# Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Main Results

**Theorem**

**1.**

- (1)
- If $0<{y}_{*}<1$, then for each fixed $n\ge 1$, it holds that $\mathbb{E}\left({e}^{\gamma {S}_{n-1}}\right)<\infty ,\phantom{\rule{4pt}{0ex}}\mathbb{E}\left({e}^{\gamma {M}_{n-1}}\right)<\infty $, and$$\begin{array}{c}\mathbb{P}({S}_{n}>x)\sim \frac{{p}_{*}h\left({y}_{*}\right)\mathbb{E}\left({e}^{\gamma {S}_{n-1}}\right)}{{y}_{*}^{\alpha -1}}l\left(x\right){x}^{\alpha -1}{e}^{-\frac{\gamma x}{{y}_{*}}},\hfill \end{array}$$$$\begin{array}{c}\mathbb{P}({M}_{n}>x)\sim \frac{{p}_{*}h\left({y}_{*}\right)\mathbb{E}\left({e}^{\gamma {M}_{n-1}}\right)}{{y}_{*}^{\alpha -1}}l\left(x\right){x}^{\alpha -1}{e}^{-\frac{\gamma x}{{y}_{*}}}.\hfill \end{array}$$
- (2)
- If ${y}_{*}=1$, then for each fixed $n\ge 1$, it holds that$$\mathbb{P}({M}_{n}>x)\sim \mathbb{P}({S}_{n}>x)\sim \frac{{p}_{*}^{n}h\left({y}_{*}\right){\gamma}^{n-1}{(\Gamma \left(\alpha \right))}^{n}}{{y}_{*}^{(\alpha -1)n}\Gamma \left(n\alpha \right)}{\left(l\left(x\right)\right)}^{n}{x}^{n\alpha -1}{e}^{-\gamma x}.$$

**Theorem**

**2.**

## 3. Proofs of Main Results

**Lemma**

**1.**

**Proof**

**of Lemma 1.**

**Proof**

**of Theorem 1.**

**Remark**

**1.**

**Proof**

**of Theorem 2.**

## 4. Simulation Study

Y | 0.2 | 0.6 | 1 |

$\mathbb{P}(Y=y)$ | 0.3 | 0.4 | 0.3 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- R. Norberg. “Ruin problems with assets and liabilities of diffusion type.” Stoch. Process. Appl. 81 (1999): 255–269. [Google Scholar] [CrossRef]
- W. Vervaat. “On a stochastic difference equation and a representation of nonnegative infiitely divisible random variables.” Adv. Appl. Probab. 11 (1979): 750–783. [Google Scholar]
- A. Brandt. “The stochastic equation Y
_{n + 1}= A_{n}Y_{n}+ B_{n}with stationary coefficients.” Adv. Appl. Probab. 18 (2010): 211–220. [Google Scholar] [CrossRef] - Y. Yang, R. Leipus, and J. Šiaulys. “On the ruin probability in a dependent discrete time risk model with insurance and financial risks.” J. Comput. Appl. Math. 236 (2012): 3286–3295. [Google Scholar] [CrossRef]
- Y. Yang, and D.G. Konstantinides. “Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks.” Scand. Actuar. J. 8 (2015): 641–659. [Google Scholar] [CrossRef]
- Y. Zhang, X. Shen, and C. Weng. “Approximation of the tail probability of randomly weighted sums and applications.” Stoch. Process. Appl. 119 (2009): 655–675. [Google Scholar] [CrossRef]
- Y. Chen. “The finite-time ruin probability with dependent insurance and financial risks.” J. Appl. Probab. 48 (2011): 1035–1048. [Google Scholar] [CrossRef]
- Y. Chen, J. Liu, and F. Liu. “Ruin with insurance and financial risks following the least risky FGM dependence structure.” Insur. Math. Econ. 62 (2015): 98–106. [Google Scholar] [CrossRef]
- D. Dufresne. “Stochastic life annuities.” N. Am. Actuar. J. 11 (2007): 136–157. [Google Scholar] [CrossRef]
- M.J. Goovaerts, and J. Dhaene. “Supermodular ordering and stochastic annuities.” Insur. Math. Econ. 24 (1999): 281–290. [Google Scholar] [CrossRef]
- Y. Sun, and L. Wei. “The finite-time ruin probability with heavy-tailed and dependent insurance and financial risks.” Insur. Math. Econ. 59 (2014): 178–183. [Google Scholar] [CrossRef]
- E. Hashorva, and J. Li. “ECOMOR and LCR reinsurance with gamma-like claims.” Insur. Math. Econ. 53 (2013): 206–215. [Google Scholar] [CrossRef][Green Version]
- B.A. Rogozin, and M.S. Sgibnev. “Banach algebras of measures on the line with given asymptotics of distributions at infinity.” Sib. Math. J. 40 (1999): 565–576. [Google Scholar] [CrossRef]
- L. De Haan. “On Regular Variation and Its Application to the Weak Convergence of Sample Extremes.” Ph.D. Thesis, Mathematisch Centrum, Amsterdam, The Netherlands, 1970. [Google Scholar]
- Q. Tang, and G. Tsitsiashvili. “Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments.” Adv. Appl. Probab. 36 (2004): 1278–1299. [Google Scholar] [CrossRef]
- Q. Tang. “On convolution equivalence with applications.” Bernoulli 12 (2006): 535–549. [Google Scholar] [CrossRef]
- Y. Yang, and K.C. Yuen. “Asymptotics for a discrete-time risk model with Gamma-like insurance risks.” Scand. Actuar. J. 6 (2016): 565–579. [Google Scholar] [CrossRef][Green Version]
- A.V. Asimit, and A.L. Badescu. “Extremes on the discounted aggregate claims in a time dependent risk model.” Scand. Actuar. J. 2 (2010): 93–104. [Google Scholar] [CrossRef]
- J. Li. “Asymptotics in a time-dependent renewal risk model with stochastic return.” J. Math. Anal. Appl. 387 (2012): 1009–1023. [Google Scholar] [CrossRef]
- J. Li, Q. Tang, and R. Wu. “Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model.” Adv. Appl. Probab. 42 (2010): 1126–1146. [Google Scholar] [CrossRef]
- Y. Yang, E. Ignatavičiūtė, and J. Šiaulys. “Conditional tail expectation of randomly weighted sums with heavy-tailed distributions.” Stat. Probab. Lett. 105 (2015): 20–28. [Google Scholar] [CrossRef]
- P. Embrechts, E. Hashorva, and T. Mikosch. “Aggregation of log-linear risks.” J. Appl. Probab. 51 (2014): 203–212. [Google Scholar] [CrossRef]
- N.H. Bingham, C.M. Goldie, and J.L. Teugels. Regular Variation. Cambridge, UK: Cambridge University Press, 1987. [Google Scholar]

**Table 1.**Comparison between the simulated estimate values and the asymptotic values in Theorems 1-(2) for $N=1.0\times {10}^{7}$.

x | Simulated Estimate Values | Asymptotic Values |
---|---|---|

100 | $2.06\times {10}^{-5}$ ($1.20451\times {10}^{-6}$) | $1.18\times {10}^{-5}$ |

110 | $1.16\times {10}^{-5}$ ($3.06396\times {10}^{-7}$) | $8.47\times {10}^{-6}$ |

120 | $4.85\times {10}^{-6}$ ($2.51962\times {10}^{-7}$) | $5.73\times {10}^{-6}$ |

130 | $2.05\times {10}^{-6}$ ($1.50271\times {10}^{-7}$) | $3.69\times {10}^{-6}$ |

140 | $1.08\times {10}^{-6}$ ($1.28344\times {10}^{-7}$) | $2.28\times {10}^{-6}$ |

150 | $3.92\times {10}^{-7}$ ($2.11470\times {10}^{-7}$) | $1.36\times {10}^{-6}$ |

**Table 2.**Comparison between the simulated estimate values and the asymptotic values in Theorem 1-(2) for $N=1.5\times {10}^{7}$.

x | Simulated Estimate Values | Asymptotic Values |
---|---|---|

100 | $2.03\times {10}^{-5}$ ($6.75224\times {10}^{-7}$) | $1.18\times {10}^{-5}$ |

110 | $1.23\times {10}^{-5}$ ($3.35691\times {10}^{-7}$) | $8.47\times {10}^{-6}$ |

120 | $5.04\times {10}^{-6}$ ($2.35759\times {10}^{-7}$) | $5.73\times {10}^{-6}$ |

130 | $2.27\times {10}^{-6}$ ($1.04678\times {10}^{-7}$) | $3.69\times {10}^{-6}$ |

140 | $1.45\times {10}^{-6}$ ($6.50245\times {10}^{-8}$) | $2.28\times {10}^{-6}$ |

150 | $6.98\times {10}^{-7}$ ($3.32866\times {10}^{-8}$) | $1.36\times {10}^{-6}$ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Huang, X.-F.; Zhang, T.; Yang, Y.; Jiang, T. Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks. *Risks* **2017**, *5*, 14.
https://doi.org/10.3390/risks5010014

**AMA Style**

Huang X-F, Zhang T, Yang Y, Jiang T. Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks. *Risks*. 2017; 5(1):14.
https://doi.org/10.3390/risks5010014

**Chicago/Turabian Style**

Huang, Xing-Fang, Ting Zhang, Yang Yang, and Tao Jiang. 2017. "Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks" *Risks* 5, no. 1: 14.
https://doi.org/10.3390/risks5010014