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.

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.

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.

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 Continuous, Discrete, and Conditional Scan Statistics

2012 ◽  
Vol 49 (01) ◽  
pp. 199-209 ◽  
Author(s):  
James C. Fu ◽  
Tung-Lung Wu ◽  
W.Y. Wendy Lou
Keyword(s):  
Rates Of Convergence ◽  
Scan Statistics ◽  
Random Permutations ◽  
Bernoulli Trials ◽  
Finite Markov Chain ◽  
Scan Statistic ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Runs And Patterns ◽  
Theoretical Results

The distributions for continuous, discrete, and conditional discrete scan statistics are studied. The approach of finite Markov chain imbedding, which has been applied to random permutations as well as to runs and patterns, is extended to compute the distribution of the conditional discrete scan statistic, defined from a sequence of Bernoulli trials. It is shown that the distribution of the continuous scan statistic induced by a Poisson process defined on (0, 1] is a limiting distribution of weighted distributions of conditional discrete scan statistics. Comparisons of rates of convergence as well as numerical comparisons of various bounds and approximations are provided to illustrate the theoretical results.

Waiting times for patterns in a sequence of multistate trials

2001 ◽  
Vol 38 (2) ◽  
pp. 508-518 ◽  
Author(s):  
Demetrios L. Antzoulakos
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variable ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns ◽  
Imbedding Method

Let Xn, n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let Xr,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of Xr,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.

Waiting times for patterns in a sequence of multistate trials

2001 ◽  
Vol 38 (02) ◽  
pp. 508-518 ◽  
Author(s):  
Demetrios L. Antzoulakos
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variable ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns ◽  
Imbedding Method

Let X n , n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let X r,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of X r,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.

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