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 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 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 Two-dimensional stochastic modeling for predicting bankruptcy for manufacturing companies

2021 ◽  
Vol 54 (2) ◽  
pp. 123-129
Author(s):  
James C. Fu ◽  
Winnie H. W. Fu
Keyword(s):  
Brownian Motion ◽  
Real Data ◽  
Bankruptcy Prediction ◽  
Boundary Crossing ◽  
Time Interval ◽  
Two Dimensional ◽  
Manufacturing Companies ◽  
Boundary Crossing Probability ◽  
Data Set ◽  
Crossing Probability

Increasing accuracy of the model prediction on business bankruptcy helps reduce substantial losses for owners, creditors, investors and workers, and, further, minimize an economic and social problem frequently. In this study, we propose a stochastic model of financial working capital and cashflow as a two-dimensional Brownian motion X(t) = (X1(t),X2(t)) on the business bankruptcy prediction. The probability of bankruptcy occurring in a time interval [0,T] is defined by the boundary crossing probability of the two-dimensional Brownian motion entering a predetermined threshold domain. Mathematically, we extend the result in Fu and Wu (2016) on the boundary crossing probability of a high dimensional Brownian motion to an unbounded convex hull. The proposed model is applied to a real data set of companies in US and the numerical results show the proposed method performs well.

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.

A boundary-crossing result for Brownian motion

1992 ◽  
Vol 29 (2) ◽  
pp. 448-453 ◽  
Author(s):  
Thomas H. Scheike
Keyword(s):  
Brownian Motion ◽  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Crossing Probability

In this paper we obtain a new crossing result for Brownian motion. The boundary studied is a piecewise linear function consisting of two lines. The expression obtained for the boundary crossing probability is of a simple directly computable form.

A boundary-crossing result for Brownian motion

1992 ◽  
Vol 29 (02) ◽  
pp. 448-453 ◽  
Author(s):  
Thomas H. Scheike
Keyword(s):  
Brownian Motion ◽  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Crossing Probability

In this paper we obtain a new crossing result for Brownian motion. The boundary studied is a piecewise linear function consisting of two lines. The expression obtained for the boundary crossing probability is of a simple directly computable form.

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.

Long-Time Solutions to Heat-Conduction Transients with Time-Dependent Inputs

1980 ◽  
Vol 102 (1) ◽  
pp. 115-120 ◽  
Author(s):  
H. T. Ceylan ◽  
G. E. Myers
Keyword(s):  
Heat Conduction ◽  
Lower Cost ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Functions ◽  
Time Dependent ◽  
Time Interval ◽  
Piecewise Linear Functions ◽  
One Dimensional ◽  
Long Time

An economical method for obtaining long-time solutions to one, two, or three-dimensional heat-conduction transients with time-dependent forcing functions is presented. The conduction problem is spatially discretized by finite differences or by finite elements to obtain a system of first-order ordinary differential equations. The time-dependent input functions are each approximated by continuous, piecewise-linear functions each having the same uniform time interval. A set of response coefficients is generated by which a long-time solution can be carried out with a considerably lower cost than for conventional methods. A one-dimensional illustrative example is included.

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