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.

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.

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.

Stochastic Boundary Crossing Probabilities for the Brownian Motion

2013 ◽  
Vol 50 (02) ◽  
pp. 419-429 ◽  
Author(s):  
Xiaonan Che ◽  
Angelos Dassios
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Simulation Experiments ◽  
Crossing Probability ◽  
Crossing Probabilities ◽  
Banks Set ◽  
Stochastic Boundary

Using martingale methods, we derive a set of theorems of boundary crossing probabilities for a Brownian motion with different kinds of stochastic boundaries, in particular compound Poisson process boundaries. We present both the numerical results and simulation experiments. The paper is motivated by limits on exposure of UK banks set by CHAPS. The central and participating banks are interested in the probability that the limits are exceeded. The problem can be reduced to the calculation of the boundary crossing probability from a Brownian motion with stochastic boundaries. Boundary crossing problems are also very popular in many fields of statistics.

scholarly journals Stochastic Boundary Crossing Probabilities for the Brownian Motion

2013 ◽  
Vol 50 (2) ◽  
pp. 419-429 ◽  
Author(s):  
Xiaonan Che ◽  
Angelos Dassios
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Simulation Experiments ◽  
Crossing Probability ◽  
Crossing Probabilities ◽  
Banks Set ◽  
Stochastic Boundary

Using martingale methods, we derive a set of theorems of boundary crossing probabilities for a Brownian motion with different kinds of stochastic boundaries, in particular compound Poisson process boundaries. We present both the numerical results and simulation experiments. The paper is motivated by limits on exposure of UK banks set by CHAPS. The central and participating banks are interested in the probability that the limits are exceeded. The problem can be reduced to the calculation of the boundary crossing probability from a Brownian motion with stochastic boundaries. Boundary crossing problems are also very popular in many fields of statistics.

Boundary crossing probability for Brownian motion and general boundaries

1997 ◽  
Vol 34 (01) ◽  
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.

Boundary crossing probability for Brownian motion

2001 ◽  
Vol 38 (1) ◽  
pp. 152-164 ◽  
Author(s):  
Klaus Pötzelberger ◽  
Liqun Wang
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Approximation Error ◽  
Simulation Method ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Approximation Errors ◽  
Crossing Probabilities ◽  
Asymptotic Upper Bound

Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n2).

Boundary crossing probability for Brownian motion

2001 ◽  
Vol 38 (01) ◽  
pp. 152-164 ◽  
Author(s):  
Klaus Pötzelberger ◽  
Liqun Wang
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Approximation Error ◽  
Simulation Method ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Approximation Errors ◽  
Crossing Probabilities ◽  
Asymptotic Upper Bound

Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n 2).

scholarly journals On the First Passage time for Brownian Motion Subordinated by a Lévy Process

2009 ◽  
Vol 46 (1) ◽  
pp. 181-198 ◽  
Author(s):  
T. R. Hurd ◽  
A. Kuznetsov
Keyword(s):  
Brownian Motion ◽  
Approximation Scheme ◽  
First Passage Time ◽  
Gaussian Model ◽  
Passage Time ◽  
Financial Mathematics ◽  
First Passage ◽  
Financial Modeling ◽  
Motion Time ◽  
Normal Inverse Gaussian

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

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