On the first passage problem for correlated Brownian motion

Metzler, Adam

doi:10.1016/j.spl.2009.11.001

Cited by 54 publications

(70 citation statements)

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“…Then the distribution function P (r, t|r 0 ) is known exactly, (see, e.g., Refs. [26,46]) and is represented by an infinite series whose leading term for t → ∞ is given, up to a normalization constant, by…”

Section: Uniformity Distribution P (ω) In a Pie-wedge Domainmentioning

confidence: 99%

First passages in bounded domains: When is the mean first passage time meaningful?

et al. 2012

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We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P (ω) distribution of the random variable ω = τ1/(τ1 +τ2), which is a measure for how similar the first passage times τ1 and τ2 are of two independent realisations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P (ω) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P (ω), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in twodimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.

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Section: Uniformity Distribution P (ω) In a Pie-wedge Domainmentioning

confidence: 99%

First passages in bounded domains: When is the mean first passage time meaningful?

et al. 2012

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“…In the multidimensional case this would correspond to the hyperplane boundary changing from a ′ x = b+ct when t ≤ t * to a ′ x = b * + c * t when t ≥ t * . Iyengar (1985) and Metzler (2010) examine the first time planar Brownian motion hits either a horizontal line or a vertical line. Using our approach, one could generalize this problem to analyze the first time planar Brownian motion hits one of two perpendicular lines.…”

Section: Resultsmentioning

confidence: 99%

Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries

Atkinson

Singham

2015

Statistics & Probability Letters

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“…The multivariate boundary crossing problem has also been studied. Metzler (2010) considers the situation where a vector of prices are observed that evolve according to the multivariate diffusion…”

Section: Static Price Limits With Brownian Price Processesmentioning

confidence: 99%

Single stock circuit breakers on the London Stock Exchange: do they improve subsequent market quality?

Brugler

Linton

2014

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractThis paper uses proprietary data to evaluate the efficacy of single-stock circuit breakers on the London Stock Exchange during July and August 2011. We exploit exogenous variation in the length of the uncrossing periods that follow a trading suspension to estimate the effect of auction length on market quality, measured by volume of trades, frequency of trading and the change in realized variance of returns. We also estimate the effect of a trading suspension in one FTSE-100 stock on the volume of trades, trading frequency and the change in realized variance of returns for other FTSE-100 stocks. We find that auction length has a significant detrimental effect on market quality for the suspended security when returns are negative but no discernible effect when returns are positive. We also find that trading suspensions help to ameliorate the spread of market microstructure noise and price inefficiency across securities during falling markets but the reverse is true during rising markets. Although trading suspensions may not improve the trading process within a particular security, they do play an important role preventing the spread of poor market quality across securities in falling markets and therefore can be effective tools for promoting market-wide stability.

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On the first passage problem for correlated Brownian motion

Cited by 54 publications

References 4 publications

First passages in bounded domains: When is the mean first passage time meaningful?

First passages in bounded domains: When is the mean first passage time meaningful?

Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries

Single stock circuit breakers on the London Stock Exchange: do they improve subsequent market quality?

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