# Differential equations for ruin probability in a special risk model with FGM copula for the claim size and the inter-claim time

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## Abstract

In this paper, we consider a risk process in which the distribution of the inter-claim time is the sum of two independent exponential random variables. We introduce a dependence structure between the claim size and the inter-claim time. The structure is based on FGM copula. An integro-differential equation for the expected discounted penalty function is derived and an explicit expression for the Laplace transform of ruin probability is given for exponential claim size.

## Keywords

Risk Model Risk Process Claim Size Surplus Process Initial Surplus## 1 Introduction

In classical risk models, the surplus process usually relies on the assumption of independence between the claim size and the inter-claim time. However, in many applications this assumption is too restrictive and unrealistic. Actually, we know that the greater the claim size is, the longer the inter-claim time is. The requirement for generalization has led to some papers on the modeling of dependence. Among them, Albrecher and Teugels [1] consider general dependence structure based on a copula for the claim size and the inter-claim time, they derive asymptotic results for both the finite and infinite time ruin probabilities. Boudreault et al. [2] consider a particular dependence structure among the inter-claim time and the claim size and derive the defective renewal equation satisfied by expected discounted penalty function. Cossette et al. [3] consider the compound Poisson risk model in which the claim size and the inter-claim time are dependent, and the dependence structure is based on Farlie-Gumbel-Morgenstern (FGM) copula. They derive the Laplace transform of the expected discounted penalty function, and give explicit expression for the Laplace transform of the time of ruin for exponential claim sizes. Barges and Cossette [4] investigate the computation of the moments of the compound Poisson risk model with FGM copula.

In this article, we consider a Sparre Andersen risk process where the claim size and the inter-claim time are dependent with FGM copula, and the distribution of the inter-claim time is the sum of two independent exponential random variables. In ruin theory, a common approach is to obtain an integro-differential equation for the expected discounted penalty function and apply it to derive the Laplace transform of the function. Dickson and Hipp [5] investigate ruin probability for $Erlang(2)$ risk process, Li and Garrido [6] consider this problem of $Erlang(n)$ risk model, Gerber and Shui [7] also do some relative works.

The paper is arranged as follows. In Section 2, we present the risk model and give some notations. An integro-differential equation of the expected discounted penalty function is formulated and the main results are presented in Section 3. In Section 4, we apply the integro-differential equation to derive the Laplace transform of the expected discounted penalty function. The special case where the claim size is exponentially distributed is considered in the final section.

## 2 The risk model

The surplus process is defined as $U(t)=u+ct-{\sum}_{i=1}^{N(t)}{X}_{i}$, where $U(0)=u$ is the initial surplus, *c* is the premium rate, $\{N(t),t\in {\mathbb{R}}^{+}\}$ is a renewal process, and ${X}_{i}$ ($i=1,2,\dots $) is the random variable (r.v.) corresponding to the amount of the *i* th claim. The time between the $(i-1)$th and *i* th claim is defined by the r.v. ${V}_{i}$ with ${V}_{1}$ being the time of the first claim.

The claim amounts $\{{X}_{i},i\in {\mathbb{N}}^{+}\}$ form a sequence of independent identically distributed (i.i.d.) random variables (r.v.s) as the r.v. *X* with probability density function (p.d.f.) ${f}_{X}$, cumulative distribution function (c.d.f.) ${F}_{X}$. The inter-claim times $\{{V}_{i},i\in {\mathbb{N}}^{+}\}$ form a sequence of independent r.v.’s identically distributed as the canonical r.v. *V*, $V={W}_{1}+{W}_{2}$, where $\{{W}_{j}\}$ are two independent exponentially distributed r.v.s with parameters ${\lambda}_{j}$, $j=1,2$, *V* has p.d.f. ${f}_{V}$, and c.d.f. ${F}_{V}$. Note that the $Erlang(2)$ model is the special case where ${\lambda}_{1}={\lambda}_{2}$. $\{({X}_{i},{V}_{i}),i\in {\mathbb{N}}^{+}\}$ form a sequence of i.i.d. random vectors distributed as the canonical random vector $(X,V)$. The joint p.d.f. of $(X,V)$ is denoted by ${f}_{X,V}(x,t)$ with $x,t\in {\mathbb{R}}^{+}$.

for every $(u,v)$ in ${[0,1]}^{2}$ and the dependence parameter *θ* takes value in $[-1,1]$.

We choose this class of copula since it provides an easy manner to construct bivariate models with a variety of dependence structures. Even if the FGM copula introduces only light dependence, it admits positive as well as negative dependence between a set of random variables and includes the independence copula when $\theta =0$ (see Nelsen [8]).

*T*denote the time to ruin, so that

*u*is defined as

where $\omega (x,y)$, for all $x,y\ge 0$, is the penalty function at the time of ruin for the surplus prior to ruin and the deficit at ruin, $I(\cdot )$ is the indicator function, and *δ* is a nonnegative parameter. We can think of *δ* either as being a force of interest or as a dummy variable in the context of the Laplace transform. A special case of the expected discounted penalty function with $\omega (x,y)=1$ is the Laplace transform of the time of ruin. The cases $\delta =0$ and $\omega (x,y)=1$ correspond to the infinite-time ruin probability.

## 3 Integro-differential equation

**Theorem 1**

*In the risk model introduced in Section*2,

*the excepted discounted penalty function*${m}_{\delta}(u)$

*satisfies the following integro*-

*differential equation*:

*where*

*I* *and* *D* *are the identity and differential operators*.

*Proof*Case 1. ${\lambda}_{1}\ne {\lambda}_{2}$, we obtain the p.d.f. and c.d.f. of r.v.

*V*,

In order to get (14), we firstly apply the operator $\beta (D)$ to both sides of (18) with respect to (w.r.t.) *u*.

Secondly, we take the operator $\alpha (D)$ to both sides of (19) w.r.t. *u*.

Hence, the integro-differential equation is true. □

## 4 The Laplace transform of ${m}_{\delta}(u)$ for $\delta =0$

Before deriving the Laplace transform of Gerber-Shiu function, one important step is to develop Lundberg’s fundamental equation and examine its properties.

To derive the expression of ${}_{m}^{{}_{\delta}\ast}(s)$, we need to know the number of roots in the right-half-plane of Lundberg’s fundamental equation (28). For $\delta =0$ and $\theta \ne 0$, by Theorem 1 of Klimenok [10], we can determine the number of roots to (28) with a positive real part. However, for $\delta >0$ and $\theta \ne 0$ we do not reach the conclusion about it.

**Lemma 1** *For* $\delta =0$ *and* $\theta \ne 0$, *Lundberg’s fundamental equation* (28) *has exactly* 4 *roots denoted by* $\{{\rho}_{i},i=1,\dots ,4\}$ *with* $Re({\rho}_{i})>0$ *and a* 5*th root* ${\rho}_{5}=0$.

*Proof* Let $z(s)=\frac{k-s}{k}$, ${C}_{k}=\{s:|z(s)|=1\}$.

*C*the limiting contour. We want to show

Because $\alpha (s)\beta (s)$ have 5 positive roots, by Theorem 1 of Klimenok [10], we can conclude that the number of solutions to (28) inside *C* is equal to 4. Finally, it is clear that the 5th root to (28) is ${\rho}_{5}=0$ with $\delta =0$. Hence, the conclusion is true. □

In the following sections, we only consider the case that the roots $\{{\rho}_{i},i=1,\dots ,5\}$ are distinct.

**Theorem 2**

*In the risk model introduced in Section*2,

*the Laplace transform of*${m}_{\delta}(u)$

*for*$\delta =0$

*denoted by*${m}_{0}^{\ast}(s)$

*is*

*where*

*and*

*Proof*First, from the properties of the Laplace transform, we have

In order to get (30), we must take the Laplace transform of both sides of (14).

where $q(s)$ is a polynomial of degree four or less, with coefficients in terms of *c*, ${\lambda}_{1}$, ${\lambda}_{2}$, and the value of ${m}_{0}(0)$ and its first 4 derivatives at $u=0$.

where ${q}_{1}(s)$ and ${q}_{2}(s)$ are polynomials of degree three or less, with coefficients in terms of *c*, ${\lambda}_{1}$, ${\lambda}_{2}$.

Thereby, we complete the proof. □

## 5 Exponential claim size

In this section, we assume that the individual claim size follows an exponential distribution with parameter *μ*.

In order to get the expression of the ruin probability, we consider a special case of the expected penalty function with $\omega (x,y)=1$ for all $x,y>0$.

**Corollary 1**

*In the risk model introduced in Section*2,

*if*$X\sim Exp(\mu )$

*and*$\omega (x,y)=1$,

*the Laplace transform of ruin probability denoted by*${\psi}^{\ast}(s)$

*is*

□

**Example 1**For the numerical results, we choose ${\lambda}_{1}=1$, ${\lambda}_{2}=1$, $\mu =1$ and the premium rate $c=1$. We can invert the Laplace transform in (35) leading to $\psi (u)$. In Table 1, the analytic expressions of $\psi (u)$ are provided for differential dependence parameters: $\theta =-1;-0.5;-0.25;0;0.25;0.5\text{and}1$, respectively.

**Analytic expressions of ruin probability**

| Expressions for the ruin probability $\mathit{\psi}\mathbf{(}\mathit{u}\mathbf{)}$ |
---|---|

−1 | 0.5201exp(−0.5214 |

−0.5 | 0.4665exp(−0.5661 |

−0.25 | 0.4372exp(−0.5910 |

0 | 0.3820exp(−0.6180 |

0.25 | 0.3557exp(−0.6474 |

0.5 | 0.3270exp(−0.6794 |

1 | 0.2600exp(−0.7534 |

## Author’s contributions

WY and HX work together to complete the composition of this paper. HX finished the draft and WY modified it.

## Notes

### Acknowledgements

This research was supported by the Preliminary Doctoral Program for Mathematics of Chongqing University of Technology. The authors would like to express their thanks to the referees for helpful suggestions.

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