# How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- (i)
- Erlang$(2,\lambda )$-distributed interarrival times with Exp$\left(\mu \right)$-distributed claim sizes,
- (ii)
- Exp$\left(\lambda \right)$-distributed interarrival times with Erlang$(2,\mu )$-distributed claim sizes.

- P1.
- The premium function behaving like a constant at infinity$$p(\infty )=c,\phantom{\rule{2.em}{0ex}}{p}^{\prime}\left(u\right)=O\left(\frac{1}{{u}^{2}}\right);$$
- P2.
- The premium function exploding at infinity, $p(\infty )=\infty $ as$$p\left(u\right)=c+\sum _{i=1}^{l}{\u03f5}_{i}{u}^{i},\phantom{\rule{2.em}{0ex}}{\u03f5}_{i},c>0.$$

## 2. Ruin Probabilities—Method

## 3. Erlang$(\mathbf{2},\mathbf{\lambda})$ Distributed Interarrival Times with Exp$\left(\mathbf{\mu}\right)$ Distributed Claim Sizes

#### 3.1. General Premium

**Theorem**

**1.**

**Remark**

**1.**

**Proof.**

#### 3.2. Linear Premium

## 4. Exp$\left(\mathbf{\lambda}\right)$-Distributed Interarrival Times with Erlang$(\mathbf{2},\mathbf{\mu})$-Distributed Claim Sizes

#### 4.1. General Premium

**Theorem**

**2.**

**Remark**

**2.**

**Proof.**

#### 4.2. Linear Premium

## 5. Asymptotic Analysis—Comparison Results

#### 5.1. Exp$\left(\lambda \right)$-Distributed Interarrival Times with Exp$\left(\mu \right)$-Distributed Claim Sizes

#### 5.2. Erlang$(2,\lambda )$-Distributed Interarrival Times with Exp$\left(\mu \right)$-Distributed Claim Sizes

#### 5.3. Exp$\left(\lambda \right)$-Distributed Interarrival Times with Erlang$(2,\mu )$-Distributed Claim Sizes

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**General premium.**The ruin probability $\psi \left(u\right)$ has the following explicit expression

**Linear premium.**The explicit form of ruin probability $\psi \left(u\right)$ is

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**MDPI and ACS Style**

Wang, J.; Palmowski, Z.; Constantinescu, C.
How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability. *Risks* **2021**, *9*, 157.
https://doi.org/10.3390/risks9090157

**AMA Style**

Wang J, Palmowski Z, Constantinescu C.
How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability. *Risks*. 2021; 9(9):157.
https://doi.org/10.3390/risks9090157

**Chicago/Turabian Style**

Wang, Jing, Zbigniew Palmowski, and Corina Constantinescu.
2021. "How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability" *Risks* 9, no. 9: 157.
https://doi.org/10.3390/risks9090157