# The Løkka–Zervos Alternative for a Cramér–Lundberg Process with Exponential Jumps

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## Abstract

**:**

## 1. Introduction

- if the cost of capital injections is low, then according to a double-barrier strategy, it is optimal to pay dividends and to inject capital, meaning ruin never occurs;
- if the cost of capital injections is high, then according to a single-barrier strategy, it is optimal to pay dividends and never inject capital, meaning ruin occurs at the first passage below zero.

#### 1.1. The Model

#### 1.2. The Problem

## 2. The Classical Dividend Problems for SNLPs

#### 2.1. De Finetti’s Problem

#### 2.2. Shreve, Lehoczky, and Gaver’s Problem

**Proposition**

**1.**

- (a)
- For fixed x and b, the function $k\mapsto {V}_{k}^{SLG}\left(x\right)$ is non-increasing.
- (b)
- For $k={k}_{f}\left(b\right)$, the value function defined in (9) can be written as follows:$$\begin{array}{c}{V}_{{k}_{f}\left(b\right)}^{0,b}\left(x\right)={k}_{f}\left(b\right)\left[{\overline{Z}}_{q}\left(x\right)+\frac{p}{q}-{Z}_{q}\left(x\right){V}^{b}\left(b\right)\right]\hfill \\ \hfill ={k}_{f}\left(b\right)\left[{Z}_{q}^{\left(1\right)}\left(x\right)+{Z}_{q}\left(x\right)\left(\frac{p}{q}-{V}^{b}\left(b\right)\right)\right],\end{array}$$$${Z}_{q}^{\left(1\right)}\left(x\right):={\int}_{0}^{x}\left({Z}_{q}\left(y\right)-p{W}_{q}\left(y\right)\right)\mathrm{d}y.$$
- (c)
- For fixed k, the barrier function ${H}_{k}^{SLG}$ has a unique point of maximum ${b}_{k}\ge 0$. It is decreasing, and thus ${b}_{k}=0$ if, and only if $k\in (1,{k}_{0}]$. Finally, if ${b}_{k}>0$, then $k={k}_{f}\left({b}_{k}\right)$.

**Remark**

**1.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 3. The Løkka–Zervos Alternative for a Cramér–Lundberg Model with Exponential Jumps

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

- (a)
- ${Z}_{q}\left(x\right)+\mu {Z}_{q}^{\left(1\right)}\left(x\right)=c{W}_{q}\left(x\right)$, for all $x>0$;
- (b)
- $\frac{1}{{k}_{f}\left({b}^{*}\right)}=\frac{c{W}_{q}^{\prime}\left({b}^{*}\right)}{\mu}$;
- (c)
- ${V}_{{k}_{f}\left({b}^{*}\right)}^{0,{b}^{*}}\left(x\right)={V}^{dF}\left(x\right)$, for all $x>0$.

**Proof.**

## 4. Conclusions and Conjecture

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | Some papers refer to this as the log-convexity of ${Z}_{q}\left(x\right)$. |

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**MDPI and ACS Style**

Avram, F.; Goreac, D.; Renaud, J.-F.
The Løkka–Zervos Alternative for a Cramér–Lundberg Process with Exponential Jumps. *Risks* **2019**, *7*, 120.
https://doi.org/10.3390/risks7040120

**AMA Style**

Avram F, Goreac D, Renaud J-F.
The Løkka–Zervos Alternative for a Cramér–Lundberg Process with Exponential Jumps. *Risks*. 2019; 7(4):120.
https://doi.org/10.3390/risks7040120

**Chicago/Turabian Style**

Avram, Florin, Dan Goreac, and Jean-François Renaud.
2019. "The Løkka–Zervos Alternative for a Cramér–Lundberg Process with Exponential Jumps" *Risks* 7, no. 4: 120.
https://doi.org/10.3390/risks7040120