scholarly journals Boundary crossing probabilities for stationary Gaussian processes and Brownian motion

1981 ◽  
Vol 263 (2) ◽  
pp. 469-469 ◽  
Author(s):  
Jack Cuzick
Keyword(s):  
Brownian Motion ◽  
Gaussian Processes ◽  
Boundary Crossing ◽  
And Brownian Motion ◽  
Stationary Gaussian Processes ◽  
Crossing Probabilities

Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

1983 ◽  
Vol 20 (03) ◽  
pp. 529-536
Author(s):  
W. J. R. Eplett
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Partial Ordering ◽  
Boundary Crossing ◽  
Conditional Probabilities ◽  
Life Distribution ◽  
Life Distributions ◽  
Bounded Below ◽  
Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

Approximations of boundary crossing probabilities for a Brownian motion

1999 ◽  
Vol 36 (4) ◽  
pp. 1019-1030 ◽  
Author(s):  
Alex Novikov ◽  
Volf Frishling ◽  
Nino Kordzakhia
Keyword(s):  
Brownian Motion ◽  
Power Series ◽  
Numerical Integration ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Square Root ◽  
Girsanov Transformation ◽  
Piecewise Linear Approximations ◽  
Infinite Power ◽  
Crossing Probabilities

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

scholarly journals Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes

2010 ◽  
Vol 47 (4) ◽  
pp. 1058-1071 ◽  
Author(s):  
James C. Fu ◽  
Tung-Lung Wu
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
First Passage Time ◽  
Passage Time ◽  
Boundary Crossing ◽  
Finite Markov Chain ◽  
Crossing Probability ◽  
Linear And Nonlinear ◽  
Absorbing States ◽  
Crossing Probabilities

We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear and nonlinear boundaries for Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain, and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundary studied in the literature are provided in order to illustrate our method.

Boundary crossing probability for Brownian motion and general boundaries

1997 ◽  
Vol 34 (1) ◽  
pp. 54-65 ◽  
Author(s):  
Liqun Wang ◽  
Klaus Pötzelberger
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Boundary Crossing ◽  
Time Interval ◽  
Linear Boundary ◽  
Piecewise Linear Functions ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Crossing Probabilities

An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.

A note on boundary - crossing probabilities for the Brownian motion

1972 ◽  
Vol 9 (04) ◽  
pp. 857-861 ◽  
Author(s):  
C. S. Smith
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Simultaneous Equations ◽  
Boundary Crossing ◽  
Direct Solution ◽  
Kolmogorov Smirnov ◽  
Alternative Procedures ◽  
Crossing Probabilities ◽  
Special Case ◽  
Smirnov Test

In his paper ‘Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test’ (J. Appl. Prob. 8, 431–453), Durbin derives an integral equation whose kernel has a singularity. Since direct solution of an approximating set of simultaneous equations would be very inaccurate, he uses probability arguments to approximate to integrals of sub-intervals. In this note, two alternative procedures are discussed. One makes a linear transformation of the original integral equation to eliminate the singularity; the other, due to Weiss and Anderssen, integrates the singular factor in the kernel over the sub-interval. Computation of a special case indicates this latter method to be the most effective of the three.

scholarly journals Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes

2010 ◽  
Vol 47 (04) ◽  
pp. 1058-1071 ◽  
Author(s):  
James C. Fu ◽  
Tung-Lung Wu
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
First Passage Time ◽  
Passage Time ◽  
Boundary Crossing ◽  
Finite Markov Chain ◽  
Crossing Probability ◽  
Linear And Nonlinear ◽  
Absorbing States ◽  
Crossing Probabilities

We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear and nonlinear boundaries for Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain, and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundary studied in the literature are provided in order to illustrate our method.

Boundary crossing probabilities for high-dimensional Brownian motion

2016 ◽  
Vol 53 (2) ◽  
pp. 543-553 ◽  
Author(s):  
James C. Fu ◽  
Tung-Lung Wu
Keyword(s):  
Brownian Motion ◽  
Nonlinear Boundary ◽  
Boundary Crossing ◽  
High Dimensional ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
One Dimensional ◽  
Markov Chain Imbedding ◽  
Dimensional Boundary ◽  
Crossing Probabilities

Abstract The two-sided nonlinear boundary crossing probabilities for one-dimensional Brownian motion and related processes have been studied in Fu and Wu (2010) based on the finite Markov chain imbedding technique. It provides an efficient numerical method to computing the boundary crossing probabilities. In this paper we extend the above results for high-dimensional Brownian motion. In particular, we obtain the rate of convergence for high-dimensional boundary crossing probabilities. Numerical results are also provided to illustrate our results.

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