The first exit time of a Brownian motion from an unbounded convex domain

Li, Wenbo V.

doi:10.1214/aop/1048516546

Cited by 50 publications

(46 citation statements)

References 39 publications

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“…In fact, the computation of the probability that a standard Brownian motion remains between two given boundaries is a complex question that has occupied many researchers over many years (see, e.g., Anderson, 1960;Robbins and Siegmund, 1970;Durbin, 1971;Lerche, 1986;Durbin, 1992;Daniels, 1996;Borodin and Salminen, 2002;Li, 2003). In this literature, explicit representations of two-sided boundary crossing probabilities are extremely rare and mostly address linear cases (Anderson, 1960;Hall, 1997), though explicit computations are available for a few nonlinear boundaries (Robbins and Siegmund, 1970;Daniels, 1996;Novikov et al, 1999).…”

Section: P(−u(t) ≤ W (T) ≤ U(t) T ∈ [0 1]) = 1 − αmentioning

confidence: 99%

Confidence bands for Brownian motion and applications to Monte Carlo simulation

2007

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Minimal area regions are constructed for Brownian paths and perturbed Brownian paths. While the theoretical optimal region cannot be obtained in closed form, we provide practical confidence regions based on numerical approximations and local time arguments. These regions are used to provide informal convergence assessments for both Monte Carlo and Markov Chain Monte Carlo experiments, via the Brownian asymptotic approximation of cumulative sums.

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Section: P(−u(t) ≤ W (T) ≤ U(t) T ∈ [0 1]) = 1 − αmentioning

confidence: 99%

Confidence bands for Brownian motion and applications to Monte Carlo simulation

2007

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“…Our aim in this article is to do the same for the first exit time of IBM over bounded domains in R n , and for the first exit time of BTBM over several domains in R n . See Bañuelos and DeBlassie [3], Li [21], Lifshits and Shi [22] and Nane [23] for a survey of results obtained for Brownian motion and iterated Brownian motion in these domains.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…In [21], using Gaussian techniques, Li studied lifetime asymptotics of Brownian motion in domains of the following form…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Lifetime asymptotics of iterated Brownian motion in $\mathbb{R}^{n}$

Nane

2007

ESAIM: PS

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Let τ D (Z) be the first exit time of iterated Brownian motion from a domain D ⊂ R n started at z ∈ D and let P z [τ D (Z) > t] be its distribution. In this paper we establish the exact asymptotics of P z [τ D (Z) > t] over bounded domains as an improvement of the results in [12,24], for z ∈ DwhereHere λ D is the first eigenvalue of the Dirichlet Laplacian 1 2 ∆ in D, and ψ is the eigenfunction corresponding to λ D .We also study lifetime asymptotics of Brownian-time Brownian motion (BTBM), Z 1 t = z+X(|Y (t)|), where X t and Y t are independent one-dimensional Brownian motions.Mathematics Subject Classification (2000): 60J65, 60K99.

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“…To our best knowledge, in most cases, the distribution is a solution of ODE or PDE, likely did by [13][14][15], and no analytical solution is available. The motivation of this paper is practical validity for applications in risk analysis and financial default probability, and our results principally provide a way to compute the distribution of first exit time of BM from a double linear time-dependent barrier by expressing the aforementioned distribution by infinite series.…”

Section: Discussionmentioning

confidence: 99%

“…As an alternative, many works fall back on estimation or approximating the distribution or density of first exit time of Brownian motion (cf [11][12][13][14][15]). In this paper, by applying Girsanov theorem iteratively, we obtained the density of first exit time for Brownian motion from a double linear time-dependent barrier in terms of infinity series.…”

Section: Introductionmentioning

confidence: 99%

On the Distribution of First Exit Time for Brownian Motion with Double Linear Time-Dependent Barriers

Zhu

2013

ISRN Applied Mathematics

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This paper focuses on the first exit time for a Brownian motion with a double linear time-dependent barrier specified by = + , = , ( > 0, < 0, > 0). We are concerned in this paper with the distribution of the Brownian motion hitting the upper barrier before hitting the lower linear barrier. The main method we applied here is the Girsanov transform formula. As a result, we expressed the density of such exit time in terms of a finite series. This result principally provides us an analytical expression for the distribution of the aforementioned exit time and an easy way to compute the distribution of first exit time numerically.

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The first exit time of a Brownian motion from an unbounded convex domain

Cited by 50 publications

References 39 publications

Confidence bands for Brownian motion and applications to Monte Carlo simulation

Confidence bands for Brownian motion and applications to Monte Carlo simulation

Lifetime asymptotics of iterated Brownian motion in $\mathbb{R}^{n}$

On the Distribution of First Exit Time for Brownian Motion with Double Linear Time-Dependent Barriers

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