# On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Formulation of the Problem

#### 2.2. The Boundary-Value Problems

## 3. Solutions to the Boundary-Value Problem

#### 3.1. The General Solution of the Ordinary Differential Equation

#### 3.2. The Solution to the Boundary-Value Problem

**(i)**Let us first consider the domain $a\vee b\le s-q\le a+b$. In this case, solving the system of equations in (26) and (27), we conclude that the candidate value function admits the representation

**(ii)**Let us now consider the domain $a\le s-q<b$. In this case, it follows from the equations in (26) and (29) that the candidate value function admits the representation

**(iii)**Let us now consider the domain $b\le s-q<a$. In this case, it follows from the equations in (27) and (28) that the candidate value function admits the representation

**(iv)**Let us now consider the domain $0\le s-q<a\wedge b$. In this case, it follows that the candidate value function admits the representation

#### 3.3. Some Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 4. The Result and Proof

**Theorem**

**1.**

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A computer drawing of the state space of the process $(X,S,Q)$, for some $q\in \mathbb{R}$ fixed and $a<b$.

**Figure 2.**A computer drawing of the state space of the process $(X,S,Q)$, for some $q\in \mathbb{R}$ fixed and $b\le a$.

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**MDPI and ACS Style**

Gapeev, P.V.; Rodosthenous, N.; Chinthalapati, V.L.R.
On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes. *Risks* **2019**, *7*, 87.
https://doi.org/10.3390/risks7030087

**AMA Style**

Gapeev PV, Rodosthenous N, Chinthalapati VLR.
On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes. *Risks*. 2019; 7(3):87.
https://doi.org/10.3390/risks7030087

**Chicago/Turabian Style**

Gapeev, Pavel V., Neofytos Rodosthenous, and V. L. Raju Chinthalapati.
2019. "On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes" *Risks* 7, no. 3: 87.
https://doi.org/10.3390/risks7030087