**1. Introduction**

This paper comprises three essays on stopping.

In Section 2, we compute the Laplace transform of the first hitting time of a fixed upper barrier for a reflected Brownian motion with drift. This expands on and corrects a result by Perry et al. (2004).

In Section 3, we show, by using an intrinsic delay differential equation, that for a diffusion process, the maximum before a fixed drawdown threshold is generically exponentially distributed, only if the diffusion characteristic *μ*/*σ*<sup>2</sup> is constant. This complements the sufficient condition formulated by Lehoczky (1977). By solving discrete delay differential equations, we further construct diffusions, where the exponential law only holds for specific drawdown sizes.

Section 4 uses Lehoczky (1977)'s argumen<sup>t</sup> to show that the maximum before a fixed drawdown threshold is exponentially distributed for any spectrally negative Lévy process, the parameter being the right-sided logarithmic derivative of the scale function. This yields an alternative proof to the original one in Mijatovi´c and Pistorius (2012) and is also similar to the one in Landriault et al. (2017).

### **2. The First Hitting Time for a Reflected Brownian Motion With Drift**

Let *X* be a reflected Brownian motion on [0, <sup>∞</sup>), with drift *μ* and volatility *σ*. Then *X* can be written as

$$X\_t = \mathbf{x} + \mu t + \sigma \mathcal{W}\_t + L\_{t\prime}$$

where *W* = ( *Wt*)*t*≥0 is a standard Brownian motion, and *L* = (*Lt*)*t*≥0 is an inon-decreasing process, such that the induced random measure *dL* is supported on {*X* = <sup>0</sup>}. Itô's formula implies that for any *f* ∈ *C*<sup>2</sup> *b* ([0, ∞)) satisfying *f* (0+) = 0, the process

$$f(\mathbf{X}\_t) - f(\mathbf{x}) - \int\_0^t \mathcal{A}\_{\mathcal{Y}} f(\mathbf{X}\_s) ds$$

is a martingale, where A*y* is the differential operator, defined by A*y f*(*y*) = *σ*22 *f* (*y*) + *μ f* (*y*).<sup>1</sup> For *δ* ≥ <sup>−</sup>*x*, we define the first hitting time:

$$\pi\_{\delta} := \inf \{ t \ge 0 \mid X\_t = \delta + x \}.$$

Since, before reaching the boundary 0, the process cannot be distinguished from a Brownian motion with drift, we may confine ourselves to computing *τδ* for barriers *δ* + *x*, where *δ* > 0. Our aim is to compute the Laplace transform:

$$\Psi(\theta;\delta,\mathfrak{x}) := \mathbb{E}[e^{-\theta\tau\_{\delta}} \mid X\_0 = \mathfrak{x}], \quad \theta \ge 0.$$

**Theorem 1.** *For δ* ≥ 0*, the Laplace transform of the first hitting time of a reflected Brownian motion with drift μ and volatility σ is given by*

$$\Psi(\theta; \mathbf{x}, \delta) := \varepsilon^{\frac{\delta \mathbf{x}}{\sigma^2}} \frac{\sqrt{\mu^2 + 2\theta \sigma^2} \cosh\left(\frac{\mathbf{x}\sqrt{\mu^2 + 2\theta \sigma^2}}{\sigma^2}\right) + \mu \sinh\left(\frac{\mathbf{x}\sqrt{\mu^2 + 2\theta \sigma^2}}{\sigma^2}\right)}{\sqrt{\mu^2 + 2\theta \sigma^2} \cosh\left(\frac{(\mathbf{x} + \delta)\sqrt{\mu^2 + 2\theta \sigma^2}}{\sigma^2}\right) + \mu \sinh\left(\frac{(\mathbf{x} + \delta)\sqrt{\mu^2 + 2\theta \sigma^2}}{\sigma^2}\right)}. \tag{1}$$

**Proof.** Pick Φ ∈ *C*<sup>∞</sup>*c* (R), such that Φ(*ξ*) = 1 for |*ξ*| ≤ *x* + *δ*. Furthermore, let *κ* ∈ R; then for any *θ* ≥ 0 and *t* ≥ 0, the function

$$F(t, \mathbf{x}) := e^{-\theta t} \Phi(\mathbf{x}) \left( e^{-\kappa \mathbf{x}} + \kappa \mathbf{x} \right),$$

satisfies *f* := *<sup>F</sup>*(*<sup>t</sup>*, ·) ∈ *C*2*b* and *f* (0) = 0. According to the introductory notes of this section, the process *<sup>F</sup>*(*<sup>t</sup>*, *Xt*) − *t*0 *∂s<sup>F</sup>*(*<sup>s</sup>*, *Xs*)*ds* − *t*0 A*y<sup>F</sup>*(*<sup>s</sup>*, *Xs*)*ds* is a uniformly bounded martingale; therefore, the stopped process

$$F(t, X\_{t \wedge \tau\_\delta}) - (e^{-\kappa x} + \kappa x) - \int\_0^{t \wedge \tau\_\delta} \partial\_t F(s, X\_s) ds - \int\_0^{t \wedge \tau\_\delta} \mathcal{A}\_y F(s, X\_s) ds$$

is also a true martingale, which starts at zero, P*<sup>x</sup>*- almost surely. Using the fact that <sup>Φ</sup>(*Xt*<sup>∧</sup>*τδ* ) = 1, we find that the stopped process satisfies for any *t* ≥ 0,

$$\begin{split} &e^{-\theta(t\wedge\tau\_{\delta})} \left( e^{-\kappa X\_{l\wedge\tau\_{\delta}}} + \kappa X\_{l\wedge\tau\_{\delta}} \right) - \left( e^{-\kappa x} + \kappa x \right) + \theta \int\_{0}^{t\wedge\tau\_{\delta}} e^{-\kappa X\_{s} - \theta s} ds + \theta \kappa \int\_{0}^{t\wedge\tau\_{\delta}} e^{-\theta s} X\_{s} ds \\ & - \mu \int\_{0}^{t\wedge\tau\_{\delta}} \left( \kappa e^{-\theta s} - \kappa e^{-\kappa X\_{s} - \theta s} \right) ds - \frac{\sigma^{2}\kappa^{2}}{2} \int\_{0}^{t\wedge\tau\_{\delta}} e^{-\kappa X\_{s} - \theta s} ds \\ & = e^{-\theta(t\wedge\tau\_{\delta})} \left( e^{-\kappa X\_{t\wedge\tau\_{\delta}}} + \kappa X\_{t\wedge\tau\_{\delta}} \right) - \left( e^{-\kappa x} + \kappa x \right) + \theta \kappa \int\_{0}^{t\wedge\tau\_{\delta}} e^{-\theta s} X\_{s} ds \\ & - \frac{\mu\kappa}{\theta} \left( 1 - e^{-\theta(t\wedge\tau\_{\delta})} \right) - \left( \frac{\sigma^{2}\kappa^{2}}{2} - \kappa\mu - \theta \right) \int\_{0}^{t\wedge\tau\_{\delta}} e^{-\kappa X\_{s} - \theta s} ds. \end{split}$$

Letting *t* → <sup>∞</sup>, we thus ge<sup>t</sup> by optional sampling,

$$\begin{split} & \left( e^{-\kappa(\mathbf{x}+\delta)} + \kappa(\mathbf{x}+\delta) \right) \mathbb{E} \left[ e^{-\theta \tau\_{\delta}} \mid X\_{0} = \mathbf{x} \right] - \left( e^{-\mathbf{x}\cdot\mathbf{x}} + \kappa\mathbf{x} \right) + \theta\mathbf{x} \mathbb{E} \left[ \int\_{0}^{\tau\_{\delta}} e^{-\theta s} X\_{s} ds \mid X\_{0} = \mathbf{x} \right] \\ & - \frac{\mu\mathsf{x}}{\theta} \left( 1 - \mathbb{E} \left[ e^{-\theta \tau\_{\delta}} \mid X\_{0} = \mathbf{x} \right] \right) - \left( \frac{\sigma^{2}\kappa^{2}}{2} - \kappa\mu - \theta \right) \mathbb{E} \left[ \int\_{0}^{\tau\_{\delta}} e^{-\kappa X\_{s} - \theta s} ds \mid X\_{0} = \mathbf{x} \right] = \mathbf{0}. \end{split}$$

<sup>1</sup> In the language of linear diffusions Borodin and Salminen (2012), *X* has infinitesimal generator A*y* acting on the domain <sup>D</sup>(A*y*) = { *f* ∈ *C*2*b* ([0, ∞)) | *f* (0+) = <sup>0</sup>}.

For the two choices *κ* ∈ {*<sup>κ</sup>*−, *<sup>κ</sup>*+}, where

$$\kappa\_{\pm} := \frac{\mu \pm \sqrt{\mu^2 + 2\theta \sigma^2}}{\sigma^2},$$

we thus obtain two equations, for two unknown moments,

$$\begin{split} & \left( \varepsilon^{-\mathbf{x}\pm(\mathbf{x}+\boldsymbol{\delta})} + \kappa\_{\pm}(\mathbf{x}+\boldsymbol{\delta}) + \frac{l\mathbb{K}\pm}{\theta} \right) \mathbb{E} [\varepsilon^{-\theta\tau\_{\mathcal{I}}} \mid \mathcal{X}\_{0} = \mathbf{x}] + \theta \kappa\_{\pm} \mathbb{E} \left[ \int\_{0}^{\mathsf{T}} \varepsilon^{-\theta\mathbf{s}} \mathcal{X}\_{\mathsf{s}} d\mathbf{s} \mid \mathcal{X}\_{0} = \mathbf{x} \right] \\ & = (\varepsilon^{-\mathsf{x}\_{\pm}\cdot\mathsf{x}} + \kappa \mathrm{x}) + \frac{l\mathbb{K}\pm}{\theta} .\end{split}$$

Solving this linear system for the involved moments yields the Laplace transform of *τδ*, Equation (1).

**Remark 1.** *This result can also be obtained from a more general result for spectrally negative Lévy processes, reflected at an upper barrier (Avram et al. 2017, Proposition 4.B and Section 10.1). In fact, the distribution of τδ is equal in distribution to the first hitting time* 0 *of the Brownian motion Xt* = *δ* + *σBt* − *μt, starting at δ* ≥ 0*, reflected at x* + *δ* > 0*. Its Laplace transform is therefore given by*

$$
\psi\_{
\theta}^{[\mathfrak{x}+\delta]}(\delta) = e^{\frac{\mu\delta}{\sigma^2}} \frac{H(\mathfrak{x})}{H(\mathfrak{x}+\delta)'},
$$

*where*

$$H(\xi) = \sqrt{2\theta\sigma^2 + \mu^2} \cosh\left(\frac{\frac{\chi}{\xi}\sqrt{2\theta\sigma^2 + \mu^2}}{\sigma^2}\right) + \mu \sinh\left(\frac{\frac{\chi}{\xi}\sqrt{2\theta\sigma^2 + \mu^2}}{\sigma^2}\right) \sigma$$

*(see Avram et al. (2017), Section 10.1).*
