# A Renewal Shot Noise Process with Subexponential Shot Marks

## Abstract

**:**

## 1. Introduction

**Assumption**

**1.**

- (i)
- the shot marks ${X}_{1}$, ${X}_{2}$, … form a sequence of independent and identically distributed (i.i.d.) real-valued random variables with a generic random variable X and distribution F;
- (ii)
- their arrival times ${\tau}_{1}$, ${\tau}_{2}$, … form a sequence of renewal epochs, so that the number of shots by time $t\ge 0$, namely,$${N}_{t}=sup\left\{k\in \mathbb{N}:{\tau}_{k}\le t\right\},$$is an ordinary renewal counting process;
- (iii)
- the two sequences $\{{X}_{1},{X}_{2},\dots \}$ and $\{{\tau}_{1},{\tau}_{2},\dots \}$ are mutually independent;
- (iv)
- the response function $h(\xb7)$ is non-increasing on $[0,\infty )$ with $0<h(0+)<\infty $.

#### A Brief Literature Review

## 2. The Main Result

**Theorem**

**1.**

## 3. Lemmas

**Lemma**

**1.**

**Lemma**

**2.**

## 4. Proof of Theorem 1

## Funding

## Conflicts of Interest

## References

- Albrecher, Hansjörg, and Søren Asmussen. 2006. Ruin probabilities and aggregate claims distributions for shot noise Cox processes. Scandinavian Actuarial Journal 2: 86–110. [Google Scholar] [CrossRef]
- Asmussen, Søren, Hanspeter Schmidli, and Volker Schmidt. 1999. Tail probabilities for non-standard risk and queueing processes with subexponential jumps. Advances in Applied Probability 31: 422–47. [Google Scholar] [CrossRef]
- Basu, Sankarshan, and Angelos Dassios. 2002. A Cox process with log-normal intensity. Insurance: Mathematics and Economics 31: 297–302. [Google Scholar] [CrossRef] [Green Version]
- Bingham, Nicholas H., Chrles M. Goldie, and Jozef L. Teugels. 1987. Regular Variation. Cambridge: Cambridge University Press. [Google Scholar]
- Brémaud, Pierre, and Laurent Massoulié. 2002. Power spectra of general shot noises and Hawkes point processes with a random excitation. Advances in Applied Probability 34: 205–22. [Google Scholar] [CrossRef]
- Brix, Anders. 1999. Generalized gamma measures and shot-noise Cox processes. Advances in Applied Probability 31: 929–53. [Google Scholar] [CrossRef]
- Campbell, Norman. 1909. The study of discontinuous phenomena. Proceedings of the Cambridge Philosophical Society 15: 117–36. [Google Scholar]
- Chen, Yiqing. 2019. A Kesten-type bound for sums of randomly weighted subexponential random variables. Statistics & Probability Letters. to appear. [Google Scholar]
- Daouia, Abdelaati, Stéphane Girard, and Gilles Stupfler. 2018. Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 80: 263–92. [Google Scholar] [CrossRef]
- Dassios, Angelos, and Ji-Wook Jang. 2003. Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance and Stochastics 7: 73–95. [Google Scholar] [CrossRef]
- Dassios, Angelos, Jiwook Jang, and Hongbiao Zhao. 2015. A risk model with renewal shot-noise Cox process. Insurance: Mathematics and Economics 65: 55–65. [Google Scholar] [CrossRef]
- Embrechts, Paul, Claudia Klüppelberg, and Thomas Mikosch. 1997. Modelling Extremal Events: For Insurance and Finance. Berlin/Heidelberg: Springer Science & Business Media. [Google Scholar]
- Foss, Sergey, Dmitry Korshunov, and Stan Zachary. 2011. An Introduction to Heavy-tailed and Subexponential Distributions. New York: Springer. [Google Scholar]
- Ganesh, Ayalvadi, Claudio Macci, and Giovanni Torrisi. 2005. Sample path large deviations principles for Poisson shot noise processes and applications. Electronic Journal of Probability 10: 1026–43. [Google Scholar] [CrossRef]
- Iksanov, Alexander. 2013. Functional limit theorems for renewal shot noise processes with increasing response functions. Stochastic Processes and their Applications 123: 1987–2010. [Google Scholar] [CrossRef]
- Iksanov, Alexander, Alexander Marynych, and Matthias Meiners. 2014. Limit theorems for renewal shot noise processes with eventually decreasing response functions. Stochastic Processes and their Applications 124: 2132–70. [Google Scholar] [CrossRef]
- Jang, Jiwook, and Angelos Dassios. 2013. A bivariate shot noise self-exciting process for insurance. Insurance: Mathematics and Economics 53: 524–32. [Google Scholar] [CrossRef]
- Jang, Ji-Wook, and Yuriy Krvavych. 2004. Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform. Insurance: Mathematics and Economics 35: 97–111. [Google Scholar] [CrossRef]
- Kelly, Bryan, and Hao Jiang. 2014. Tail risk and asset prices. The Review of Financial Studies 27: 2841–71. [Google Scholar] [CrossRef]
- Klüppelberg, Claudia, and Thomas Mikosch. 1995. Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1: 125–47. [Google Scholar] [CrossRef]
- Klüppelberg, Claudia, Thomas Mikosch, and Anette Schärf. 2003. Regular variation in the mean and stable limits for Poisson shot noise. Bernoulli 9: 467–96. [Google Scholar] [CrossRef]
- Kočetova, Jelena, Remigijus Leipus, and Jonas Šiaulys. 2009. A property of the renewal counting process with application to the finite-time ruin probability. Lithuanian Mathematical Journal 49: 55–61. [Google Scholar] [CrossRef]
- Landsman, Zinoviy, Udi Makov, and Tomer Shushi. 2016. Tail conditional moments for elliptical and log-elliptical distributions. Insurance: Mathematics and Economics 71: 179–88. [Google Scholar] [CrossRef]
- Liang, Xiaoqing, and Yi Lu. 2017. Indifference pricing of a life insurance portfolio with risky asset driven by a shot-noise process. Insurance: Mathematics and Economics 77: 119–32. [Google Scholar] [CrossRef]
- Li, Xiaohu, and Jintang Wu. 2014. Asymptotic tail behavior of Poisson shot-noise processes with interdependence between shock and arrival time. Statistics & Probability Letters 88: 15–26. [Google Scholar]
- Lowen, Steven B., and Malvin C. Teich. 1991. Doubly stochastic Poisson point process driven by fractal shot noise. Physical Review A 43: 4192–215. [Google Scholar] [CrossRef]
- Lund, Robert B., Ronald W. Butler, and Robert L. Paige. 1999. Prediction of shot noise. Journal of Applied Probability 36: 374–88. [Google Scholar] [CrossRef]
- Lund, Robert, William P. McCormick, and Yuanhui Xiao. 2004. Limiting properties of Poisson shot noise processes. Journal of Applied Probability 41: 911–18. [Google Scholar] [CrossRef]
- McCormick, William P. 1997. Extremes for shot noise processes with heavy tailed amplitudes. Journal of Applied Probability 34: 643–56. [Google Scholar] [CrossRef]
- Møller, Jesper. 2003. Shot noise Cox processes. Advances in Applied Probability 35: 614–40. [Google Scholar] [CrossRef]
- Møller, Jesper, and Giovanni Luca Torrisi. 2005. Generalised shot noise Cox processes. Advances in Applied Probability 37: 48–74. [Google Scholar] [CrossRef]
- Resnick, Sidney I. 1987. Extreme Values, Regular Variation and Point Processes. New York: Springer. [Google Scholar]
- Samorodnitsky, Gennady. 1998. Tail behavior of some shot noise processes. In A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Boston: Birkhäuser Boston, Inc., pp. 473–86. [Google Scholar]
- Scherer, Matthias, Ludwig Schmid, and Thorsten Schmidt. 2012. Shot-noise driven multivariate default models. European Actuarial Journal 2: 161–86. [Google Scholar] [CrossRef]
- Schmidt, Thorsten. 2014. Catastrophe insurance modeled by shot-noise processes. Risks 2: 3–24. [Google Scholar] [CrossRef]
- Schottky, Walter. 1918. Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern. Annalen der physik 362: 541–67. [Google Scholar] [CrossRef]
- Takács, Lajos. 1956. On secondary stochastic processes generated by recurrent processes. Acta Mathematica Hungarica 7: 17–29. [Google Scholar] [CrossRef]
- Tang, Qihe. 2006. On convolution equivalence with applications. Bernoulli 12: 535–49. [Google Scholar] [CrossRef]
- Tang, Qihe, Zhaofeng Tang, and Yang Yang. 2019. Sharp asymptotics for large portfolio losses under extreme risks. European Journal of Operational Research 276: 710–22. [Google Scholar] [CrossRef]
- Tang, Qihe, and Zhongyi Yuan. 2014. Randomly weighted sums of subexponential random variables with application to capital allocation. Extremes 17: 467–93. [Google Scholar] [CrossRef]
- Weng, Chengguo, Yi Zhang, and Ken Seng Tan. 2013. Tail behavior of Poisson shot noise processes under heavy-tailed shocks and Actuarial applications. Methodology and Computing in Applied Probability 15: 655–82. [Google Scholar] [CrossRef]
- Yang, Yang, and Yuebao Wang. 2013. Tail behavior of the product of two dependent random variables with applications to risk theory. Extremes 16: 55–74. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Chen, Y.
A Renewal Shot Noise Process with Subexponential Shot Marks. *Risks* **2019**, *7*, 63.
https://doi.org/10.3390/risks7020063

**AMA Style**

Chen Y.
A Renewal Shot Noise Process with Subexponential Shot Marks. *Risks*. 2019; 7(2):63.
https://doi.org/10.3390/risks7020063

**Chicago/Turabian Style**

Chen, Yiqing.
2019. "A Renewal Shot Noise Process with Subexponential Shot Marks" *Risks* 7, no. 2: 63.
https://doi.org/10.3390/risks7020063