Three essays on stopping

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic μ...

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Bibliographic Details
Published inRisks (Basel) Vol. 7; no. 4; pp. 1 - 10
Main Author Mayerhofer, Eberhard
Format Journal Article
LanguageEnglish
Published Basel MDPI 01.12.2019
MDPI AG
Subjects
Brownian motion
Diffusion
drawdown
Laplace transforms
linear diffusions
reflected Brownian motion
spectrally negative Lévy processes
Theorems
Volatility
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Summary:First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic μ/σ2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum before a fixed drawdown is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatovi´c and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017).
ISSN:2227-9091
2227-9091
DOI:10.3390/risks7040105