位置参数变点的非参数检验及其渐近性质

本文基于Ｕ－统计量,对于位置参数模型,讨论了位置参数变点的检验问题,给出了检验统计量并研究它的分市的极限性质,证明了检验统计量的极限分布是ｓｕｐ|Ｂ(ｔ)|,其中{Ｂ(ｔ),０＜ｔ＜１}是一个Ｂｒｏｗｎ桥．将此结果应用到了双参数指数分布和Ｗｅｉｂｕｌｌ分布尺度参数变点的检验问题中．     

Dirichlet外问题的概率数值方法

针对定解区域是无界区域的Dirichlet外问题,提出了一种新的有效的概率数值方法,它是从解的随机表达式出发,将无界区域上的问题转化成区域边界上的问题.此时,只要在边界上进行剖分,将问题离散化,然后在无界区域外的有界区域内构作一个辅助球,并且利用布朗运动、漂移布朗运动从球外一点出发,首中球面的位置和时间的分布等,就可以获得Dirichlet外问题的数值解.     

An Extension of Vervaat's Transformation and Its Consequences

Vervaat⁽¹⁸⁾ proved that by exchanging the pre-minimum and post-minimum parts of a Brownian bridge one obtains a normalized Brownian excursion. Let s (0, 1), then we extend this result by determining a random time m _s such that when we exchange the pre-m _s-part and the post-m _s-part of a Brownian bridge, one gets a Brownian bridge conditioned to spend a time equal to s under 0. This transformation leads to some independence relations between some functionals of the Brownian bridge and the time it spends under 0. By splitting the Brownian motion at time m _s in another manner, we get a new path transformation which explains an identity in law on quantiles due to Port. It also yields a pathwise construction of a Brownian bridge conditioned to spend a time equal to s under 0.     

Probability density function of the local score position

We calculate the probability density function of the local score position on complete excursions of a reflected Brownian motion. We use the trajectorial decomposition of the standard Brownian bridge to derive two different expressions of the density: the first one is based on a series and an integral while the second one is free off the series.     

Computational approaches to estimation in the principal component analysis of a stochastic process

After performing a review of the classical procedures for estimation in the principal component analysis (PCA) of a second order stochastic process, two alternative procedures have been developed to approach such estimates. The first is based on the orthogonal projection method and uses cubic interpolating splines when the data are discrete. The second is based on the trapezoidal method. The accuracy of both procedures is tested by simulating approximated sample-functions of the Brownian motion and the Brownian bridge. The real principal factors of these stochastic processes, which can be evaluated directly, are compared with those estimated by means of the two mentioned algorithms. An application for estimation in the PCA of tourism evolution in Spain from real data is also included.     

Chung's law for integrated Brownian motion

The small ball problem for the integrated process of a real-valued Brownian motion is solved. In sharp contrast to more standard methods, our approach relies on the sample path properties of Brownian motion together with facts about local times and Lévy processes.

     

时间序列中参数变点的检测(英文)

The paper considers the problem of testing for a change point in the parameters of AR（p） models.It is shown that the asymptotically limiting distribution of the residual CUSUM of squares test（RCUSQ） is still the sup of a standard Brownian bridge under null hypothesis.We also show via simulations that our asymptotic results provide good approximations in finite samples.     

A note about the Gaussian property of the Brownian bridge

The Gaussian property of the Brownian bridge is characterized as an application of Ramachandran's theorem in terms of the independence of the random variables that appear in the Karhunen-Loéve expansion of the process. A reference about the construction of the Brownian bridge by means of functional transformations is also included.     

广义布朗单的容度估计

研究了既没有平稳增量性,也没有scaling性质的N指标d维广义布朗单的容度问题．证明了广义布朗单“好象”一个局部平稳增量过程,应用Cairoli极大不等式和多参数鞅的方法得到了广义布朗单的碰撞概率与容度之间的关系,给出了其碰撞概率的确切容度估计．所得结果包含了布朗单和可加布朗运动的相应结果．     

Optimal stopping of a Brownian bridge with an unknown pinning point

The problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. A few general properties, such as continuity and various bounds of the value function, are established. However, structural properties of the optimal stopping region are shown to crucially depend on the prior, and we provide a general condition for a one-sided stopping region. Moreover, a detailed analysis is conducted in the cases of the two-point and the mixed Gaussian priors, revealing a rich structure present in the problem.     

Statistical Analysis of the Fractional Ornstein–Uhlenbeck Type Process

We consider the fractional analogue of the Ornstein–Uhlenbeck process, that is, the solution of a one-dimensional homogeneous linear stochastic differential equation driven by a fractional Brownian motion in place of the usual Brownian motion. The statistical problem of estimation of the drift and variance parameters is investigated on the basis of a semimartingale which generates the same filtration as the observed process. The asymptotic behaviour of the maximum likelihood estimator of the drift parameter is analyzed. Strong consistency is proved and explicit formulas for the asymptotic bias and mean square error are derived. Preparing for the analysis, a change of probability method is developed to compute the Laplace transform of a quadratic functional of some auxiliary process. This revised version was published online in August 2006 with corrections to the Cover Date.     

Test for Parameter Change in Diffusion Processes by Cusum Statistics Based on One-step Estimators

In this paper, we consider the problem of testing for a parameter change using the cusum test based on one-step estimators in diffusion processes. It is shown that under regularity conditions the cusum test statistic has the limiting distribution of a functional of Brownian bridge.     

The Distribution of the Sojourn Time for the Brownian Excursion

The solutions of various problems in the theories of queuing processes, branching processes, random graphs and others require the determination of the distribution of the sojourn time (occupation time) for the Brownian excursion. However, no standard method is available to solve this problem. In this paper we approximate the Brownian excursion by a suitably chosen random walk process and determine the moments of the sojourn time explicitly. By using a limiting approach, we obtain the corresponding moments for the Brownian excursion. The moments uniquely determine the distribution, enabling us to derive an explicit formula.     

Characterization of the least concave majorant of brownian motion,conditional on a vertex point,with application to construction

The characterization of the least concave majorant of brownian motion by Pitman (1983,Seminar on Stochastic Processes, 1982 (eds. E. Cinlar, K. L. Chung and R. K. Getoor), 219–228, Birkhäuser, Boston) is tweaked, conditional on a vertex point. The joint distribution of this vertex point is derived and is shown to be generated with extreme ease. A procedure is then outlined by which one can construct the least concave majorant of a standard Brownian motion path over any finite, closed subinterval of (0, ∞). This construction is exact in distribution. One can also construct a linearly interpolated version of the Brownian motion path (i.e. we construct the Brownian motion path over a grid of points and linearly interpolate) corresponding to this least concave majorant over the same finite interval. A discussion of how to translate the aforementioned construction to the least concave majorant of a Brownian bridge is also presented.     

Bridge representation and modal-path approximation

The article shows a bridge representation for the joint density of a system of stochastic processes consisting of a Brownian motion with drift coupled with a correlated fractional Brownian motion with drift. As a result, a small time approximation of the joint density is readily obtained by substituting the conditional expectation under the bridge measure by a single path: the modal-path from the initial point to the terminal point.     

On Rescaled Poisson Processes and the Brownian Bridge

The process obtained by rescaling a homogeneous Poisson process by the maximum likelihood estimate of its intensity is shown to have surprisingly strong self-correcting behavior. Formulas for the conditional intensity and moments of the rescaled Poisson process are derived, and its behavior is demonstrated using simulations. Relationships to the Brownian bridge are explored, and implications for point process residual analysis are discussed.     

带趋势项的ARCH误差模型参数变化的残量检验

This paper analyzes the problem of testing for parameters change in ARCH errors models with deterministic trend based on residual cusum test. It is shown that the asymptotically limiting distribution of the residual cusum test statistic is still the sup of a standard Brownian bridge under null hypothesis. In order to check this, we carry out a Monte Carlo simulation and examine the return of IBM data. The results from both simulation and real data analysis support our claim. We also can explain this phenomenon from a theoretical viewpoint that the variance in ARCH model in mainly determined by its parameters.     

How do path generation methods affect the accuracy of quasi-Monte Carlo methods for problems in finance?

Quasi-Monte Carlo (QMC) methods are important numerical tools in computational finance. Path generation methods (PGMs), such as Brownian bridge and principal component analysis, play a crucial role in QMC methods. Their effectiveness, however, is problem-dependent. This paper attempts to understand how a PGM interacts with the underlying function and affects the accuracy of QMC methods. To achieve this objective, we develop efficient methods to assess the impact of PGMs. The first method is to exploit a quadratic approximation of the underlying function and to analyze the effective dimension and dimension distribution (which can be done analytically). The second method is to carry out a QMC error analysis on the quadratic approximation, establishing an explicit relationship between the QMC error and the PGM. Equalities and bounds on the QMC errors are established, in which the effect of the PGM is separated from the effect of the point set (in a similar way to the Koksma–Hlawka inequality). New measures for quantifying the accuracy of QMC methods combining with PGMs are introduced. The usefulness of the proposed methods is demonstrated on two typical high-dimensional finance problems, namely, the pricing of mortgage-backed securities and Asian options (with zero strike price). It is found that the success or failure of PGMs that do not take into account the underlying functions (such as the standard method, Brownian bridge and principal component analysis) strongly depends on the problem and the model parameters. On the other hand, the PGMs that take into account the underlying function are robust and powerful. The investigation presents new insight on PGMs and provides constructive guidance on the implementation and the design of new PGMs and new QMC rules.     

Invariance of Wiener processes and of Brownian bridges by integral transforms and applications

We find all possible integral equations among a given family which conserve in distribution the Brownian bridge and the Wiener process. The result implies that the Parzen statistic for goodness of fit can be used if and only if the distribution is exponential, direct or reverse, or uniform.     

Karhunen-Loeve expansions for the m-th order detrended Brownian motion

The m-th order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion onto the subspace spanned by polynomials of degree up to m. We obtain the Karhunen-Loeve expansion for the process and establish a connection with the generalized (m-th order) Brownian bridge developed by MacNeill (1978) in the study of distributions of polynomial regression. The resulting distribution identity is also verified by a stochastic Fubini approach. As applications, large and small deviation asymptotic behaviors for the L ₂ norm are given.