A note on boundary - crossing probabilities for the Brownian motion

1972 ◽  
Vol 9 (4) ◽  
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.

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 Fractional Relaxation Equations and Brownian Crossing Probabilities of a Random Boundary

2012 ◽  
Vol 44 (2) ◽  
pp. 479-505 ◽  
Author(s):  
L. Beghin
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Boundary Crossing ◽  
Random Boundary ◽  
Fractional Equations ◽  
Gamma Density ◽  
Fractional Relaxation ◽  
Crossing Probabilities ◽  
Special Case

In this paper we analyze different forms of fractional relaxation equations of order ν ∈ (0, 1), and we derive their solutions in both analytical and probabilistic forms. In particular, we show that these solutions can be expressed as random boundary crossing probabilities of various types of stochastic process, which are all related to the Brownian motion B. In the special case ν = ½, the fractional relaxation is shown to coincide with Pr{sup0≤s≤tB(s) < U} for an exponential boundary U. When we generalize the distributions of the random boundary, passing from the exponential to the gamma density, we obtain more and more complicated fractional equations.

scholarly journals Fractional Relaxation Equations and Brownian Crossing Probabilities of a Random Boundary

2012 ◽  
Vol 44 (02) ◽  
pp. 479-505 ◽  
Author(s):  
L. Beghin
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Boundary Crossing ◽  
Random Boundary ◽  
Fractional Equations ◽  
Gamma Density ◽  
Fractional Relaxation ◽  
Crossing Probabilities ◽  
Special Case

In this paper we analyze different forms of fractional relaxation equations of order ν ∈ (0, 1), and we derive their solutions in both analytical and probabilistic forms. In particular, we show that these solutions can be expressed as random boundary crossing probabilities of various types of stochastic process, which are all related to the Brownian motionB. In the special case ν = ½, the fractional relaxation is shown to coincide with Pr{sup0≤s≤tB(s) &lt;U} for an exponential boundaryU. When we generalize the distributions of the random boundary, passing from the exponential to the gamma density, we obtain more and more complicated fractional equations.

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 Approximations for weighted Kolmogorov–Smirnov distributions via boundary crossing probabilities

2016 ◽  
Vol 27 (6) ◽  
pp. 1513-1523 ◽  
Author(s):  
Nino Kordzakhia ◽  
Alexander Novikov ◽  
Bernard Ycart
Keyword(s):  
Boundary Crossing ◽  
Kolmogorov Smirnov ◽  
Crossing Probabilities

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.

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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