The first-passage density of the Brownian motion process to a curved boundary

1992 ◽  
Vol 29 (02) ◽  
pp. 291-304 ◽  
Author(s):  
J. Durbin ◽  
D. Williams
Keyword(s):  
Brownian Motion ◽  
Brownian Bridge ◽  
Sample Path ◽  
Practical Method ◽  
Time Interval ◽  
Brownian Motion Process ◽  
First Passage ◽  
Curved Boundary ◽  
Multiple Integrals ◽  
High Degree

An expression for the first-passage density of Brownian motion to a curved boundary is expanded as a series of multiple integrals. Bounds are given for the error due to truncation of the series when the boundary is wholly concave or wholly convex. Extensions to the Brownian bridge and to continuous Gauss–Markov processes are given. The series provides a practical method for calculating the probability that a sample path crosses the boundary in a specified time-interval to a high degree of accuracy. A numerical example is given.

The first-passage density of the Brownian motion process to a curved boundary

1992 ◽  
Vol 29 (2) ◽  
pp. 291-304 ◽  
Author(s):  
J. Durbin ◽  
D. Williams
Keyword(s):  
Brownian Motion ◽  
Brownian Bridge ◽  
Sample Path ◽  
Practical Method ◽  
Time Interval ◽  
Brownian Motion Process ◽  
First Passage ◽  
Curved Boundary ◽  
Multiple Integrals ◽  
High Degree

An expression for the first-passage density of Brownian motion to a curved boundary is expanded as a series of multiple integrals. Bounds are given for the error due to truncation of the series when the boundary is wholly concave or wholly convex. Extensions to the Brownian bridge and to continuous Gauss–Markov processes are given. The series provides a practical method for calculating the probability that a sample path crosses the boundary in a specified time-interval to a high degree of accuracy. A numerical example is given.

Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test

1971 ◽  
Vol 8 (03) ◽  
pp. 431-453 ◽  
Author(s):  
J. Durbin
Keyword(s):  
Brownian Motion ◽  
Integral Equations ◽  
Sample Path ◽  
Distribution Functions ◽  
Finite Sample ◽  
Brownian Motion Process ◽  
Sample Distribution ◽  
Motion Process ◽  
Kolmogorov Smirnov ◽  
Smirnov Test

Let w(t), 0 ≦ t ≦ ∞, be a Brownian motion process, i.e., a zero-mean separable normal process with Pr{w(0) = 0} = 1, E{w(t 1)w(t 2)}= min (t 1, t 2), and let a, b denote the boundaries defined by y = a(t), y = b(t), where b(0) < 0 < a(0) and b(t) < a(t), 0 ≦ t ≦ T ≦ ∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis and optional stopping is that of calculating the probability that a sample path of w(t) crosses a or b before t = T. This paper shows how this probability may be computed for sufficiently smooth boundaries by numerical solution of integral equations for the first-passage distribution functions. The technique used is to approximate the integral equations by linear recursions whose coefficients are estimated by linearising the boundaries within subintervals. The results are extended to cover the tied-down process subject to the condition w(1) = 0. Some related results for the Poisson process and the sample distribution function are given. The procedures suggested are exemplified numerically, first by computing the probability that the tied-down Brownian motion process crosses a particular curved boundary for which the true probability is known, and secondly by computing the finite-sample and asymptotic powers of the Kolmogorov-Smirnov test against a shift in mean of the exponential distribution.

A reconciliation of two different expressions for the first-passage density of brownian motion to a curved boundary

1988 ◽  
Vol 25 (04) ◽  
pp. 829-832 ◽  
Author(s):  
J. Durbin
Keyword(s):  
Brownian Motion ◽  
Markov Processes ◽  
First Passage ◽  
Curved Boundary

An expression for the first-passage density of Brownian motion to a curved boundary due to Daniels and Lerche is shown to give the same result as a different form due to the author. The equivalence is extended to continuous Gaussian Markov processes.

A reconciliation of two different expressions for the first-passage density of brownian motion to a curved boundary

1988 ◽  
Vol 25 (4) ◽  
pp. 829-832 ◽  
Author(s):  
J. Durbin
Keyword(s):  
Brownian Motion ◽  
Markov Processes ◽  
First Passage ◽  
Curved Boundary

An expression for the first-passage density of Brownian motion to a curved boundary due to Daniels and Lerche is shown to give the same result as a different form due to the author. The equivalence is extended to continuous Gaussian Markov processes.

Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test

1971 ◽  
Vol 8 (3) ◽  
pp. 431-453 ◽  
Author(s):  
J. Durbin
Keyword(s):  
Brownian Motion ◽  
Integral Equations ◽  
Sample Path ◽  
Distribution Functions ◽  
Finite Sample ◽  
Brownian Motion Process ◽  
Sample Distribution ◽  
Motion Process ◽  
Kolmogorov Smirnov ◽  
Smirnov Test

Let w(t), 0 ≦ t ≦ ∞, be a Brownian motion process, i.e., a zero-mean separable normal process with Pr{w(0) = 0} = 1, E{w(t1)w(t2)}= min (t1, t2), and let a, b denote the boundaries defined by y = a(t), y = b(t), where b(0) < 0 < a(0) and b(t) < a(t), 0 ≦ t ≦ T ≦ ∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis and optional stopping is that of calculating the probability that a sample path of w(t) crosses a or b before t = T. This paper shows how this probability may be computed for sufficiently smooth boundaries by numerical solution of integral equations for the first-passage distribution functions. The technique used is to approximate the integral equations by linear recursions whose coefficients are estimated by linearising the boundaries within subintervals. The results are extended to cover the tied-down process subject to the condition w(1) = 0. Some related results for the Poisson process and the sample distribution function are given. The procedures suggested are exemplified numerically, first by computing the probability that the tied-down Brownian motion process crosses a particular curved boundary for which the true probability is known, and secondly by computing the finite-sample and asymptotic powers of the Kolmogorov-Smirnov test against a shift in mean of the exponential distribution.

scholarly journals On the first-passage time of integrated Brownian motion

2005 ◽  
Vol 2005 (3) ◽  
pp. 237-246
Author(s):  
Christian H. Hesse
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Stopping Time ◽  
Passage Time ◽  
Accurate Approximation ◽  
Brownian Motion Process ◽  
First Passage ◽  
Conditional Mean ◽  
Conditional Moments ◽  
Motion Process

Let (Bt;t≥0) be a Brownian motion process starting from B0=ν and define Xν(t)=∫0tBsds. For a≥0, set τa,ν:=inf{t:Xν(t)=a} (with inf φ=∞). We study the conditional moments of τa,ν given τa,ν<∞. Using martingale methods, stopping-time arguments, as well as the method of dominant balance, we obtain, in particular, an asymptotic expansion for the conditional mean E(τa,ν|τa,ν<∞) as ν→∞. Through a series of simulations, it is shown that a truncation of this expansion after the first few terms provides an accurate approximation to the unknown true conditional mean even for small ν.

scholarly journals Oscillating Brownian motion

1978 ◽  
Vol 15 (02) ◽  
pp. 300-310 ◽  
Author(s):  
Julian Keilson ◽  
Jon A. Wellner
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
Brownian Motion Process ◽  
First Passage ◽  
Special Cases ◽  
Motion Process ◽  
Transition Densities ◽  
Oscillating Random Walk

An ‘oscillating' version of Brownian motion is defined and studied. ‘Ordinary' Brownian motion and ‘reflecting' Brownian motion are shown to arise as special cases. Transition densities, first-passage time distributions, and occupation time distributions for the process are obtained explicitly. Convergence of a simple oscillating random walk to an oscillating Brownian motion process is established by using results of Stone (1963).

scholarly journals Oscillating Brownian motion

1978 ◽  
Vol 15 (2) ◽  
pp. 300-310 ◽  
Author(s):  
Julian Keilson ◽  
Jon A. Wellner
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
Brownian Motion Process ◽  
First Passage ◽  
Special Cases ◽  
Motion Process ◽  
Transition Densities ◽  
Oscillating Random Walk

An ‘oscillating' version of Brownian motion is defined and studied. ‘Ordinary' Brownian motion and ‘reflecting' Brownian motion are shown to arise as special cases. Transition densities, first-passage time distributions, and occupation time distributions for the process are obtained explicitly. Convergence of a simple oscillating random walk to an oscillating Brownian motion process is established by using results of Stone (1963).

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