**Eberhard Mayerhofer**

Department of Mathematics and Statistics, University of Limerick, Limerick V94TP9X, Ireland; eberhard.mayerhofer@ul.ie

Received: 27 September 2019; Accepted: 16 October 2019; Published: 18 October 2019

**Abstract:** 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*<sup>2</sup> 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).

**Keywords:** reflected Brownian motion; linear diffusions; spectrally negative Lévy processes; drawdown

**MSC (2010):** 60J65; 60J75
