On the infimum of the local time of a Wiener process

Let {W(t), t0} be a standard Wiener process, and let L(x, t) be its jointly continuous local time. Define

Distribution of the time of first arrival of a Wiener process at a given boundary

An exact expression is derived for the characteristic function of the distribution of the first arrival time of a Wiener process at a given boundary, which is fixed in time. An approximate expression is obtained for the same characteristic function assuming that the given boundary is a function of time. The properties of the distribution of the first arrival time are analyzed.Translated from Statisticheskie Metody, pp. 31–42, 1980.     

A Wiener process in a curvilinear strip

One investigates the representation of the probabilities of nonexit of a Brownian motion from a curvilinear strip in the form of an expansion with respect to a system of eigenfunctions. It is shown that the coefficients of this series satisfy a Volterra integral equation in ₂. For thin strips, with bounded first derivatives of the boundaries, the successive iterations of the equation determine a complete asymptotic expansion of the nonexit probabilities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 561–563, April, 1990.     

A martingale approach for detecting the drift of a Wiener process

Lerchez (Ann. Statist. 14, 1986b, 1030–1048) considered a sequential Bayes-test problem for the drift of the Wiener process. In the case of a normal prior an o(c)-optimal test could be constructed. In this paper a new martingale approach is presented, which provides an expansion of the Bayes risk for a one-sided SPRT. Relations to the optimal Bayes risk are given, which show the o(c)-optimality for suitable nonnormal priors.     

Asymptotic expansions in the central limit theorem for a branching Wiener process

Science China Mathematics - We consider a branching Wiener process in ?d, in which particles reproduce as a super-critical Galton-Watson process and disperse according to a Wiener process....     

Extremes of Shepp statistics for the Wiener process

Define , where W(·) is a standard Wiener process. We study the maximum of Y up to time T: and de termine an asymptotic expression for when u→ ∞. Further we establish the limiting Gumbel distribution of M _T when T→ ∞ and present the corresponding normalization sequence.      

Gradient-based iterative identification for MISO Wiener nonlinear systems: Application to a glutamate fermentation process

This paper deals with modeling and parameter identification of multiple-input single-output Wiener nonlinear systems. The basic idea is to construct a multiple-input single-output Wiener nonlinear model and to derive the gradient-based iterative algorithm for the proposed model. The proposed method has been applied to identify the parameters of a glutamate fermentation process. The results of real data simulation show that this method is effective.     

On large increments of a two-parameter fractional Wiener process

In this paper, how big the increments are and some liminf behaviors are studied of a two-parameter fractional Wiener process. The results are based on some inequalities on the suprema of this process, which also are of independent interest     

Probability of a stationary stochastic process staying in a strip

We consider the distribution of the duration of a stochastic process staying in a strip. An example is given of the calculation of the distribution for a Gaussian cosine process.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 128–130, 1988.     

On the maximum of a Wiener process and its location

Summary Consider a Wiener process {W(t),t0}, letM(t)=max |W(s)| andv(t) be the location of the maximum of the absolute value of in [0,t] i.e.|W(v(t))|=M(t). We study the limit points of ( _t M(t), _t v(t)) ast where _t and _t are positive, decreasing normalizing constants. Moreover, a lim inf result is proved for the length of the longest flat interval ofM(t).Research supported by Hungarian National Foundation for Scientific Research Grant n. 1808     

An integral test for the supremum of Wiener local time

Summary The upper classes of the supremum of Wiener (Brownian) local time are characterized by convergence or divergence of a certain integral.Dedicated to Klaus Krickeberg on the occasion of his 60th birthdayResearch supported by Hungarian National Foundation for Scientific Research Grant No. 1808     

On the long time behavior of the TCP window size process

The TCP window size process appears in the modeling of the famous transmission control protocol used for data transmission over the Internet. This continuous time Markov process takes its values in [0,∞)$[0,\infty )$, and is ergodic and irreversible. It belongs to the additive increase–multiplicative decrease class of processes. The sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. Several aspects of this process have already been investigated in the literature. In the present paper, we mainly get quantitative estimates for the convergence to equilibrium, in terms of the _W1$W_1$ Wasserstein coupling distance, for the process and also for its embedded chain.     

On the increments of the local time of a Wiener sheet

{W(x, y), x≥0, y≥0} be a Wiener process and let η(u, (x, y)) be its local time. The continuity of η in (x, y) is investigated, i.e., an upper estimate of the process η(μ, [x, x + α) × [y, y + β)) is given when αβ is small.     

Upper classes for the increments of the fractional Wiener process

Summary LetX(t) be a fractional Wiener process, i.e., a centered Gaussian process on [0, ) with stationary increments and varianceEX ² (t)=t ², anda(t) a positive nondecreasing function witha(t)t. We investigate the a.s. asymptotic behaviour of the incrementsI(t, a (t))=max{X{u+a(t))–X(u): 0ut–a(t)} (and some others that are similarly defined) ast.