# De Finetti’s Control Problem with Parisian Ruin for Spectrally Negative Lévy Processes

## Abstract

**:**

## 1. Introduction and Main Result

#### 1.1. Problem Formulation

**Remark**

**1.**

#### 1.2. Main Result and Organization of the Paper

**Theorem**

**1.**

## 2. More on the Value Function

**Remark**

**2.**

## 3. Horizontal Barrier Strategies

#### 3.1. Second Family of Scale Functions

#### 3.2. Value Function of a Barrier Strategy

**Proposition**

**1.**

**Proof.**

#### 3.3. Optimal Barrier Level

**Proposition**

**2.**

- (a)
- $\sigma >0$ and ${\left(\Phi (p+q)\right)}^{2}/p<2/{\sigma}^{2}$;
- (b)
- $\sigma =0$ and $\nu (0,\infty )=\infty $;
- (c)
- $\sigma =0$, $\nu (0,\infty )<\infty $ and$$\frac{c\Phi (p+q)}{p}\left(\Phi (p+q)-\frac{p}{c}\right)<\frac{q+\nu (0,\infty )}{c},$$

**Proof.**

**Remark**

**3.**

## 4. Verification Lemma and Proof of the MainResult

**Lemma**

**1.**

**Proof.**

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

## Appendix B. Proof of Proposition 2

## References

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Renaud, J.-F.
De Finetti’s Control Problem with Parisian Ruin for Spectrally Negative Lévy Processes. *Risks* **2019**, *7*, 73.
https://doi.org/10.3390/risks7030073

**AMA Style**

Renaud J-F.
De Finetti’s Control Problem with Parisian Ruin for Spectrally Negative Lévy Processes. *Risks*. 2019; 7(3):73.
https://doi.org/10.3390/risks7030073

**Chicago/Turabian Style**

Renaud, Jean-François.
2019. "De Finetti’s Control Problem with Parisian Ruin for Spectrally Negative Lévy Processes" *Risks* 7, no. 3: 73.
https://doi.org/10.3390/risks7030073