**1. Introduction**

The classical Poisson model is attractive in the sense that the memoryless property of the exponential distribution makes calculations easy. Then the research was extended to ordinary Sparre-Andersen renewal risk models where the inter-claim times have other distributions than the exponential distribution. Dickson and Hipp [1, 2] considered the Erlang-2 distribution, Li and Garrido [3] the Erlang-n distribution, Gerber and Shiu [4] the generalized Erlang-n distribution (a sum of n independent exponential distributions with different scale parameters) and Li and Garrido [5] looked into the Coxian class distributions. One difficulty with these models is that we have to assume that a claim occurs at time 0, which is not the case in usual setting.

Albrecher and Teugels [6] considered modeling dependence with the use of an arbitrary copula. In a similar dependence model to Albrecher and Teugels as well, Asimit and Badescu [7] considered a constant force of interest and heavy tailed claim amounts.

Barges et al. [8] followed the idea of Albrecher and Teugels [6] and supposed that the dependence is introduced by a copula, the Farlie-Gumbel-Morgenstern (GGM) copula, between a claim inter-arrival time and its subsequent claim amount. ii. The *kth* random claim is given by *Xk*, and

*DOI: http://dx.doi.org/10.5772/intechopen.88699*

� � of *X*<sup>1</sup> exist.

*Z*0ðÞ¼ *t*

where *Z*0ðÞ¼ *t Zd*ðÞ¼ *t* 0 if *N*0ðÞ¼ *t Nd*ðÞ¼ *t* 0.

*N* X*o*ð Þ*t*

*k*¼1 *e*

We introduce a specific structure of dependence based on the

ð Þþ *v θFXi*

ð Þ *x ;* F*Ti* ð Þ ð Þ*v f Xi*

ð Þþ *v θ f Xi*

where *f Xi* and *f* <sup>τ</sup>*<sup>i</sup>* are the p.d.f.'s of *Xi* and τ*<sup>i</sup>* respectively.

ð Þ *x F*τ*<sup>i</sup>*

for ð Þ *u; v* ∈½ � 0*;* 1 ∗ ½ � 0*;* 1 so that the joint probability density function (p.d.f.) of

ð Þ *x f Ti*

It is often easier to calculate the moments of the random variable f g *Zd*ð Þ*t ; t* ≥0 than finding its distribution. If the probability generation function of f g *Zd*ð Þ*t ; t*≥0 or its moment generating function (mgf ) exists, it is possible to obtain the

With these hypotheses, we present in Section 3 recursive formula of the higher moments of this present value risk process, for a constant instantaneous

ð Þ *x f Ti* ð Þ*v*

ð Þ *x ; F*τ*<sup>i</sup>* ð Þ ð Þ*v*

ð Þ *x F*τ*<sup>i</sup>*

respectively. Recall that the density of the FGM copula is

for ð Þ *x; v* ∈ R<sup>þ</sup> ∗ R<sup>þ</sup> and where *FXi*ð Þ *x* and *F*τ*<sup>i</sup>*

*c FGM*

*FGM <sup>θ</sup> FXi*

ð Þ *x f Ti*

**3. Recursive expression for higher moments**

¼ *f Xi*

Farlie-Gumbel-Morgenstern (FGM) copula. The advantage of using the FGM copula and its generalizations lies in its mathematical manageability. The joint cumulative distribution function (c.d.f.) of *Xi* ð Þ *;* τ*<sup>i</sup>* , the *i*th claim and its

<sup>μ</sup>*<sup>k</sup>* <sup>¼</sup> *E X<sup>k</sup>* 1

case:

**2.1 The dependence**

occurrence time is

*FXi,* <sup>τ</sup>*<sup>i</sup>*

*Xi* ð Þ *;* τ*<sup>i</sup>* is

interest rate.

**273**

*f Xi,Ti*

ð Þ¼ *x; v c*

ð Þ¼ *x; v C FXi*

¼ *FXi*

• f g *Xk; k*∈ N are independent and identically distributed (i.i.d),

*Moments of the Discounted Aggregate Claims with Delay Inter-Occurrence Distribution…*

• f g *Xk;* τ*k; k*∈ N are mutually independent; and the higher moments,

iii. The discounted aggregate value at time *t* ¼ 0 of the claims recorded over the period 0½ � *; t* yields, respectively, for the ordinary and the delayed renewal

�<sup>δ</sup>*TkXk, Zd*ðÞ¼ *<sup>t</sup>*

*N* X*d*ð Þ*t*

*k*¼1 *e*

ð Þ*<sup>v</sup>* ð Þ <sup>1</sup>‐*FXi*

<sup>θ</sup> ð Þ¼ *<sup>u</sup>; <sup>v</sup>* <sup>1</sup> <sup>þ</sup> <sup>θ</sup>ð Þ <sup>1</sup>‐2*<sup>u</sup>* ð Þ <sup>1</sup>‐2*<sup>v</sup> ,* (3)

ð Þ*<sup>v</sup>* ð Þ <sup>1</sup>‐2F*Xi*

�<sup>δ</sup>*TkXk,* (1)

ð Þ *<sup>x</sup>* <sup>1</sup>‐*F*τ*<sup>i</sup>* ð Þ ð Þ*<sup>v</sup> ,* (2)

ð Þ *<sup>x</sup>* <sup>1</sup>‐2F*Ti* ð Þ ð Þ*<sup>v</sup> ,* (4)

ð Þ*v* are the marginals of *Xi* and τ*<sup>i</sup>*

Adékambi and Dziwa [9] and Adékambi [10] provide a direct point of extension but assuming that the claim counting process to follow an unknown general distribution in a framework of dependence with random force of interest to calculate the first two moments of the present value of aggregate random cash flows or random dividends.

The discounted aggregate sum has also been applied in many other fields. For example, it can be used in health cost modeling, see Govorun and Latouche [11], Adékambi [12], or in reliability, in civil engineering, see Van Noortwijk and Frangopol [13].

The delayed or modified renewal risk model solves this problem by assuming that the time until the first claim has a different distribution than the rest of the interclaim times. Not much research has been done for this model at this stage. Among the first works was Willmot [14] where a mixture of a "generalized equilibrium" distribution and an exponential distribution is considered for the distribution of the time until the first claim. Special cases of the model include the stationary renewal risk model and the delayed renewal risk model with the time until the first claim exponentially distributed. Our focus is to extend the work of Bargès et al. [8], Adékambi and Dziwa [9] and Adékambi [10] by allowing the counting process to follow a delay renewal risk process and thus derive a recursive formula of the moments of this subsequent Discounted Compound Delay Poisson Risk Value (DCDPRV).

For example, young performer companies typically have a high growth rate at the beginning, but as they mature their growth rate may decrease with the increasing scarcity of investment opportunities. That makes dividends dependent on the economic climate at the dividend occurrence time. Obviously the distribution of inter-dividends time in times of economic expansion and in times of economic crisis cannot be identically distributed. So it would be appropriate to use a delayed renewal model to model the distribution of the inter-dividend time. A delayed renewal process is just like an ordinary renewal process, except that the first arrival time is allowed to have a different distribution than the other inter-dividends times.

The chapter is organized as follows: In the second section, we present the model of the continuous time discounted compound delay-Poisson risk process that we use and give some notation. In Section 3, we present a general formula for all the moments of the DCDPRV process. A numerical example of the first two moments will then follow in Section 4.
