On Exact Simulation Algorithms for Some Distributions Related to Brownian Motion and Brownian Meanders

Devroye, Luc

doi:10.1007/978-3-7908-2598-5_1

Cited by 15 publications

(32 citation statements)

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“…The study of Brownian extrema date back to the founders of the field (Lèvy 1948). Our brief discussion follows Devroye (2010) with additional results taken from Shepp (1979); Durrett et al (1977a, b), Imhof (1985), Borodin and Salminen (2002). See also (Bertoin and Pitman 1994;Karatzas and Shreve 1998;Pitman and Yor 1996;Yor 1997).…”

Section: Distributions Of Brownian Extremamentioning

confidence: 97%

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The value of the high, low and close in the estimation of Brownian motion

Riedel

2020

Stat Inference Stoch Process

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The conditional density of Brownian motion is considered given the max, $$B(t|\max )$$ B ( t | max ) , as well as those with additional information: $$B(t|close, \max )$$ B ( t | c l o s e , max ) , $$B(t|close, \max , \min )$$ B ( t | c l o s e , max , min ) where the close is the final value: $$B(t=1)=c$$ B ( t = 1 ) = c and $$t \in [0,1]$$ t ∈ [ 0 , 1 ] . The conditional expectation and conditional variance of Brownian motion are evaluated subject to one or more of the statistics: the close (final value), the high (maximum), the low (minimum). Computational results displaying both the expectation and variance in time are presented and compared with the theoretical values. We tabulate the time averaged variance of Brownian motion conditional on knowing various extremal properties of the motion. The final table shows that knowing the high is more useful than knowing the final value among other results. Knowing the open, high, low and close reduces the time averaged variance to $$42\%$$ 42 % of the value of knowing only the open and close (Brownian bridge).

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Section: Distributions Of Brownian Extremamentioning

confidence: 97%

“…Let B(t) be a Brownian motion on [0,1] and B c (t) be the Brownian motion restricted to B c (t = 1) = c. We allow an arbitrary const variance, E[B(s) 2 ] = σ 2 . Our notation tracks the excellent compendium of results by Devroye (2010). Many of the results summarized in Sect.…”

Section: Introductionmentioning

confidence: 93%

The value of the high, low and close in the estimation of Brownian motion

Riedel

2020

Stat Inference Stoch Process

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“…In this appendix, we describe the algorithms that we have used here to simulate various constrained Brownian motions. We refer the interested reader to [52] for an extended discussion of these algorithms.…”

Section: E Numerical Simulations Of Constrained Brownian Motionmentioning

confidence: 99%

On Certain Functionals of the Maximum of Brownian Motion and Their Applications

Perret¹,

Comtet²,

Majumdar³

et al. 2015

J Stat Phys

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We consider a Brownian motion (BM) x(τ ) and its maximal value x max = max 0≤τ ≤t x(τ ) on a fixed time interval [0, t]. We study functionals of the maximum of thecan be any arbitrary function and develop various analytical tools to compute their statistical properties. These tools rely in particular on (i) a "counting paths" method and (ii) a path-integral approach. In particular, we focus on the case where V (x) = δ(x − r ), with r a real parameter, which is relevant to study the density of near-extreme values of the BM (the so called density of states), ρ(r, t), which is the local time of the BM spent at given distance r from the maximum. We also provide a thorough analysis of the family of functionals T α (t) = t 0 (x max − x(τ )) α dτ , corresponding to V (x) = x α , with α real. As α is varied, T α (t) interpolates between different interesting observables. For instance, for α = 1, T α=1 (t) is a random variable of the "area", or "Airy", type while for α = −1/2 it corresponds to the maximum time spent by a ballistic particle through a Brownian random potential. On the other hand, for α = −1, it corresponds to the cost of the optimal algorithm to find the maximum of a discrete random walk, proposed by Odlyzko. We revisit here, using tools of theoretical physics, the statistical properties of this algorithm which had been studied before using probabilistic methods. Finally, we extend our methods to constrained BM, including in particular the Brownian bridge, i.e., the Brownian motion starting and ending at the origin.

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“…imum over [κ i , κ i+1 ] conditioned on τ a as the maximum of a Brownian meander; see maxmeander algorithm in Devroye (2010).…”

Section: Remark 52)mentioning

confidence: 99%

“…, W κ b , W ∆ ), sampling sup 0≤t≤∆ W t and the location of maximum time is similar to Step 6 of Subroutine 1. It is sufficient to sample (µ i , t µ i ) jointly, where µ i = sup κ i ≤t≤κ i+1 W t , and t µ i is the location of the maximum over [κ i , κ i+1 ] conditional on τ a as the maximum of a Brownian meander; see the maxmeander algorithm in Devroye (2010).…”

Section: Exact Sampling Of Time-dependent Drift Brownian Bridgementioning

confidence: 99%

Exact Simulation of Non-Stationary Reflected Brownian Motion

Mousavi

2013

SSRN Journal

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This paper develops the first method for the exact simulation of reflected Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The running time of generating exact samples of non-stationary RBM at any time t is uniformly bounded by O(1/γ 2 ) whereγ is the average drift of the process. The method can be used as a guide for planning simulations of complex queueing systems with non-stationary arrival rates and/or service time.

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On Exact Simulation Algorithms for Some Distributions Related to Brownian Motion and Brownian Meanders

Cited by 15 publications

References 50 publications

The value of the high, low and close in the estimation of Brownian motion

The value of the high, low and close in the estimation of Brownian motion

On Certain Functionals of the Maximum of Brownian Motion and Their Applications

Exact Simulation of Non-Stationary Reflected Brownian Motion

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