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Open AccessArticle

Three Essays on Stopping

Department of Mathematics and Statistics, University of Limerick, Limerick V94TP9X, Ireland
Risks 2019, 7(4), 105; https://doi.org/10.3390/risks7040105
Received: 27 September 2019 / Revised: 16 October 2019 / Accepted: 16 October 2019 / Published: 18 October 2019
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 / σ 2 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 Mijatović and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017). View Full-Text
Keywords: reflected Brownian motion; linear diffusions; spectrally negative Lévy processes; drawdown reflected Brownian motion; linear diffusions; spectrally negative Lévy processes; drawdown
MDPI and ACS Style

Mayerhofer, E. Three Essays on Stopping. Risks 2019, 7, 105.

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