Standard Wiener Process Research Articles

Adapted solution of a backward stochastic nonlinear Volterra integral equation1

This paper studies the existence and uniqueness of the following kind of backward stochastic nonlinear Volterra integral equation under global Lipschitz condition, where {W t ;t∈[0,T]} is a standard k-dimensional Wiener process defined on a probability space {Ω,F,F t ,P}, and X is {F T } measurable d-dimensional random vector. The problem is to look for an adapted pair of processes {X(t),Z(t,s);t∈[0,T],s∈[t,T]} with values in R d and R d×k respectively, which solves the above equation. This paper also generalize our results to the following equation: under rather restrictive assumptions on g. *This research was supported by the National Natural Science Foundation of China, program no. 79790130.

THE INVARIANCE PRINCIPLE FOR LINEAR PROCESSES WITH APPLICATIONS

Let Xt be a linear process defined by Xt = [sum ]k=0∞ ψkεt−k, where {ψk, k ≥ 0} is a sequence of real numbers and {εk, k = 0,±1,±2,...} is a sequence of random variables. Two basic results, on the invariance principle of the partial sum process of the Xt converging to a standard Wiener process on [0,1], are presented in this paper. In the first result, we assume that the innovations εk are independent and identically distributed random variables but do not restrict [sum ]k=0∞ |ψk| < ∞. We note that, for the partial sum process of the Xt converging to a standard Wiener process, the condition [sum ]k=0∞ |ψk| < ∞ or stronger conditions are commonly used in previous research. The second result is for the situation where the innovations εk form a martingale difference sequence. For this result, the commonly used assumption of equal variance of the innovations εk is weakened. We apply these general results to unit root testing. It turns out that the limit distributions of the Dickey–Fuller test statistic and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test statistic still hold for the more general models under very weak conditions.

Relationships of convolution products, generalizedtransforms, and the first variation on function space

We use a generalized Brownian motion process to define the generalized Fourier‐Feynman transform, the convolution product, and the first variation. We then examine the various relationships that exist among the first variation, the generalized Fourier‐Feynman transform, and the convolution product for functionals on function space that belong to a Banach algebra S(Lab[0, T]). These results subsume similar known results obtained by Park, Skoug, and Storvick (1998) for the standard Wiener process.

On the Hausdorff Dimension of the Set Generated by Exceptional Oscillations of a Two-Parameter Wiener Process

S. Orey and S. J. Taylor (1974, Proc. London Math. Soc.28, 174–192) proved that for 0⩽λ⩽1 the set E(λ)={t∈[0, 1]:limsuph↓0(2hlog(1/h))−1/2(W′(t+h)−W′(t))⩾λ} has Hausdorff dimension dimE(λ)=1−λ2 a.s. where W′(t) is a standard Wiener process. A corresponding result is obtained when W′ is replaced by a two-parameter Wiener process.

Addendum to 'Asymptotic Inference for a Linear Stochastic Differential Equation with Time Delay'

In Gushchin and Kiichler (1999) we considered the stochastic differential equation dX(t) = aX(t)dt + bX(t - 1)dt + dW(t), t > 0, (1) with initial condition X(t) = Xo(t), t E [-1, 0], where (Xo(t), t E [-1, 0]) is a given continuous stochastic process independent of a standard Wiener process (W(t), t > 0). The parameter 9 = (a, b)* is unknown and supposed to be estimated based on (X(t), t < T). (The sign * denotes matrix or vector transposition.) The solutions (X(t), t E [-1, T]) of the differential equation (1) generate a family (Pt, 9 E Rl2) of distributions on C([-1, T]). We studied local asymptotic properties of this family and corresponding properties of the maximum likelihood estimators of 9 for T -- oc.

CHARACTERISATION OF LINEAR MINI-MAX ESTIMATORS FOR LOSS FUNCTIONS OF ARBITRARY POWER

Let Y(t), t∈[0,1], be a stochastic process modelled as dYt=θ(t)dt+dW(t), where W(t) denotes a standard Wiener process, and θ(t) is an unknown function assumed to belong to a given set Θ⊂L2[0,1]. We consider the problem of estimating the value ℒ(θ), where ℒ is a continuous linear function defined on Θ, using linear estimators of the form &lt;m,y&gt;=∫m(t)dY(t), m∈L2[0,1]. The distance between the quantity ℒ(θ) and the estimated value is measured by a loss function. In this paper, we consider the loss function to be an arbitrary even power function. We provide a characterisation of the best linear mini-max estimator for a general power function which implies the characterisation for two special cases which have previously been considered in the literature, viz. the case of a quadratic loss function and the case of a quartic loss function.

Bivariate Sign Tests Based on the Sup, L1and L2Norms

The bivariate location problem is considered. The sup, L1 and L2 norms are used to construct bivariate sign tests from the univariate sign statistics computed on the projected observations on all lines passing through the origin. The tests so obtained are affine-invariant and distribution-free under the null hypothesis. The sup-norm gives rise to Hodges' test. A class of tests derived from the L2-norm, with Blumen's test as a member, is seen to be related to a class proposed by Oja and Nyblom (1989, J. Amer. Statist. Assoc., 84, 249-259). The L1-norm gives rise to a new test. Its asymptotic null distribution is seen to be the same as that of the L1-norm of a certain normal process related to the standard Wiener process. An explicit expression of its cumulative distribution function is given. A simulation study will examine the merits of the three approaches.

Weak approximation for a class of Gaussian processes

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

Weak approximation for a class of Gaussian processes

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

A Jurdjevic-Quinn-type theorem for stochastic nonlinear control systems

We consider stochastic nonaffine nonlinear control systems x/sub t/=x/sub 0/+/spl int//sub 0//sup t/ f(x/sub s/,u) ds+/spl int//sub 0//sup t/ g(x/sub s/,u) d/spl omega//sub s/ (written in the sense of Ito), /spl omega/ being a standard Wiener process, for which we give a sufficient condition for global stabilization by a bounded smooth state feedback that is explicitly given. This condition generalizes the well-known Jurdjevic-Quinn result for deterministic affine control systems.

Absolute-in-Nonlinearity-and-Delay Stability in Quadratic Mean of Stochastic Differential-Difference Systems with Poisson Switchings

Algebraic coefficient conditions of absolute asymptotic stability in quadratic mean of the equilibrium of a stochastic dynamic automatic control system are obtained in the present paper. A mathematical model of this system is a system of Ito-Skorokhod stochastic differential-difference delay equations with several standard Wiener processes, Poisson perturbations, and a finite number of delays.

Quadratic variation for Gaussian processes and application to time deformation

We are interested in the functional convergence in distribution of the process of quadratic variations taken along a regular partition for a large class of Gaussian processes indexed by [0,1], including the standard Wiener process as a particular case. This result is applied to the estimation of a time deformation that makes a non-stationary Gaussian process stationary.

Large deviations of theL p-norm of a wiener process with drift

In this paper we calculate the exact asymptotics of the probability P{‖w(t)+uct‖ p >u},u→∞, wherew(t) is the standard Wiener process and ‖x‖ p is the ordinary norm in the spaceL p[0,1],p≥2. The result is obtained on the basis of a general theorem due to the author on the asymptotics of the Gaussian measureP(uD),u→∞, for a Borel setD belonging to a separable Banach space.

Bayesian testing simple hypotheses on The drift of a wiener process With randomly relayed observations

We present a Byesian test of simple hypotheses on the drift of the observation process Xt = θt + Wtwhere θis a bernouli random variable (wt) is a standard Wiener process. sequential tests δ = (T, d) are used, where T is a (FX t)-stopping time, d is a (FX t+n)-measurable decision function, n is a postive random variable with a given distribution. The cost function V(π, T, d) of any sequential test θ = (T, d) is based on costs of wrong decisions and observation, respectively.

Existence results for viability problem associated to nonconvex stochastic differentiable inclusions

For the stochastic viability problem of the form where K, F are nonconvex-valued mapsg is a single-valued functionW is a standard Wiener process, we establish the existence of solutions under the assumption that F and g possess Lipschitz property and satisfy the tangential condition introduced by J Aubin and G Da Prato

On the occupation time of an iterated process having no local time

We study the asymptotic behaviour of the occupation time process ∫ t 0 I A ( W 1( L 2( s)))d s, t ⩾ 0, where W 1 is a standard Wiener process and L 2 is a Wiener local time process at zero that is independent from W 1. We prove limit laws, as well as almost sure upper and lower class theorems. Possible extensions of the obtained results are also discussed.

On functions transforming a Wiener process into a semimartingale

The description of the class of all functions f = (f(t,x), t≥ 0,x∈R) is given, for which the transformed process (f(t,W t ),t≥ 0) (where W is a standard Wiener process) is a semimartingale.

On first-passage-time and transition densities for strongly symmetric diffusion processes

One dimensional diffusion processes have been increasingly invoked to model a variety of biological, physical and engineering systems subject to random fluctuations (cf., for instance, Blake, I. F. and Lindsey, W. C. [2], Abrahams, J. [1], Giorno, V. et al [10] and references therein). However, usually the knowledge of the ‘free’ transition probability density function (pdf) is not sufficient; one is thus led to the more complicated task of determining transition functions in the presence of preassigned absorbing boundaries, or first-passage-time densities for time-dependent boundaries (see, for instance, Daniels, H. E. [6], [7], Giorno, V. et al. [10]). Such densities are known analytically only in some special instances so that numerical methods have to be implemented in general (cf., for instance, Buono-core, A. et al [3], [4], Giorno, V. et al [11]). The analytical approach becomes particularly effective when the diffusion process exhibits some special features, such as the symmetry of its transition pdf. For instance, in [10] special symmetry conditions on the transition pdf of one-dimensional time-homogeneous diffusion process with natural boundaries are investigated to derive closed form results concerning the transition pdf’s and the first-passage-time pdf for particular time-dependent boundaries. On the other hand, by using the method of images, in [6] Daniels has obtained a closed form expression for the transition pdf of the standard Wiener process in the presence of a particular time-dependent absorbing boundary. It is interesting to remark that such density cannot be obtained via the methods described in [10], even though the considered process exhibits the kind of symmetry discussed therein.

Franchissement et temps local pour l'oscillateur harmonique

We consider the second order Stochastic Differential Equation dP t β = V t β dt with P 0 β = p 0, dV t β = β V t β dt − βω 2 P t β + β dW t with V 0 β = v 0, where W stands for a standard Wiener process and where ω is a real constant. It is well-known that P β converges, as β goes to infinity, to an Ornstein-Uhlenbeck process P. In this Note, we study the convergence of the crossings of P β at level u during the time interval [0, t] · ( N t P β ( u)) to the local time of P( L t P ( u)).

THE TAIL BEHAVIOR OF THE STATIONARY DISTRIBUTION OF A FLUID QUEUE WITH A GAUSSIAN-TYPE INPUT RATE PROCESS

This paper deals with a fluid ueue with a Gausszan-type input rate process. The Gaussian-type 7 processes are ones defined as Rt = m + f_ h(t - s)dws, where m is a positive constant, wt is a standard Wiener process and h(t) is an integrable function such that h(tl2 and H(t) = fw h(s)ds are also integrable. The class of Gaussian-type processes is wide enough to contain most of continuous time stochastic processes proposed so far for coded video traffic. For the model, in this paper, the exponential decay property of the tail of the buffer content distribution is studied, and an upper bound and a lower one are given for the tail probability P(Qw > X) of the buffer content distribution in the steady state. These bounds show that the tail probability decays exponentially with rate -*, where H(0) = f: h(t)dt and C is the output rate of the fluid queue. This result guarantees, in a sense, the plausibility of the approximation formula P(Qw > X) % Bexp{-*X} i-f(0I2/2 proposed in the previous paper (Performance Evaluation, 19951.