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# The distribution of some extremum on the risk process whose income depend on the current reserve

- Jingmin He
^{1}, - Zaiming Liu
^{2}and - Wei Zhang
^{2}Email author

**Received:**23 April 2016**Accepted:**7 November 2016**Published:**15 November 2016

## Abstract

This paper considers the distribution of some extremum on the risk process whose income depend on the current reserve. We first construct the defective renewal sequence and obtain the density function of them. By the presented renewal measure and the strong Markov property, the distribution of the first hitting time is obtained explicitly. Then, the ruin probability and the probability that the surplus process less than *x* is obtained. Furthermore, the distribution of supreme profits before ruin, the joint distributions of the supreme profit and the deficit before the time of the surplus process first up-crossing level zero after ruin, and the joint distributions of the supreme profit and the deficit before the surplus process leave zero ultimately are derived. Finally, the exact calculating results for them are obtained when the individual claim amounts in the compound Poisson risk model are exponentially distributed.

## Keywords

- Ruin probability
- Strong Markov property
- Supreme profit and deficit
- The time of ruin

## Mathematics Subject Classification

- 60J25
- 91B30

## Background

*f*and mean value \(\mu\). Then the well-known classical compound Poisson risk model is given by

*u*denotes the initial capital of an insurance company, \(c>0\) is the premium income rate. If the company charges a constant premium rate

*c*, but invests its money at interest rate \(\delta\), we get the compound Poisson risk model with constant interest. The dynamics of the surplus process can be described by

*u*, the risk process run in accordance with a differential equation. When the claim occurs, the risk process run from the new initial value, according to the same differential equation, and the procedure goes on and on. We use \(\phi (t,x)\) to denote the deterministic path starting from the initial value

*x*in between jumps, which satisfies

*x*generated on \((\Omega ,{\mathscr {F}}_{\infty })\).

For the reason why the model (1) is important, the readers are referred to Cai et al. (2009) and Chapter 8 of Asmussen and Albrecher (2010), where some well-known important risk models are given by taking different kinds of functions of \(g(\cdot )\).

In fact the above model was considered by many authors: such as Asmussen (2000), Asmussen and Albrecher (2010), Cai et al. (2009), Dassios and Embrechts (1989), Das and Mahavier (2012), Embrechts and Schmidli (1994), Egídio dos Reis (2002), Li and Lu (2013) and Wang et al. (2003). For this model, Cai et al. (2009) investigate various applications of the total discounted operating costs up to default. Dassios and Embrechts (1989) or Embrechts and Schmidli (1994) showed in general how to use the theory of piecewise deterministic Markov processes for solving insurance risk problems. Das and Mahavier (2012) study the joint distribution of the surplus immediately before ruin and the deficit at ruin for the compound Poisson risk model with constant interest. Li and Lu (2013) consider the generalized expected discounted penalty function in a risk process with credit and debit interests. It is worthwhile pointing out that a deep review and details on applications of this model can be found at Chapter 8 of Asmussen and Albrecher (2010). From Dassios and Embrechts (1989) or Embrechts and Schmidli (1994), we know that \(\{U(t)\}_{t\ge 0}\) is a piecewise deterministic Markov process. We use \(P(t,x,\Gamma )\) to denote the transition function of the model (1), for any \(\Gamma \in {\mathscr {B}}(\mathbb {R})\) (the Borel \(\sigma\)-algebra on \(\mathbb {R}\)). Throughout this paper, it is assumed that \(P(t,x,\Gamma )\) has a density function *p*(*t*, *x*, *y*) for \(y<\phi (t,x)\).

To the best of our knowledge, there are only a few papers exclusively concerned with the extremum of the risk model (1). The distribution of extremum is very important in risk theory, which can portray the best and worst condition of an insurance company and provide early warning for the development of the insurance company. It is worth pointing out that our method is different from the traditional method, and we obtain the distribution of the first hitting time by the method of constructing the renewal sequence. And then, the distributions of some extreme value are investigated.

Inspired by Wu et al. (2003), we mainly study the first hitting time and some extremum of the model (1). The rest of this paper is organized as follows. We intend to introduce in “Preliminaries” section the renewal measure of the defective renewal sequence. In “The first hitting time” section, the expression of the renewal measure is derived. Thus, the distribution function on the first hitting time is obtained. Furthermore, the ruin probability and the probability that the surplus process is less than *x* is obtained. In “The supreme profit and the deficit” section, the distribution of supreme profits before ruin, the joint distributions of the supreme profit and deficit before the time of the first up-crossing level zero after ruin, and the joint distributions of the supreme profit and deficit before the time of the surplus process leaving zero ultimately are derived. As a validation of all results’ applications, we give the explicit expressions for the compound Poisson risk model with the claim amount being exponentially distributed.

## Preliminaries

*T*be the time of ruin, \(T_{x}\) the time of the surplus process below the level

*x*, i.e. the time of default (see Cai et al. 2009) for the first time and

*L*the time of the surplus process leaving zero ultimately, then

*L*.

*x*points on the time scale of the surplus process as follows:

*x*points before

*t*(and including

*t*). Therefore, \(\{N_t^{x}\}_{t\ge 0}\) is a counting process and \(N_\infty ^{x}=\sup \{k>0: T_k^{x}<+\infty \}\quad (N_\infty ^{x}=0\) if the set is empty) is the total number of the

*x*points of the process. Putting \({\mathscr {F}}_t=\sigma \{U(s),s\le t\}\), then

*T*, \(T_{x}\) and \(\{T_k^{x}\}_{k\ge 1}\) are all \({\mathscr {F}}_t\)-stopping times.

It is known that (see, for example, Gerber and Shiu 1998) the stopping times play an important role in many risk portfolio. Among others, we only mention a few, for example Dickson and Li (2013), Gerber (1990), Gerber and Shiu (1998), Zacks (2007), Xu (2012), Kyprianou (2013), Landriault and Shi (2014), Li and Lu (2014), Li et al. (2015) and references therein.

*u*, and \(\Phi (u)=1-\Psi (u)\) be the survival probability. We define the probability the surplus process falls below the level

*x*, i.e. the probability of default, as

*x*can be expressed as

*k*-th renewal epoch is \(T_k^{x}=\sum \nolimits _{n=1}^{k}S_n^{x}\). Let \(F_{x}\) be the common distribution of \(\{S_{k}^{x}\}_{k\ge 2}\), and \(F_{u}^{x}\) be the distribution of \(S_{1}^{x}\). Then the renewal measure \(G_{u}^{x}\) is defined by

*n*-fold convolution of \(F_{x}(t)\). Thus we have

*I*denotes a general interval. Let \(g_{u}^{x}(\cdot )\) and \(f_{u}^{x}(\cdot )\) be the density functions of \(G_{u}^{x}\) and \(F_{u}^{x}\) respectively, if they exist.

## The first hitting time

We now show in detail how the renewal measure \(G_{u}^{x}(\cdot )\) can be used to express the first hitting time. The key point is to obtain \(G_{u}^{x}(\cdot )\). We first give the following lemma, which plays an important role in getting the expression of \(G_{u}^{x}(\cdot )\).

###
**Lemma 1**

*Let*\(X_{t}\)

*satisfy the ordinary differential equation*

*where*\(h(\cdot )\)

*is a continuously differentiable Lipschitz continuous function. If there exists a*\(t^{*}\in [0,T]\),

*such that*\(X_{t^{*}}=x\),

*then we have*

*where constant*\(K_{1}\)

*depends only on*

*T*

*and*

*x*

*and constant*

*K*

*depends on*\(h(\cdot )\).

###
*Proof*

*T*and

*x*, such that for any \(0\le s\le T\), \(|h(X_{s})|\le K_{1}\). Hence, for any \(s\in [0,T]\), we obtain

By the description just after the model (1), we know that the function \(g(\cdot )\) of model (1) can be considered as special cases of the function \(h(\cdot )\) of Lemma 1.

###
**Theorem 1**

*For*\(s\ge 0\),

*we have*

- (1)
*When*\(\phi (s,u)>x\),$$\begin{aligned} g_{u}^{x}(s)=\left\{ \begin{array}{ll} g(x)p(s,u,x) &{}\quad {\mathrm{if}}\ s>0,\\ 0 &{}\quad \mathrm{{if}}\ s=0,\\ \end{array}\right. \end{aligned}$$(3)*where*\(g(\cdot )\)*is given by*(1). - (2)
*When*\(\phi (s,u)<x\),$$\begin{aligned} g_{u}^{x}(s)=0. \end{aligned}$$ - (3)
*When*\(\phi (s,u)=x\),- (a)
*If*\(s>0\),*then*\(G_{u}^{x}(t)\)*will jump at time**s*,*that is*,$$\begin{aligned} G_{u}^{x}(t)=0,\quad \mathrm{{for}}\ 0\le t<s,\quad G_{u}^{x}(s)=e^{-\lambda s}. \end{aligned}$$ - (b)
*If*\(s=0\),*i.e.*, \(s=0, u=x\),*then*$$\begin{aligned} g_{u}^{x}(s)=0. \end{aligned}$$

- (a)

###
*Proof*

- (1)When \(\phi (s,u)>x\), we haveBy its probability meaning, we obtain$$\begin{aligned} g_{u}^{x}(s)ds&= {} \sum \limits _{k=1}^{\infty }P^{u}\left( T_{k}^{x}\in (s,s+ds]\right) =\sum \limits _{k=1}^{\infty }P^{u}\left( T_{k}^{x}\in ds\right) \\&= {} \sum \limits _{k=1}^{\infty }P^{u}\left( T_{k}^{x}\in ds, N(s,s+ds]=0\right) +\sum \limits _{k=1}^{\infty }P^{u}\left( T_{k}^{x}\in ds, N(s,s+ds]\ge 1\right) . \end{aligned}$$Note that \(T_{N_{s+ds}^{x}}^{x}\in (s, s+ds]\) and \(\frac{dU(t)}{dt}=g(U(t))\), for any \(t\in (s,s+ds]\). By Lemma 1, we have$$\begin{aligned} \sum \limits _{k=1}^{\infty }P^{u}\left( T_{k}^{x}\in ds, N(s,s+ds]=0\right) =P^{u}\left( U(s)<x, U(s+ds)\ge x, N(s,s+ds]=0\right) . \end{aligned}$$Hence, when \(s>0\), we get$$\begin{aligned} |U(s+ds)-U(s)-g(x)ds)|\le KK_{1}d^{2}s. \end{aligned}$$and$$\begin{aligned}&P^{u}(U(s)<x, U(s+ds)\ge x, N(s,s+ds]=0)\nonumber \\&\quad =P^{u}(x-g(x)ds+O(d^{2}s)\le U(s)<x, N(s,s+ds]=0)\nonumber \\&\quad =P^{u}(x-g(x)ds+O(d^{2}s)\le U(s)<x)e^{-\lambda ds}\nonumber \\&\quad = g(x)p(s,u,x)ds+O(d^{2}s), \end{aligned}$$Then we have$$\begin{aligned} \sum \limits _{k=1}^{\infty }P^{u}\left( T_{k}^{x}\in ds, N(s,s+ds]\ge 1\right) =O(d^{2}s). \end{aligned}$$When \(s=0\), we get$$\begin{aligned} g_{u}^{x}(s)=g(x)p(s,u,x). \end{aligned}$$(4)Hence we arrive at$$\begin{aligned} g_{u}^{x}(0)ds&= {} \sum \limits _{k=1}^{\infty }P^{u}\left( T_{k}^{x}\in (0,ds]\right) =\sum \limits _{k=1}^{\infty }P^{u}\left( T_{k}^{x}\in ds\right) \\&= {} \sum \limits _{k=1}^{\infty }P^{u}\left( T_{k}^{x}\in ds, N(0,ds]=0\right) +\sum \limits _{k=1}^{\infty }P^{u}\left( T_{k}^{x}\in ds, N(0,ds]\ge 1\right) \\&= {} O(d^{2}s). \end{aligned}$$Combining (4) with (5), we obtain (3) immediately.$$\begin{aligned} g_{u}^{x}(0)=0. \end{aligned}$$(5)
- (2)
when \(\phi (s,u)<x\), \(P^{u}(T_{1}^{x}>s)=1\). This follows that \(g_{u}^{x}(s)=0\).

- (3)when \(\phi (s,u)=x\). (a) If \(0\le t<s\), then \(\phi (t,u)<x\), thus \(G_{u}^{x}(t)=0\). If \(t=s\), then there is no jump before
*s*, that is(b) When \(s=0, u=x\), we have$$\begin{aligned} G_{u}^{x}(s)=P^{u}(T_{1}^{x}=s)=P^{u}(S_{1}>s)=e^{-\lambda s}. \end{aligned}$$Hence, \(g_{u}^{x}(0)=0\). So the proof is completed. \(\square\)$$\begin{aligned} g_{u}^{x}(0)ds&= {} \sum \limits _{k=1}^{\infty }P^{u}(T_{k}^{x}\in (0,ds])=\sum \limits _{k=1}^{\infty }P^{u}(T_{k}^{x}\in (0, ds], N(0,ds]\ge 1)=O(d^{2}s). \end{aligned}$$

###
*Remark 1*

When \(g(x)=c\), the risk model is reduced to the classical risk model, Theorem 1 coincides exactly with Lemma 3.1 in Wu et al. (2003).

###
**Lemma 2**

*There exists a constant*

*M*,

*such that*

*for any*\(v\ge M\).

###
*Proof*

By Theorem 1, we see that the expression of the renewal measure \(G_{u}^{x}\) can be derived once *u* and *x* are fixed. Next, we consider the explicit expressions of the distribution on the first hitting time, which is expressed in terms of \(G_{u}^{x}\).

###
**Theorem 2**

*For*\(s>0\),

*we have*

###
*Proof*

###
**Corollary 1**

###
*Proof*

Since \(P^{u}(\lim \nolimits _{t\rightarrow \infty }U(t)=\infty )=1\), then \(\Psi (u)=P^{u}(T<\infty )=P^{u}(T_{1}^{0}<\infty )\) and \(\Psi _{x}(u)=P^{u}(T_{x}<\infty )=P^{u}(T_{1}^{x}<\infty )\) can be obtained directly. \(\square\)

## The supreme profit and the deficit

In this section, some distributions on the maximum surplus and the maximal severity of ruin are given. Before proceeding with the next Theorem, we will give a simple explanation of shift operators first. For \(t\ge 0\), let \(\theta _{t}\) be the shift operators from \(\Omega\) to itself defined by \(U(s)\circ \theta _{t}= U(s+t)\). For stopping time *T*, conditioning on \(\{T<\infty \}\) the map \(\theta _{T}\) is defined by \(U(t)\circ \theta _{T}=U(t+T)\) (see Revuz and Yor 1991, pp. 34, 37 and 74). Let \(G(u,a)=P^{u}(\sup \nolimits _{0\le t<T}U(t)>a,T<\infty )\) to denote the probability distribution of the supreme profit of an insurance company before the time of ruin. First we will give the explicit expression of *G*(*u*, *a*).

###
**Theorem 3**

*For*\(u\ge 0\),

*we have*

*where*\(\Psi (u), \Phi (u)\)

*are given by*(7).

###
*Proof*

###
**Corollary 2**

*For*\(a>u\ge 0\),

*we have*

###
**Theorem 4**

*For*\(a>u\),

*we have*

*where*\(\Phi (u)\)

*can be obtained by*(7).

###
*Proof*

In the following, we consider the maximum surplus and the maximal severity of ruin before the time of recovery. To some extent, as ‘indexes’, they can describe the ‘best’ situation and the ‘worst’ situation the company would experience before the surplus process up-crossing level zero after ruin for the first time. Their joint distributions are derived.

###
**Theorem 5**

*For*\(a>u\ge 0\)

*and*\(b>0\),

*we have*

*where*\(\Phi (u), \Phi _{-b}(u)\)

*are presented by*(7)

*and*(8).

###
*Proof*

###
*Remark 2*

- 1.
When \(g(x)=c\), the risk model simplifies to the classical compound Poisson risk model, Theorem 5 is the same as Lemma 3.5 in Wu et al. (2003).

- 2.When \(g(x)=c, a=\infty\), Theorem 5 simplifies towhich coincides with Theorem 1 in Picard (1994).$$\begin{aligned} P^{u}\Big (\inf \limits _{0\le t<T_{1}^{0}}U(t)\ge -b\Big )=\frac{\Phi (u+b)-\Phi (u)}{\Phi (b)}, \end{aligned}$$
- 3.When \(g(x)=c+\delta x\) for \(x \ge 0\) and \(g(x)=c+r x\) for \(x<0\), the risk model is reduced to the risk model with credit and debit interests. Let \(a=\infty\), Theorem 5 simplifies towhich is the same as (6.2) in Li and Lu (2013).$$\begin{aligned} P^{u}\Big (\inf \limits _{0\le t<T_{1}^{0}}U(t)\ge -b\Big )=\frac{\Phi _{-b}(u)-\Phi (u)}{\Phi _{-b}(0)}, \end{aligned}$$

Next, we consider the maximum surplus and the maximal severity of ruin before the time of the surplus process leaving zero ultimately, which describe the best situation and the worst situation the company would experience before the time of the surplus process leaving zero ultimately. We obtain their explicit expression in the following theorem.

###
**Theorem 6**

*For*\(a>u\ge 0\)

*and*\(b>0\),

*we have*

*In particular*,

*where*\(\Phi (u), \Phi _{-b}(u)\)

*are given by*(7) and (8).

###
*Proof*

###
*Example 1*

*W*(0,

*a*,

*b*) in Wu et al. (2003). By Theorem 6, we have

## Conclusions

In order to make a reasonably realistic description of the actual behavior, we investigate the risk model whose income depend on the current reserve.

The distribution of extremum is very important in risk theory, which can portray the best and worst condition of an insurance company. The research of extremum is necessary and meaningful whether theoretically or practically. In this paper, the distribution of supreme profits before ruin, the joint distributions of the supreme profit and the deficit before the time of the surplus process first up-crossing level zero after ruin, and the joint distributions of the supreme profit and the deficit before the surplus process leave zero ultimately are derived. All these results provide early warning for the development of the insurance company.

Concretely speaking, the distributions of some extremum can be converted to the problem of hitting time, so we study the first hitting time of this model. With the help of the strong Markov property, we construct the renewal measure of the defective renewal sequence, and obtain the distribution of the renewal measure. The method that we used to solve the stopping time problem is innovative. By the presented renewal measure and the Laplace–Stieltjes transforms, the distribution of the first hitting time is obtained explicitly. Then, the ruin probability and the probability that the surplus process less than *x* is obtained.

## Declarations

### Authors' contributions

All the authors have contributed to the manuscript equally. All the authors have read and approved the final manuscript.

### Acknowledgements

The authors are grateful to the anonymous referee for carefully reading, valuable comments and suggestions to improve the earlier version of the paper. This research is supported by MOE (Ministry of Education in China) Youth Project of Humanities and Social Sciences (Project Nos. 14YJCZH048, 15YJCZH204), Humanities and Social Sciences Foundation of Ministry of Education of China(No. 12YJAZH173), National Natural Science Foundation of China (Grant Nos. 11401436, 11101434, 11371374, 11571372, 11271373) and National Social Science Fund of China (No. 15BJY122), partly supported by Hunan Provincial Natural Science Foundation of China (Grant No. 13JJ5043) and Mathematics and Interdisciplinary Sciences Project, Central South University. ZW and HJM would show their great gratitude to Professor Wu Rong for her kindness and patience. ZW would like to show his thanks to Professor Feng Lihua for his advice in dealing with the tex format of the manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Asmussen S (2000) Ruin probabilities. World Scientific, SingaporeView ArticleMATHGoogle Scholar
- Asmussen S, Albrecher H (2010) Ruin probabilities. World Scientific Publishing, SingaporeView ArticleMATHGoogle Scholar
- Cai J, Feng RH, Willmot G (2009) On the expectation of total discounted operating costs up to default and its applications. Adv Appl Probab 41:495–522MathSciNetView ArticleMATHGoogle Scholar
- Das KP, Mahavier WT (2012) Further results for the joint distribution of the surplus immediately before and after ruin under force of interest. J Stat Theory Pract 6:344–353MathSciNetView ArticleGoogle Scholar
- Dassios A, Embrechts P (1989) Martingales and insurance risk. Commun Stat Stoch Models 5(2):181–217MathSciNetView ArticleMATHGoogle Scholar
- Dickson D, Li S (2013) The distributions of the time to reach a given level and the duration of negative surplus in the Erlang(2) risk model. Insur Math Econ 52(3):490–497MathSciNetView ArticleMATHGoogle Scholar
- Egídio dos Reis AD (2002) How many claims does it take to get ruined and recovered. Insur Math Econ 31:235–248MathSciNetView ArticleMATHGoogle Scholar
- Embrechts P, Schmidli H (1994) Ruin estimation for a general insurance risk model. Adv Appl Probab 26(2):404–422MathSciNetView ArticleMATHGoogle Scholar
- Gerber HU (1990) When does the surplus reach a given target. Insur Math Econ 9:115–119MathSciNetView ArticleMATHGoogle Scholar
- Gerber HU, Shiu ESW (1998) On the time value of ruin. N Am Actuar J 2(1):48–78MathSciNetView ArticleMATHGoogle Scholar
- Kyprianou AE (2013) Gerber–Shiu risk theory, EAA series. Springer, BerlinMATHGoogle Scholar
- Landriault D, Shi T (2014) First passage time for compound Poisson processes with diffusion: ruin theoretical and financial applications. Scand Actuar J 4:368–382MathSciNetView ArticleGoogle Scholar
- Li S, Lu Y (2013) On the generalized Gerber–Shiu function for surplus processes with interest. Insur Math Econ 52(2):127–134MathSciNetView ArticleMATHGoogle Scholar
- Li S, Lu Y (2014) The density of the time of ruin in the classical risk model with a constant dividend barrier. Ann Actuar Sci 8(1):63–78View ArticleGoogle Scholar
- Li S, Lu Y, Jin C (2015) Number of jumps in two-sided first-exit problems for a compound Poisson process. Methodol Comput Appl Probab. doi:10.1007/s11009-015-9453-8 MathSciNetMATHGoogle Scholar
- Picard P (1994) On some measures of the severity ruin in the classical Poisson model. Insur Math Econ 14(2):107–117MathSciNetView ArticleMATHGoogle Scholar
- Revuz D, Yor M (1991) Continuous Martingales and Brownian motion. Springer, BerlinView ArticleMATHGoogle Scholar
- Wang GJ, Zhang CS, Wu R (2003) Ruin theory for the risk process described by PDMPs. Acta Math Appl Sin Engl Ser 19(1):59–70MathSciNetView ArticleMATHGoogle Scholar
- Wu R, Wang GJ, Wei L (2003) Joint distributions of some actuarial random vectors containing the time of ruin. Insur Math Econ 33(1):147–161MathSciNetView ArticleMATHGoogle Scholar
- Xu Y (2012) First exit times of compound Poisson processes with parallel boundaries. Seq Anal: Des Methods Appl 31(2):135–144MathSciNetView ArticleMATHGoogle Scholar
- Zacks S (2007) Review of some functionals of compound Poisson processes and related stopping times. Methodol Comput Appl Probab 9:343–356MathSciNetView ArticleMATHGoogle Scholar