Positional Error Model of Line Segments with Modeling and Measuring Errors Using Brownian Bridge

Spatial linear features are often represented as a series of line segments joined by measured endpoints in surveying and geographic information science.There are not only the measuring errors of the endpoints but also the modeling errors between the line segments and the actual geographical features.This paper presents a Brownian bridge error model for line segments combining both the modeling and measuring errors.First,the Brownian bridge is used to establish the position distribution of the actual geographic feature represented by the line segment.Second,an error propagation model with the constraints of the measuring error distribution of the endpoints is proposed.Third,a comprehensive error band of the line segment is constructed,wherein both the modeling and measuring errors are contained.The proposed error model can be used to evaluate line segments’overall accuracy and trustability influenced by modeling and measuring errors,and provides a comprehensive quality indicator for the geospatial data.

The CUSUM statistic of change point under NA sequences

In this paper,we investigate the CUSUM statistic of change point under the neg-atively associated(NA)sequences.By establishing the consistency estimators for mean and covariance functions respectively,the limit distribution of the CUSUM statistic is proved to be a standard Brownian bridge,which extends the results obtained under the case of an indepen-dent normal sample and the moving average processes.Finally,the finite sample properties of the CUSUM statistic are given to show the efficiency of the method by simulation studies and an application on a real data analysis.

Variational Approach for the Adapted Solution of Backw ard Stochastic Differential Equations with Locally Lipschitz Diffusion Coefficients

One existence integral condition was obtained for the adapted solution of the general backward stochastic differential equations(BSDEs). Then by solving the integral constraint condition, and using a limit procedure, a new approach method is proposed and the existence of the solution was proved for the BSDEs if the diffusion coefficients satisfy the locally Lipschitz condition. In the special case the solution was a Brownian bridge. The uniqueness is also considered in the meaning of "F0-integrable equivalent class" . The new approach method would give us an efficient way to control the main object instead of the "noise".

Tests for Parameter Changes in Time Series

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.

Information-based approach:Pricing of a credit risky asset in the presence of default time

We extend the information-based asset-pricing framework by Brody,Hughston&Macrina to incorporate a stochastic bankruptcy time for the writer of the asset.Our model introduces a non-defaultable cash flow Zr to be made at time T,alongside the time T of a possible bankruptcy of the writer of the asset are in line with the filtration generated by a Brownian random bridge with length v=T^T and pinning point ZT,where is a constant.Quantities Z and T are not necessarily independent.The model does not depend crucially on the interpretation of as a bankruptcy time.We derived the price process of the asset and compute the prices of associated options.The dynamics of the price process satisfy a diffusion equation.Employing the approach of P.-A.Meyer,we provide the explicit computation of the compensator of v.Leveraging special properties of the bridge process,we also provide the explicit expression of the compensator of Zr I(v,+o).The resulting conclusion highlights the totally inaccessible property of the stopping time v.This characteristic is particularly suitable for financial markets where the time of default of a writer cannot be predictable from any other signal in the system until default happens.

Inference of change-point in single index models

Single index models are widely used in medicine, econometrics and some other fields. In this paper, we consider the inference of a change point problem in single index models. Based on density-weighted average derivative estimation (ADE) method, we propose a statistic to test whether a change point exists or not. The null distribution of the test statistic is obtained using a permutation technique. The permuted statistic is rigorously shown to have the same distribution in the limiting sense under both null and alternative hypotheses. After the null hypothesis of no change point is rejected, an ADE-based estimate of the change point is proposed under assumption that the change point is unique. A simulation study confirms the theoretical results.

Test for autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes

In this paper, we consider the problem of testing for an autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Levy processes. For a test, we propose a class of test statistics constructed by an iterated cumulative sums of squares of the difference between two adjacent observations. It is shown that each of the test statistics weakly converges to the supremum of the square of a Brownian bridge. The test statistics are evaluated by some empirical results.

Stochastic Systems Arising from Markov Modulated Empirical Measures

This work is devoted to stochastic systems arising from empirical measures of random sequences(termed primary sequences) that are modulated by another Markov chain. The Markov chain is used to model random discrete events that are not represented in the primary sequences. One novel feature is that in lieu of the usual scaling in empirical measure sequences, the authors consider scaling in both space and time, which leads to new limit results. Under broad conditions, it is shown that a scaled sequence of the empirical measure converges weakly to a number of Brownian bridges modulated by a continuous-time Markov chain. Ramifications and special cases are also considered.

On the compensator of the default process in an information-based model

This paper provides sufficient conditions for the time of bankruptcy(of a company or a state)for being a totally inaccessible stopping time and provides the explicit computation of its compensator in a framework where the flow of market information on the default is modelled explicitly with a Brownian bridge between 0 and 0 on a random time interval.

Distribution Function of the Blow up Time of the Solution of an Anticipating Random Fatigue Equation

In this paper,we study the distribution function of the time of explosion of a stochastic differential equation modeling the length of the dominant crack due to fatigue.The main novelty is that initial condition is regarded as an anticipating random variable and the stochastic integral is in the forward sense.Under suitable conditions,we use the substitution formula from Russo and Vallois to find the local solution of this equation.Then,we find the law of blow up time by proving some results on barrier crossing probabilities of Brownian bridge.

A General Law of Moment Convergence Rates for Uniform Empirical Process

Let {Xn; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process Fn（t） = n^-1/2 ∑^ni=1 （I{xi≤t} - t）, 0 ≤ t 〈 1, ││Fn││ = sup0≤t≤ 1 │Fn（t）│. In this paper, the exact convergence rates of a general law of weighted infinite series of E{││Fn││ -εg^s（n）}＋ are obtained.

Modified Testing for Structural Changes in Autoregressive Processes

In this paper, we consider the problem of detecting for structural changes in the autoregressive processes including AR（p） process. In performing a test, we employ the conventional residual CUSUM of squares test （RCUSQ） statistic. The RCUSQ test is based on the subsampling method introduced by Jach and Kokoszka [J. Methodology and Computing in Applied Probability 25（2004）]. It is shown that under regularity conditions, the asymptotic distribution of the test statistic is the function of a standard Brownian bridge. Simulation results as to AR（1） process and an example of real data analysis are provided for illustration.

Residual Cusum Test for Parameters Change in ARCH Errors Models with Deterministic Trend

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.