brownian scenery. Received: 17 April 2002 / Revised version: 11 October 2002 / Published online: 15 April 2003 Research supported by NSFC (10131040). Mathematics Subject Classification (2002): 60J55, 60J15, 60J65 Key words or phrases: Weak convergence – Random walk in random scenery – Local time – Fractional Brownian motion in Brownian scenery     

Linear filtering with fractional brownian motion

A Kalman type system of integral equations is obtained for the linear filtering problem in which the noise generating the signal is a fractional Brownian motion with long-range dependence. The error in applying the usual Kalman filter to this problem is determined explicitly for a simple example     

Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion

We consider a random walk that converges weakly to a fractional Brownian motion with Hurst index H > 1/2. We construct an integral-type functional of this random walk and prove that it converges weakly to an integral constructed on the basis of the fractional Brownian motion. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1040–1046, August, 2007.     

Weak convergence to the fractional Brownian sheet in Besov spaces

In this paper we study the problem of the approximation in law of the fractional Brownian sheet in the topology of the anisotropic Besov spaces. We prove the convergence in law of two families of processes to the fractional Brownian sheet: the first family is constructed from a Poisson procces in the plane and the second family is defined by the partial sums of two sequences of real independent fractional brownian motions.     

Weak convergence to the fractional Brownian sheet using martingale differences

We establish an invariance principle for the fractional Brownian sheet, starting from discrete random fields constructed from two-parameter strong martingales. This is an approximation in law of the fractional Brownian sheet in Skorohord space in the plane.     

Rate of convergence of uniform transport processes to brownian motion and application to stochastic integrals

A rate of convergence of a sequence of uniform transport processes to Brownian motion is derived, and a correspondmg rate for the Wong and Zakai approximation of stochastic integrals is given     

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.     

On generalized local time for the process of brownian motion

We prove that the functionals of a d-dimensional Brownian process are Hida distributions, i.e., generalized Wiener functionals. Here, δ_Γ(·) is a generalization of the δ-function constructed on a bounded closed smooth surface Γ⊂R ^d , k≥1 and acting on finite continuous functions φ(·) in R ^d according to the rule where ι(·) is a surface measure on Γ.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

Local field brownian motion

A local field is any locally compact, non-discrete field other than the field of real numbers or the field of complex numbers. There is a natural notion of Gaussian measures on a local field vector space. We construct and study a specific local field Gaussian stochastic process taking values in a finite dimensional local field vector space and indexed by another finite dimensional local field vector space. This process has a structure that strongly reflects the algebraic and geometric structure of the underlying index space and, as such, plays the same role in the local field setting that standard Brownian motion and the related multiparameter processes such as Lévy's multiparameter Brownian motion play in a Euclidean context. We investigate the theory of additive functionals and the related potential theory for this process and show that it strongly resembles the Euclidean prototype. As a particular consequence of this investigation, we find that a local time process exists when the process hits points. We give two intrinsic constructions of the local time at a given level. These constructions are analogous to the dilation construction of Kingman and the Hausdorff measure construction of Taylor and Wendel in the Euclidean case. Finally, the local time is shown to be continuous as a measure valued stochastic process indexed by the levèl at which it is evaluated.Research supported in part by an NSF Grant and Presidential Young Investigator Award.     

Weak convergence of random polygonal lines to a Gaussian process

We obtain a limit theorem of convergence in distribution for random polygonal lines defined by sums of independent random variables with replacements. In a particular case, the limit is the Gaussian Ornstein-Uhlenbeck process.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 33–44, January–March, 2005.Translated by V. Mackeviius     

Weak convergence to multifractional Brownian motion of Riemann-Liouville type in Besov spaces

We study the weak convergence of the family of processes {V _n (t)} _n??? defined by $$V_n(t)=\int_{0}^t(t-u)^{H(t)-\frac{1}{2}}\theta_n(u)du,$$ where {?? _n (u)} _n??? is a family of processes converging in law to a Brownian motion, as n????. We consider two cases of {?? _n }. First, we construct ?? _n based on the well-known Donsker??s theorem and show that {V _n (t)} _n??? converges in law to a multifractional Brownian motion of Riemann-Liouville type, as n????. Second, we construct ?? _n based on a Poisson process, and then show that a multifractional Brownian motion of Riemann-Liouville type can be approximated in law by {V _n (t)} _n???.     

Weak convergence of the past and future of Brownian motion given the present

In this paper, we show that for t > 0, the joint distribution of the past {W _t?s : 0 ≤ s ≤ t} and the future {W _{t + s} :s ≥ 0} of a d-dimensional standard Brownian motion (W _s ), conditioned on {W _t ∈ U}, where U is a bounded open set in ? ^d , converges weakly in C[0,∞)×C[0,∞) as t→∞. The limiting distribution is that of a pair of coupled processes Y + B ¹,Y + B ² where Y,B ¹,B ² are independent, Y is uniformly distributed on U and B ¹,B ² are standard d-dimensional Brownian motions. Let σ _t ,d _t be respectively, the last entrance time before time t into the set U and the first exit time after t from U. When the boundary of U is regular, we use the continuous mapping theorem to show that the limiting distribution as t → ∞ of the four dimensional vector with components \((W_{\sigma _{t}},t-\sigma _{t},W_{d_{t}},d_{t}-t)\), conditioned on {W _t ∈U}, is the same as that of the four dimensional vector whose components are the place and time of first exit from U of the processes Y + B ¹ and Y + B ² respectively.     

Weak convergence of compound stochastic process,I

Compound stochastic processes are constructed by taking the superpositive of independent copies of secondary processes, each of which is initiated at an epoch of a renewal process called the primary process. Suppose there are M possible k-dimensional secondary processes {ξ^v(t):t?0}, v=1,2,…,M. At each epoch of the renewal process {A(t):t?0} we initiate a random number of each of the M types. Let m_l:l?1} be a sequence of M-dimensional random vectors whose components specify the number of secondary processes of each type initiated at the various epochs. The compound process we study is

, where the ξ^v_lj() are independent copies of ξ^v,m_lv is the vth component of m and {τ_l:l?1} are the epochs of the renewal process. Our interest in this paper is to obtain functional central limit theorems for {Y(t):t?0} after appropriately scaling the time parameter and state space. A variety of applications are discussed.     

Weak convergence of the tail empirical process for dependent sequences

This paper proves weak convergence in D$D$ of the tail empirical process–the renormalized extreme tail of the empirical process–for a large class of stationary sequences. The conditions needed for convergence are (i) moment restrictions on the amount of clustering of extremes, (ii) restrictions on long range dependence (absolute regularity or strong mixing), and (iii) convergence of the covariance function. We further show how the limit process is changed if exceedances of a nonrandom level are replaced by exceedances of a high quantile of the observations. Weak convergence of the tail empirical process is one key to asymptotics for extreme value statistics and its wide range of applications, from geoscience to finance.