# Three Essays on Stopping

## Abstract

**:**

## 1. Introduction

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

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

#### Remarks On Perry et al. (2004)

## 3. Diffusions with Exponentially Distributed Gains Before Fixed Drawdowns

**Proposition**

**1.**

**Theorem**

**2.**

- 1.
- $\mu \left(x\right)/{\sigma}^{2}\left(x\right)$ is a constant on $[-a,\infty )$.
- 2.
- For each $\delta >0$, ${M}^{\delta}$ is exponentially distributed.

**Proof of**

**the Theorem.**

- ($a=-\infty $): Brownian motion with drift $\sigma {B}_{t}+\mu t$.
- $(a<\infty $): Reflected Brownian motion with drift, reflected at $-a$.
- Similar examples as in 1 and 2 can be constructed, where $\mu \left(x\right)/{\sigma}^{2}\left(x\right)$ is constant. These include reflected diffusions.

## 4. Lehoczky’s Proof for Spectrally Negative Lévy Martingales

**Theorem**

**3.**

**Theorem**

**4.**

**Proof**

**of Theorem 4.**

**Remark**

**2.**

#### Examples

**Example**

**1**

**.**Assume we have a compound Poisson process with negative exponentially distributed jumps,

**Example**

**2**

**.**A Brownian motion with drift $\mu >0$ and volatility σ,

**Example**

**3**

**.**This is a Lévy process without diffusion component, defined by its Lévy measure

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | In the language of linear diffusions Borodin and Salminen (2012), X has infinitesimal generator ${\mathcal{A}}_{y}$ acting on the domain $\mathcal{D}\left({\mathcal{A}}_{y}\right)=\{f\in {C}_{b}^{2}\left([0,\infty )\right)\mid {f}^{\prime}(0+)=0\}$. |

2 | |

3 | This random gain is called “overshoot” in Golub et al. (2016). In this section, we refrain from using this terminology due to its established meaning in the field of Lévy processes—it is the discrepancy between a certain threshold, and a jump processes’ value, passing beyond that threshold. |

4 | It goes without saying that the first time this maximum is attained is not a stopping time; otherwise, one could devise arbitrage strategies that short-sell the asset at the maximum. |

5 |

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Mayerhofer, E. Three Essays on Stopping. *Risks* **2019**, *7*, 105.
https://doi.org/10.3390/risks7040105

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Mayerhofer E. Three Essays on Stopping. *Risks*. 2019; 7(4):105.
https://doi.org/10.3390/risks7040105

**Chicago/Turabian Style**

Mayerhofer, Eberhard. 2019. "Three Essays on Stopping" *Risks* 7, no. 4: 105.
https://doi.org/10.3390/risks7040105