**2. The model**

We use the same model as the one in Bargès et al. [8], where the instantaneous interest rate δ is constant.

Define our risk model as follows:

	- the positive claim occurrence times are given by *Tk*,
	- the positive claim inter-arrival times are given by τ*<sup>k</sup>* ¼ *Tk* � *Tk*�<sup>1</sup>*, k*∈ N, and *T*<sup>0</sup> ¼ 0*:*
	- ð Þ τ*<sup>k</sup> <sup>k</sup>* <sup>≥</sup><sup>2</sup> � τ<sup>2</sup> are independent and identically distributed (i.i.d),

*Moments of the Discounted Aggregate Claims with Delay Inter-Occurrence Distribution… DOI: http://dx.doi.org/10.5772/intechopen.88699*

ii. The *kth* random claim is given by *Xk*, and


$$Z\_0(t) = \sum\_{k=1}^{N\_\sigma(t)} e^{-\delta T\_k} X\_{k\sigma} Z\_d(t) = \sum\_{k=1}^{N\_d(t)} e^{-\delta T\_k} X\_{k\sigma} \tag{1}$$

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

#### **2.1 The dependence**

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. 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

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

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

We use the same model as the one in Bargès et al. [8], where the instantaneous

i. The number of claims f g *N t*ð Þ*; t*≥0 and f g *Nd*ð Þ*t ; t*≥0 form, respectively, an ordinary and a delayed renewal process and, for *k*∈ N ¼ f g 1*;* 2*;* 3*;* … :

• ð Þ τ*<sup>k</sup> <sup>k</sup>* <sup>≥</sup><sup>2</sup> � τ<sup>2</sup> are independent and identically distributed (i.i.d),

• the positive claim inter-arrival times are given by τ*<sup>k</sup>* ¼ *Tk* � *Tk*�<sup>1</sup>*, k*∈ N,

• the positive claim occurrence times are given by *Tk*,

dividends.

Frangopol [13].

*Probability, Combinatorics and Control*

will then follow in Section 4.

interest rate δ is constant.

Define our risk model as follows:

and *T*<sup>0</sup> ¼ 0*:*

**2. The model**

**272**

We introduce a specific structure of dependence based on the 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 occurrence time is

$$\begin{split} F\_{X\_i, \mathsf{r}\_i}(\mathsf{x}, \boldsymbol{\nu}) &= \mathsf{C}(F\_{X\_i}(\boldsymbol{\mathcal{x}}), F\_{\mathsf{r}\_i}(\boldsymbol{\nu})) \\ &= F\_{X\_i}(\boldsymbol{\mathfrak{x}}) F\_{\mathsf{r}\_i}(\boldsymbol{\nu}) + \theta F\_{X\_i}(\boldsymbol{\mathfrak{x}}) F\_{\mathsf{r}\_i}(\boldsymbol{\nu}) \, (\mathtt{1} \cdot F\_{X\_i}(\boldsymbol{\mathfrak{x}})) \, (\mathtt{1} \cdot F\_{\mathsf{r}\_i}(\boldsymbol{\nu})), \end{split} \tag{2}$$

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>* ð Þ*v* are the marginals of *Xi* and τ*<sup>i</sup>* respectively. Recall that the density of the FGM copula is

$$
\mathcal{L}\_{\boldsymbol{\theta}}^{\rm FGM}(\boldsymbol{u}, \boldsymbol{v}) = \mathbf{1} + \boldsymbol{\theta}(\mathbf{1} \cdot \mathbf{2}\boldsymbol{u}) \, (\mathbf{1} \cdot \mathbf{2}\boldsymbol{v}), \tag{3}
$$

for ð Þ *u; v* ∈½ � 0*;* 1 ∗ ½ � 0*;* 1 so that the joint probability density function (p.d.f.) of *Xi* ð Þ *;* τ*<sup>i</sup>* is

$$\begin{split} f\_{X\_i, T\_i}(\mathbf{x}, \boldsymbol{\nu}) &= c\_{\theta}^{\mathrm{FGM}} \left( F\_{X\_i}(\mathbf{x}), \mathbf{F}\_{T\_i}(\boldsymbol{\nu}) \right) f\_{X\_i}(\mathbf{x}) f\_{T\_i}(\boldsymbol{\nu}) \\ &= f\_{X\_i}(\mathbf{x}) f\_{T\_i}(\boldsymbol{\nu}) + \theta f\_{X\_i}(\mathbf{x}) f\_{T\_i}(\boldsymbol{\nu}) (\mathbf{1} \cdot \mathbf{2} \mathbf{F}\_{X\_i}(\mathbf{x})) \left( \mathbf{1} \cdot \mathbf{2} \mathbf{F}\_{T\_i}(\boldsymbol{\nu}) \right)^{\mathsf{T}} \end{split} \tag{4}$$

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.

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

## **3. Recursive expression for higher moments**

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

corresponding distribution by inversion of its mgf. Since, there is relatively little research devoted to the study of the distribution of the discounted compound renewal sums. We could then think about another technique other than the one proposed by the above authors by studying the moments of f g *Zd*ð Þ*t ; t*≥0 .

We have

*E Xm*�*<sup>j</sup>*

We let,

π*m Zd*

**275**

μ0

<sup>j</sup>τ<sup>1</sup> <sup>¼</sup> *<sup>s</sup>* � � <sup>¼</sup>

∞ð

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

*xm*�*<sup>j</sup>*

<sup>¼</sup> *E Xm*�*<sup>j</sup>* � � <sup>þ</sup> <sup>θ</sup>

<sup>¼</sup> *E X<sup>m</sup>*�*<sup>j</sup>* � � <sup>þ</sup> <sup>θ</sup>

� θ ∞ð

0

*<sup>m</sup>*�*<sup>j</sup>* <sup>¼</sup> *E X*0*m*�*<sup>j</sup>* h i <sup>¼</sup>

, ∞ð

such that the above equation becomes

*X*<sup>1</sup> þ *e* �*δs Zo*ð Þ *<sup>t</sup>* � *<sup>s</sup>* � �*<sup>m</sup>* <sup>j</sup>τ<sup>1</sup> <sup>¼</sup> *<sup>s</sup>* � � � �

*E X<sup>m</sup>*�*<sup>j</sup>*

ðÞ¼ *<sup>t</sup> E Z<sup>m</sup>* ½ � ð Þ*<sup>t</sup>*

<sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�<sup>1</sup> *j*¼0

> þ ð*t*

0 *f* τ1 ð Þ*s e* �*m*δ*s* π*m Zo*

<sup>¼</sup> *EE e*�*δ<sup>s</sup>*

0 @ *m*

1 A ð*t*

0 *f* τ1 ð Þ*s e*

*j*

0

þ θ 1 � 2*F*τ<sup>1</sup> ð Þ ð Þ*s*

*<sup>f</sup> <sup>X</sup>*jτ1¼*<sup>s</sup>*

ð Þ *x dx*

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

∞ð

*xm*�*<sup>j</sup>*

*x<sup>m</sup>*�*<sup>j</sup>*

*xm*�*<sup>j</sup>* <sup>1</sup> � <sup>2</sup>*F*τ<sup>1</sup> ð Þ ð Þ*<sup>s</sup> <sup>f</sup> <sup>X</sup>*ð Þ *<sup>x</sup> dx*

∞ð

0

ð Þ *<sup>m</sup>* � *<sup>j</sup> xm*�*j*�<sup>1</sup>

�*m*δ*<sup>s</sup>* <sup>μ</sup>*<sup>m</sup>*�*<sup>j</sup>* <sup>þ</sup> <sup>θ</sup> <sup>1</sup> � <sup>2</sup>*F*<sup>τ</sup><sup>1</sup> ð Þ ð Þ*<sup>s</sup>* <sup>μ</sup><sup>0</sup>

0

∞ð

0

<sup>¼</sup> *E X<sup>m</sup>*�*<sup>j</sup>* � � <sup>1</sup> � <sup>θ</sup> <sup>1</sup> � <sup>2</sup>*F*τ<sup>1</sup> ð Þ ð Þ ð Þ*<sup>s</sup>*

∞ð

0

<sup>j</sup>τ<sup>1</sup> <sup>¼</sup> *<sup>s</sup>* � � <sup>¼</sup> <sup>μ</sup>*<sup>m</sup>*�*<sup>j</sup>* <sup>þ</sup> <sup>θ</sup> <sup>1</sup> � <sup>2</sup>*F*<sup>τ</sup><sup>1</sup> ð Þ ð Þ*<sup>s</sup>* <sup>μ</sup><sup>0</sup>

ð Þ *<sup>m</sup>* � *<sup>j</sup> xm*�*j*�<sup>1</sup>

ð Þ *t* � *s ds:*

*<sup>x</sup>m*�*<sup>j</sup>* <sup>1</sup> <sup>þ</sup> <sup>θ</sup>ð Þ <sup>1</sup> � <sup>2</sup>*FX*ð Þ *<sup>x</sup>* <sup>1</sup> � <sup>2</sup>*F*<sup>τ</sup><sup>1</sup> f g ð Þ ð Þ*<sup>s</sup> <sup>f</sup> <sup>X</sup>*ð Þ *<sup>x</sup> dx*

ð Þ 1 � 2*FX*ð Þ *x* 1 � 2*F*<sup>τ</sup><sup>1</sup> ð Þ ð Þ*s f <sup>X</sup>*ð Þ *x dx*

ð Þ 2 � 2*FX*ð Þ *x* 1 � 2*F*τ<sup>1</sup> ð Þ ð Þ*s f <sup>X</sup>*ð Þ *x dx*

(8)

(9)

ð Þ *<sup>m</sup>* � *<sup>j</sup> xm*�*<sup>j</sup>* <sup>1</sup> � *<sup>F</sup>*τ<sup>1</sup> ð Þ ð Þ*<sup>s</sup> dx:*

ð Þ 1 � *FX*ð Þ *x*

ð Þ <sup>1</sup> � *FX*ð Þ *<sup>x</sup> dx* <sup>¼</sup> *E X<sup>m</sup>*�*<sup>j</sup>* � � , <sup>∞</sup>

n o � � <sup>π</sup>

2 *dx*

*<sup>m</sup>*�*<sup>j</sup>* � μ*<sup>m</sup>*�*<sup>j</sup>*

*<sup>m</sup>*�*<sup>j</sup>* � μ*<sup>m</sup>*�*<sup>j</sup>*

� �*:* (10)

*j Zo*

ð Þ *t* � *s ds*

0

0

¼ ∞ð

#### **3.1 Delay renewal case**

The mathematical expectation of total claims plays an important role in the determination of the pure premium, in addition to giving a measure of the central tendency of its distribution. The moments centered at the average of order 2, 3 and 4 are the other moments usually considered because they usually give a good indication of the pace of distribution, and these give us respectively a measure of the dispersion of the distribution around its mean, a measure of the asymmetry and flattening of the distribution considered.

Moments, whether simple, joined or conditional, may eventually be used to construct approximations of the distribution of the DCDPRV.

#### **Theorem 3.1**

The Laplace transform of the *m*th moment of f g *Zd*ð Þ*t ; t* ≥0 is given by:

$$\begin{aligned} \bar{\pi}\_{Z\_d}^m(r) &= \left( 1 + \frac{\lambda\_2}{r + m\delta} + \frac{\lambda\_1 - \lambda\_2}{r + m\delta + \lambda\_1} \right) \bar{u}\_m(r) \\ &= \lambda\_1 \left( 1 + \frac{\lambda\_2}{r + m\delta} + \frac{\lambda\_1 - \lambda\_2}{r + m\delta + \lambda\_1} \right) \\ &\times \sum\_{j=0}^{m-1} \binom{m}{j} \left\{ \frac{\left( \mu\_{m-j} - \Theta \left( \mu\_{m-j}' - \mu\_{m-j} \right) \right)}{\lambda\_1 + m\delta + r} + \frac{2\Theta \left( \mu\_{m-j}' - \mu\_{m-j} \right)}{2\lambda\_1 + m\delta + r} \right\} \bar{\pi}\_{Z\_d}^j(r) \end{aligned} \tag{5}$$

where

$$\tilde{\pi}\_{\mathcal{Z}\_d}^m(r) = \ddot{u}\_m(r) + \frac{\lambda\_2}{m\delta} \ddot{u}\_m(r) \times L\_{\mathfrak{r}1}(m, \mathfrak{d}, r) + \frac{\lambda\_1 - \lambda\_2}{m\delta + \lambda\_1} \ddot{u}\_m \times L\_{\mathfrak{r}1}(m\delta + \lambda\_1, r). \tag{6}$$

#### *Proof*

Conditioning on the arrival of the first claim leads to

$$\begin{split} \pi\_{Z\_4}^m(t) &= E[Z^m(t)] \\ &= E\left[ E\left( \left( e^{-\delta t} X\_1 + e^{-\delta s} Z\_o(t-s) \right)^m | \pi\_1 = s \right) \right] \\ &= \sum\_{j=0}^{m-1} \binom{m}{j} \int\_0^t f\_{\pi\_1}(s) e^{-m\delta s} E\left[ X^{m-j} | \pi\_1 = s \right] \pi\_{Z\_o}^j(t-s) ds \\ &\quad + \int\_0^t f\_{\pi\_1}(s) e^{-m\delta s} \pi\_{Z\_o}^m(t-s) ds. \end{split} \tag{7}$$

*Moments of the Discounted Aggregate Claims with Delay Inter-Occurrence Distribution… DOI: http://dx.doi.org/10.5772/intechopen.88699*

We have

corresponding distribution by inversion of its mgf. Since, there is relatively little research devoted to the study of the distribution of the discounted compound renewal sums. We could then think about another technique other than the one proposed by the above authors by studying the moments of

The mathematical expectation of total claims plays an important role in the determination of the pure premium, in addition to giving a measure of the central tendency of its distribution. The moments centered at the average of order 2, 3 and 4 are the other moments usually considered because they usually give a good indication of the pace of distribution, and these give us respectively a measure of the dispersion of the distribution around its mean, a measure of the asymmetry and

Moments, whether simple, joined or conditional, may eventually be used to

*u*~*m*ð Þ*r*

*<sup>m</sup>*�*<sup>j</sup>* � μ*<sup>m</sup>*�*<sup>j</sup>*

þ

λ<sup>1</sup> � λ<sup>2</sup> *m*δ þ λ<sup>1</sup>

*E X<sup>m</sup>*�*<sup>j</sup>*

ð Þ *t* � *s ds:*

<sup>j</sup>τ<sup>1</sup> <sup>¼</sup> *<sup>s</sup>* � �<sup>π</sup> *<sup>j</sup>*

*Zo*

ð Þ *t* � *s ds*

2θ μ<sup>0</sup>

*<sup>m</sup>*�*<sup>j</sup>* � μ*<sup>m</sup>*�*<sup>j</sup>* � �

*u*~*<sup>m</sup>* � *L*τ<sup>1</sup> ð Þ *m*δ þ λ1*;r :* (6)

9 = ;~π *j Zo* ð Þ*r*

(5)

(7)

2λ<sup>1</sup> þ *m*δ þ *r*

� � � �

λ<sup>1</sup> þ *m*δ þ *r*

The Laplace transform of the *m*th moment of f g *Zd*ð Þ*t ; t* ≥0 is given by:

construct approximations of the distribution of the DCDPRV.

λ<sup>1</sup> � λ<sup>2</sup> *r* þ *m*δ þ λ<sup>1</sup>

μ*<sup>m</sup>*�*<sup>j</sup>* � θ μ<sup>0</sup>

*u*~*m*ð Þ� *r L*τ<sup>1</sup> ð Þþ *m;* δ*;r*

*X*<sup>1</sup> þ *e* �δ*s Zo*ð Þ *<sup>t</sup>* � *<sup>s</sup>* � �*<sup>m</sup>* <sup>j</sup>τ<sup>1</sup> <sup>¼</sup> *<sup>s</sup>* � � � �

Conditioning on the arrival of the first claim leads to

*m*

1 A ð*t*

0 *f* τ1 ð Þ*s e* �*m*δ*s*

*j*

λ<sup>1</sup> � λ<sup>2</sup> *r* þ *m*δ þ λ<sup>1</sup>

f g *Zd*ð Þ*t ; t*≥0 .

**3.1 Delay renewal case**

*Probability, Combinatorics and Control*

**Theorem 3.1**

ð Þ¼ *r* 1 þ

¼ λ<sup>1</sup> 1 þ

�X*<sup>m</sup>*�<sup>1</sup> *j*¼0

ð Þ¼ *r u*~*m*ð Þþ *r*

π*m Zd*

~π*m Zd*

where

~π*m Zd*

*Proof*

**274**

flattening of the distribution considered.

λ2 *<sup>r</sup>* <sup>þ</sup> *<sup>m</sup>*<sup>δ</sup> <sup>þ</sup>

*m*

0 @

*j*

λ2 *m*δ

ðÞ¼ *<sup>t</sup> E Z<sup>m</sup>* ½ � ð Þ*<sup>t</sup>*

<sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�<sup>1</sup> *j*¼0

> þ ð*t*

0 *f* τ1 ð Þ*s e* �*m*δ*s* π*m Zo*

<sup>¼</sup> *EE e*�δ*<sup>s</sup>*

0 @

λ2 *<sup>r</sup>* <sup>þ</sup> *<sup>m</sup>*<sup>δ</sup> <sup>þ</sup>

> 1 A

8 < :

� �

� �

$$\begin{split} E\left[X^{m^{-j}}|\tau\_{1}=s\right] &= \int\_{0}^{\infty} \mathbf{x}^{m^{-j}} f\_{X|\tau\_{1}=s}(\mathbf{x}) d\mathbf{x} \\ &= \int\_{0}^{\infty} \mathbf{x}^{m^{-j}} \{1 + \theta(1 - 2F\_{X}(\mathbf{x}))(1 - 2F\_{\tau\_{1}}(s))\} f\_{X}(\mathbf{x}) d\mathbf{x} \\ &= E\left[X^{m^{-j}}\right] + \theta \underbrace{\Big\|\mathbf{x}^{m^{-j}} (1 - 2F\_{X}(\mathbf{x}))(1 - 2F\_{\tau\_{1}}(s))f\_{X}(\mathbf{x}) d\mathbf{x}} \\ &= E\left[X^{m^{-j}}\right] + \theta \underbrace{\Big\|\mathbf{x}^{m^{-j}} (2 - 2F\_{X}(\mathbf{x}))(1 - 2F\_{\tau\_{1}}(s))f\_{X}(\mathbf{x}) d\mathbf{x}} \\ &\qquad - \theta \Big\|\mathbf{x}^{m^{-j}} (1 - 2F\_{\tau\_{1}}(s))f\_{X}(\mathbf{x}) d\mathbf{x} \\ &= E\left[X^{m^{-j}}\right] (1 - \theta(1 - 2F\_{\tau\_{1}}(s))) \\ &\qquad + \theta(1 - 2F\_{\tau\_{1}}(s)) \underbrace{\Big\|\mathbf{x}^{m^{-j}} (1 - \theta)\mathbf{x}^{m^{-j}} (1 - F\_{\tau\_{1}}(s)) d\mathbf{x} .\end{split} \tag{8.10}$$

We let,

$$\begin{split} \mathfrak{u}'\_{m-j} &= E\left[\mathbf{X}'^{m-j}\right] = \int\_0^\infty (m-j)\mathbf{x}^{m-j-1} (\mathbf{1} - F\_X(\mathbf{x}))^2 d\mathbf{x} \\ &< \int\_0^\infty (m-j)\mathbf{x}^{m-j-1} (\mathbf{1} - F\_X(\mathbf{x})) d\mathbf{x} = E\left[\mathbf{X}'^{m-j}\right] < \infty \end{split} \tag{9}$$

such that the above equation becomes

$$E\left[\mathbf{X}^{m-j}|\tau\_1=s\right] = \mu\_{m-j} + \Theta(\mathbf{1} - 2F\_{\tau\_1}(s))\left(\mu\_{m-j}' - \mu\_{m-j}\right). \tag{10}$$

$$\begin{split} \pi\_{Z\_{4}}^{m}(t) &= E[Z^{m}(t)] \\ &= E\left[E\left[\left(e^{-\delta t}X\_{1} + e^{-\delta t}Z\_{o}(t-s)\right)^{m}|\pi\_{1}=s\right]\right] \\ &= \sum\_{j=0}^{m-1} \binom{m}{j} \left\{f\_{\pi\_{1}}(s)e^{-m\delta s} \left\{\mu\_{m-j} + \Theta(1-2F\_{\pi\_{1}}(s))\left(\mu\_{m-j}^{\prime}-\mu\_{m-j}\right)\right\}\pi\_{Z\_{s}}^{j}(t-s)ds \\ &\quad + \int\_{0}^{t} f\_{\pi\_{1}}(s)e^{-m\delta s}\pi\_{Z\_{s}}^{m}(t-s)ds. \end{split}$$

$$\begin{aligned} \text{Let us } \int\_{\delta}^{\delta} f\_{\tau\_{1}}(s)e^{-ms^{\delta}} ds &= H\_{\delta}(t), \int\_{0}^{\delta} f\_{\tau\_{1}}(s)e^{-ms^{\delta}} ds = I\_{\delta}(t) \text{ then} \\ \pi\_{\varpi\_{\omega}}^{m}(t) &= \sum\_{j=0}^{m-1} \binom{m}{j} \Big\{f\_{\tau\_{1}}(s)e^{-ms^{\delta}} \Big\{\left[\mu\_{m-j} + \theta(1 - 2\varpi\_{\tau\_{1}}(s)) \left(\mu\_{m-j}^{\prime} - \mu\_{m-j} \right) \right] \pi\_{Z\_{\omega}}^{j}(t - s) ds \\ &+ H\_{m\delta} \ast \pi\_{Z\_{\omega}(.)}^{m} \\ &= u\_{m} + H\_{m\delta} \ast \Big\{u\_{m} + I\_{m\delta} \ast \pi\_{Z\_{\omega}(.)}^{m} \Big\} \\ &= u\_{m} + H\_{m\delta} \ast u\_{m} + H\_{m\delta} \ast I\_{m\delta} \ast \pi\_{Z\_{\omega}(.)}^{m} \\ &= u\_{m} + H\_{m\delta} \ast u\_{m} + H\_{m\delta} \ast I\_{m\delta} \ast \Big\{u\_{m} + I\_{m\delta} \ast \pi\_{Z\_{\omega}(.)}^{m} \Big\} \\ &= u\_{m} + H\_{m\delta} \ast u\_{m} + u\_{m} \ast \sum\_{k=1}^{\infty} H\_{m\delta} \ast I\_{k}^{\delta\_{m}}(t) = u\_{m} + u\_{m} \ast \sum\_{k=0}^{\infty} H\_{m\delta} \ast I\_{m}^{\delta\_{m}}(t) \\ &= u\_{m} + \Big[\mu\_{m}(t - s)e^{-ms^{\delta}} dm\_{d}(s), \end{aligned} \tag{11}$$

But,

*um*ðÞ¼ *<sup>t</sup>* <sup>X</sup>*m*�<sup>1</sup>

*j*¼0

<sup>¼</sup> <sup>X</sup>*m*�<sup>1</sup> *j*¼0

¼

þ 2θ

*u*~*m*ð Þ¼ *r* λ<sup>1</sup>

λ2 *<sup>r</sup>* <sup>þ</sup> *<sup>m</sup>*<sup>δ</sup> <sup>þ</sup>

¼ λ<sup>1</sup> 1 þ

~π*m Zo* ð Þ¼ *r*

**277**

*m*

1 A ð*t*

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

0 *f* τ1 ð Þ*s e*

0 λ1*e* �λ1*s e*

λ<sup>1</sup> þ *m*δ

2λ<sup>1</sup> þ *m*δ

*<sup>j</sup>* � μ*<sup>j</sup>* � � � �

> X*m*�1 *j*¼0

Then the Laplace transform of *um*ð Þ*t* , at *r*, will give:

� � μ*<sup>j</sup>* � θ μ<sup>0</sup>

ð Þ¼ *r* 1 þ

*j*¼0

þ

� � <sup>λ</sup>2ð Þ *<sup>r</sup>* <sup>þ</sup> *<sup>δ</sup><sup>m</sup>*

ð Þ *r* þ *δm* þ λ<sup>2</sup> ð Þ *r* þ *δm* þ 2λ<sup>2</sup>

~π*m <sup>Z</sup>*<sup>0</sup> ð Þ*r*

8 < :

�*m*δ*<sup>s</sup>* <sup>μ</sup>*<sup>j</sup>* <sup>þ</sup> <sup>θ</sup> <sup>1</sup> � <sup>2</sup>*F*<sup>τ</sup><sup>1</sup> ð Þ ð Þ*<sup>s</sup>* <sup>μ</sup><sup>0</sup>

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

�*m*δ*<sup>s</sup>* <sup>μ</sup>*<sup>j</sup>* <sup>þ</sup> <sup>θ</sup> <sup>2</sup>*<sup>e</sup>*

*m*

1 A ð*t*

0

*<sup>j</sup>* � μ*<sup>j</sup>* � � � �

ð Þ 2λ<sup>1</sup> þ *m*δ *e*

0 @

*j*

0

λ<sup>1</sup> þ *m*δ þ *r*

λ2 *<sup>r</sup>* <sup>þ</sup> *<sup>m</sup>*<sup>δ</sup> <sup>þ</sup>

*m*

1 A

8 < :

Solving the above equation for the ordinary case, where ð Þ τ<sup>2</sup> *<sup>k</sup>*≥<sup>2</sup> � τ2, we have:

*r r*ð Þ þ *δm* þ λ<sup>2</sup> ð Þ *r* þ *δm* þ 2λ<sup>2</sup>

0 @

*j*

λ2 ð Þ *r* þ *δm* þ λ<sup>2</sup>

X*m*�1 *j*¼0

*m*

1 A ð*t*

0 @

*j*

n o � � <sup>π</sup>

�λ1*<sup>s</sup>* � <sup>1</sup> � � <sup>μ</sup><sup>0</sup>

ð Þ λ<sup>1</sup> þ *m*δ *e*

þ

λ<sup>1</sup> � λ<sup>2</sup> *r* þ *m*δ þ λ<sup>1</sup> � �*u*~*m*ð Þ*<sup>r</sup>*

μ*<sup>j</sup>* � θ μ<sup>0</sup>

X*m*�1 *k*¼1 *Ck*

> X*m*�1 *k*¼1 *Ck <sup>m</sup>* μ<sup>0</sup> *<sup>k</sup>* � μ*<sup>k</sup>* � �~πð Þ *<sup>m</sup>*�*<sup>k</sup>*

n o � � <sup>π</sup>

*<sup>j</sup>* � μ*<sup>j</sup>*

*<sup>j</sup>* � μ*<sup>j</sup>*

�ð Þ λ1þ*m*δ *s* π *m*�*j Zo* ð Þ *t* � *s ds*

> π *m*�*j Zo* ð Þ *t* � *s ds*

*<sup>j</sup>* � μ*<sup>j</sup>* � � 9 = ;~π*<sup>m</sup>*�*<sup>j</sup>*

> 2θ μ<sup>0</sup> *<sup>j</sup>* � μ*<sup>j</sup>* � �

2λ<sup>1</sup> þ *m*δ þ *r*

*Zo* ð Þ*r*

*Zo* ð Þ*r* (16)

9 = ;~π*<sup>m</sup>*�*<sup>j</sup> Zo* ð Þ*r*

(17)

(18)

2λ<sup>1</sup> þ *m*δ þ *r*

*<sup>j</sup>* � μ*<sup>j</sup>* � � � �

> *<sup>m</sup>* <sup>μ</sup>*k*~πð Þ *<sup>m</sup>*�*<sup>k</sup> Zo* ð Þ*r*

<sup>λ</sup><sup>1</sup> <sup>þ</sup> *<sup>m</sup>*<sup>δ</sup> <sup>þ</sup> *<sup>r</sup>* <sup>þ</sup>

�ð Þ 2λ1þ*m*δ *s*

2θ μ<sup>0</sup>

*m*�*j Zo* ð Þ *t* � *s ds*

*m*�*j Zo* ð Þ *t* � *s ds*

(15)

0 @

*j*

*m*

1 A ð*t*

0 @

*j*

λ<sup>1</sup> μ*<sup>j</sup>* � θ μ<sup>0</sup>

λ<sup>1</sup> μ<sup>0</sup> *<sup>j</sup>* � μ*<sup>j</sup>* � �

> X*m*�1 *j*¼0

*m j*

Substituting Eq. (14) into Eq. (13), we have:

~π*m Zd*

λ2μ*<sup>m</sup> r r*ð Þ þ *δm* þ λ<sup>2</sup>

*<sup>m</sup>* � μ*<sup>m</sup>*

λ2 ð Þ *r* þ *δm* þ λ<sup>2</sup>

<sup>þ</sup> <sup>θ</sup> <sup>λ</sup>2ð Þ *<sup>r</sup>* <sup>þ</sup> *<sup>δ</sup><sup>m</sup>*

þ θ μ<sup>0</sup>

þ

� �X*<sup>m</sup>*�<sup>1</sup>

λ<sup>1</sup> � λ<sup>2</sup> *r* þ *m*δ þ λ<sup>1</sup>

$$\begin{aligned} \text{where } u\_m(t) &= \sum\_{j=0}^{m-1} \binom{m}{j} \int\_0^t f\_{\tau\_1}(s) e^{-m\delta s} \left\{ \mu\_j + \Theta(1 - 2F\_{\tau\_1}(s)) \left( \mu\_j' - \mu\_j \right) \right\} \pi\_{Z\_o}^{m-j} \\ \text{or } \begin{array}{ccccc} (t-s)d\text{s.} & & & \\ \dots & \dots & \dots & \dots & \dots & \dots \end{array} \end{aligned}$$

We consider the case where the canonical random variable τ<sup>2</sup> has an Exponential distribution with parameter λ<sup>2</sup> . 0 and τ<sup>1</sup> has an Exponential distribution with parameter λ<sup>1</sup> . 0.

That is, we have:

$$f\_{\tau\_1}(t) = \lambda\_1 e^{-\lambda\_1 t}, f\_{\tau\_2}(t) = \lambda\_2 e^{-\lambda\_2 t}, L\_{\tau\_1}(\lambda\_1, s) = \int\_0^\infty e^{-u} f\_{\tau\_1}(v) dv = \left(\frac{\lambda\_1}{\lambda\_1 + s}\right), L\_{\tau\_2}(\lambda\_2, s) = \left(\frac{\lambda\_2}{\lambda\_2 + s}\right).$$

$$m\_d(t) = \lambda\_2 t + \frac{\lambda\_1 - \lambda\_2}{\lambda\_1} \left(1 - e^{\lambda\_1 t}\right) \tag{12}$$

The *m*th moment of *Zd*ð Þ*t* is then given by,

$$\begin{aligned} \pi\_{\Sigma\_{\mathsf{L}}}^{m}(t) &= u\_{m} + \int\_{0}^{t} u\_{m}(t-s)e^{-m\delta s}dm\_{d}(s) \\ &= u\_{m} + \lambda\_{2} \begin{bmatrix} t \\ u\_{m}(t-s)e^{-m\delta s}d(s) + (\lambda\_{1} - \lambda\_{2}) \end{bmatrix} \Big| u\_{m}(t-s)e^{-(m\delta + \lambda\_{1})s}d(s) \\ &= u\_{m} + \frac{\lambda\_{2}}{m\delta} \begin{bmatrix} u\_{m}(t-s)m\delta s e^{-m\delta s}d(s) + \frac{\lambda\_{1} - \lambda\_{2}}{m\delta + \lambda\_{1}} \end{bmatrix} \Big| u\_{m}(t-s)(m\delta + \lambda\_{1})e^{-(m\delta + \lambda\_{1})s}d(s) \end{aligned} \tag{13}$$

Taking the Laplace transform of the above equation, we get:

$$\tilde{\pi}\_{\mathcal{Z}\_d}^m(r) = \tilde{u}\_m(r) + \frac{\lambda\_2}{m\delta} \tilde{u}\_m(r) \times L\_{\text{tr}}(m\delta, r) + \frac{\lambda\_1 - \lambda\_2}{m\delta + \lambda\_1} \tilde{u}\_m \times L\_{\text{tr}}(m\delta + \lambda\_1, r) \tag{14}$$

*Moments of the Discounted Aggregate Claims with Delay Inter-Occurrence Distribution… DOI: http://dx.doi.org/10.5772/intechopen.88699*

But,

Let us Ð*<sup>t</sup>*

ðÞ¼ *<sup>t</sup>* <sup>X</sup>*m*�<sup>1</sup> *j*¼0

π*m Zd* <sup>0</sup> *f* <sup>τ</sup><sup>1</sup>

¼ *um* þ

where *um*ðÞ¼ *<sup>t</sup>* <sup>P</sup>*<sup>m</sup>*�<sup>1</sup>

ð Þ *t* � *s ds*.

*f* τ1

π*m Zd*

parameter λ<sup>1</sup> . 0. That is, we have:

ðÞ¼ *t um* þ

¼ *um* þ λ<sup>2</sup>

¼ *um* þ

~π*m Zd*

**276**

ð*t*

0

ð*t*

0

ð*t*

0

λ2 *m*δ

ð Þ¼ *r u*~*m*ð Þþ *r*

�λ1*t , f* <sup>τ</sup><sup>2</sup>

ðÞ¼ *t* λ1*e*

ð Þ*<sup>s</sup> <sup>e</sup>*�*m*δ*<sup>s</sup>*

*m*

*Probability, Combinatorics and Control*

!ð*<sup>t</sup>*

*j*

<sup>þ</sup> *Hm*<sup>δ</sup> <sup>∗</sup> <sup>π</sup>*<sup>m</sup>*

0 *f* τ1 ð Þ*s e*

*Zo*ð Þ*:* <sup>¼</sup> *um* <sup>þ</sup> *Hm*<sup>δ</sup> <sup>∗</sup> *um* <sup>þ</sup> *Im*<sup>δ</sup> <sup>∗</sup> <sup>π</sup>*<sup>m</sup>*

¼ *um* þ *Hm*<sup>δ</sup> ∗ *um* þ *um* ∗

*um*ð Þ *t* � *s e*

*j*¼0

ðÞ¼ *t* λ2*e*

*um*ð Þ *t* � *s e*

*um*ð Þ *t* � *s e*

λ2 *m*δ

*um*ð Þ *t* � *s m*δ*e*

�λ2*t*

The *m*th moment of *Zd*ð Þ*t* is then given by,

�*m*δ*s*

�*m*δ*s*

ð*t*

0

<sup>¼</sup> *um* <sup>þ</sup> *Hm*<sup>δ</sup> <sup>∗</sup> *um* <sup>þ</sup> *Hm*<sup>δ</sup> <sup>∗</sup> *Im*<sup>δ</sup> <sup>∗</sup> <sup>π</sup>*<sup>m</sup>*

*ds* ¼ *H*δð Þ*t* ,

Ð*t* <sup>0</sup> *f* <sup>τ</sup><sup>2</sup>

*Zo*ð Þ*:*

n o

<sup>¼</sup> *um* <sup>þ</sup> *Hm*<sup>δ</sup> <sup>∗</sup> *um* <sup>þ</sup> *Hm*<sup>δ</sup> <sup>∗</sup> *Im*<sup>δ</sup> <sup>∗</sup> *um* <sup>þ</sup> *Im*<sup>δ</sup> <sup>∗</sup> <sup>π</sup>*<sup>m</sup>*

�*m*δ*s*

*m j* � � <sup>Ð</sup>*<sup>t</sup>*

X∞ *k*¼1

*dmd*ð Þ*s ,*

<sup>0</sup> *f* <sup>τ</sup><sup>1</sup>

*, L*<sup>τ</sup><sup>1</sup> ð Þ¼ λ1*; s*

*md*ðÞ¼ *t* λ2*t* þ

*dmd*ð Þ*s*

�*m*δ*s*

Taking the Laplace transform of the above equation, we get:

*u*~*m*ð Þ� *r L*<sup>τ</sup><sup>1</sup> ð Þþ *m*δ*;r*

ð Þ*<sup>s</sup> <sup>e</sup>*�*m*δ*<sup>s</sup>*

�*m*δ*<sup>s</sup>* <sup>μ</sup>*m*�*<sup>j</sup>* <sup>þ</sup> <sup>θ</sup> <sup>1</sup> � <sup>2</sup>*F*<sup>τ</sup><sup>1</sup> ð Þ ð Þ*<sup>s</sup>* <sup>μ</sup><sup>0</sup>

*Zo*ð Þ*:*

*Hm*<sup>δ</sup> ∗ *I*

*ds* ¼ *I*δð Þ*t* then

*Zo*ð Þ*:*

<sup>δ</sup> <sup>∗</sup> ð Þ*<sup>k</sup> <sup>m</sup>* ðÞ¼ *<sup>t</sup> um* <sup>þ</sup> *um* <sup>∗</sup>

ð Þ*<sup>s</sup> <sup>e</sup>*�*m*δ*<sup>s</sup>* <sup>μ</sup>*<sup>j</sup>* <sup>þ</sup> <sup>θ</sup> <sup>1</sup> � <sup>2</sup>*F*τ<sup>1</sup> ð Þ ð Þ*<sup>s</sup>* <sup>μ</sup><sup>0</sup>

ð Þ*<sup>v</sup> dv* <sup>¼</sup> <sup>λ</sup><sup>1</sup>

1 � *e*

We consider the case where the canonical random variable τ<sup>2</sup> has an Exponential

λ<sup>1</sup> � λ<sup>2</sup> λ1

ð*t*

*um*ð Þ *t* � *s e*

ð*t*

0

λ<sup>1</sup> � λ<sup>2</sup> *m*δ þ λ<sup>1</sup>

0

distribution with parameter λ<sup>2</sup> . 0 and τ<sup>1</sup> has an Exponential distribution with

∞ð

0 *e* �*svf* <sup>τ</sup><sup>1</sup>

*d s*ðÞþ ð Þ λ<sup>1</sup> � λ<sup>2</sup>

*d s*ð Þþ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *m*δ þ λ<sup>1</sup>

n o � �

λ<sup>1</sup> þ *s* � �

<sup>λ</sup>1*<sup>t</sup>* � � (12)

�ð Þ *m*δþλ<sup>1</sup> *s*

*um*ð Þ *t* � *s* ð Þ *m*δ þ λ<sup>1</sup> *e*

*d s*ð Þ

*u*~*<sup>m</sup>* � *L*<sup>τ</sup><sup>1</sup> ð Þ *m*δ þ λ1*;r* (14)

�ð Þ *m*δþλ<sup>1</sup> *s*

*d s*ð Þ

(13)

n o

n o � �

*<sup>m</sup>*�*<sup>j</sup>* � μ*m*�*<sup>j</sup>*

X∞ *k*¼0

*Hm*<sup>δ</sup> ∗ *I*

*<sup>j</sup>* � μ*<sup>j</sup>*

π *m*�*j Zo*

*, L*<sup>τ</sup><sup>2</sup> ð Þ¼ <sup>λ</sup>2*; <sup>s</sup>* <sup>λ</sup><sup>2</sup>

π *j Zo*

ð Þ *t* � *s ds*

<sup>δ</sup> <sup>∗</sup> ð Þ*<sup>k</sup> <sup>m</sup>* ð Þ*<sup>t</sup>*

(11)

λ<sup>2</sup> þ *s* � �

*:*

$$\begin{split} u\_m(t) &= \sum\_{j=0}^{m-1} \binom{m}{j} \left\{ t\_{\tau\_1}(s) e^{-m\delta s} \left\{ \mu\_j + \Theta(1 - 2F\_{\tau\_1}(s)) \left( \mu\_j' - \mu\_j \right) \right\} \pi\_{Z\_o}^{m-j}(t-s) ds \right. \\ &= \sum\_{j=0}^{m-1} \binom{m}{j} \left\{ \lambda\_1 e^{-\lambda\_1 t} e^{-m\delta s} \left\{ \mu\_j + \Theta(2e^{-\lambda\_1 t} - 1) \left( \mu\_j' - \mu\_j \right) \right\} \pi\_{Z\_o}^{m-j}(t-s) ds \right. \\ &= \frac{\lambda\_1 \left( \mu\_j - \Theta\left( \mu\_j' - \mu\_j \right) \right)}{\lambda\_1 + m\delta} \sum\_{j=0}^{m-1} \binom{m}{j} \left\{ (\lambda\_1 + m\delta) e^{-(\lambda\_1 + m\delta)s} \pi\_{Z\_o}^{m-j}(t-s) ds \right. \\ &\left. \tag{15} \right. \\ &\left. + 2\Phi \frac{\lambda\_1 \left( \mu\_j' - \mu\_j \right)}{2\lambda\_1 + m\delta} \sum\_{j=0}^{m-1} \binom{m}{j} \int\_0^t (2\lambda\_1 + m\delta) e^{-(2\lambda\_1 + m\delta)s} \pi\_{Z\_o}^{m-j}(t-s) ds \right. \end{split} \tag{16}$$

Then the Laplace transform of *um*ð Þ*t* , at *r*, will give:

$$\bar{u}\_m(r) = \lambda\_1 \sum\_{j=0}^{m-1} \binom{m}{j} \left\{ \frac{\left(\mu\_j - \Theta\left(\mu\_j' - \mu\_j\right)\right)}{\lambda\_1 + m\delta + r} + \frac{2\Theta\left(\mu\_j' - \mu\_j\right)}{2\lambda\_1 + m\delta + r} \right\} \hat{\pi}\_{Z\_\nu}^{m-j}(r) \tag{16}$$

Substituting Eq. (14) into Eq. (13), we have:

$$\ddot{\pi}\_{Z\_d}^m(r) = \left(\mathbf{1} + \frac{\lambda\_2}{r + m\delta} + \frac{\lambda\_1 - \lambda\_2}{r + m\delta + \lambda\_1}\right)\ddot{u}\_m(r)$$

$$= \lambda\_4 \left(\mathbf{1} + \frac{\lambda\_2}{r + m\delta} + \frac{\lambda\_1 - \lambda\_2}{r + m\delta + \lambda\_1}\right)\sum\_{j=0}^{m-1} \binom{m}{j} \left\{\frac{\left(\mu\_j - \theta\left(\mu\_j' - \mu\_j\right)\right)}{\lambda\_1 + m\delta + r} + \frac{2\theta\left(\mu\_j' - \mu\_j\right)}{2\lambda\_1 + m\delta + r}\right\}\ddot{\pi}\_{Z\_\*}^{m-j}(r) \tag{17}$$

Solving the above equation for the ordinary case, where ð Þ τ<sup>2</sup> *<sup>k</sup>*≥<sup>2</sup> � τ2, we have:

$$\begin{split} \tilde{\pi}\_{Z\_{\nu}}^{m}(r) &= \frac{\lambda\_{2}\mu\_{m}}{r(r+\delta m+\lambda\_{2})} + \frac{\lambda\_{2}}{(r+\delta m+\lambda\_{2})} \sum\_{k=1}^{m-1} C\_{m}^{k} \ln \tilde{\pi}\_{Z\_{\nu}}^{(m-k)}(r) \\\\ &+ \Theta \left(\mu\_{m}^{\prime} - \mu\_{m}\right) \frac{\lambda\_{2}(r+\delta m)}{r(r+\delta m+\lambda\_{2})(r+\delta m+2\lambda\_{2})} \\\\ &+ \Theta \frac{\lambda\_{2}(r+\delta m)}{(r+\delta m+\lambda\_{2})(r+\delta m+2\lambda\_{2})} \sum\_{k=1}^{m-1} C\_{m}^{k} \left(\mu\_{k}^{\prime} - \mu\_{k}\right) \tilde{\pi}\_{Z\_{\nu}}^{(m-k)}(r) \\\\ &+ \frac{\lambda\_{2}}{(r+\delta m+\lambda\_{2})} \tilde{\pi}\_{Z\_{0}}^{m}(r) \end{split} \tag{18}$$

Rearranging the above equation, we will get

$$\begin{split} \bar{\pi}\_{Z\_{o}}^{m}(r) &= \frac{\lambda\_{2}\mu\_{m}}{r(r+\delta m)} + \frac{\lambda\_{2}}{(r+\delta m)} \sum\_{k=1}^{m-1} \mathsf{C}\_{m}^{k} \mu\_{k} \bar{\pi}\_{Z\_{s}}^{(m-k)}(r) \\ &+ \theta \frac{\lambda\_{2} \left(\mu\_{m}^{\prime} - \mu\_{m}\right)}{r(r+\delta m+2\lambda\_{2})} + \theta \frac{\lambda\_{2}}{(r+\delta m+2\lambda\_{2})} \sum\_{k=1}^{m-1} \mathsf{C}\_{m}^{k} \left(\mu\_{k}^{\prime} - \mu\_{k}\right) \bar{\pi}\_{Z\_{s}}^{(m-k)}(r) \end{split} \tag{19}$$

*λ*1*λ*<sup>2</sup> rð Þ *r* þ *δ* ð Þ *r* þ *δ* þ *λ*<sup>1</sup>

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

λ2 *r r*ð Þ þ δ þ λ<sup>1</sup> ð Þ *r* þ δ þ 2λ<sup>2</sup>

*r* þ δ *r r*ð Þ þ δ þ λ<sup>1</sup> ð Þ *r* þ δ þ 2λ<sup>1</sup>

> λ1 ð Þ δ þ λ<sup>1</sup>

<sup>1</sup> � μ<sup>1</sup> � �

> <sup>1</sup> � μ<sup>1</sup> � �

λ1λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup>

Rearranging the above equation, will give

<sup>1</sup> � μ<sup>1</sup>

1 δ þ 2λ<sup>2</sup>

> 1 δ þ 2λ<sup>1</sup>

<sup>1</sup> � μ<sup>1</sup> � � λ<sup>1</sup> � λ<sup>2</sup>

þ θ μ<sup>0</sup>

þ θλ<sup>1</sup> μ<sup>0</sup>

þ μ<sup>1</sup>

π*Zd* ðÞ¼ *t* θ*λ*<sup>1</sup> μ<sup>0</sup>

þ θ*λ*<sup>1</sup> μ<sup>0</sup>

� θ*λ*<sup>1</sup>

� 2θ*λ*<sup>1</sup>

**279**

*:* 1 *r*

8 >>>>><

>>>>>:

~π*Zd* ð Þ¼ *r* μ<sup>1</sup>

<sup>¼</sup> *<sup>λ</sup>*1*λ*<sup>2</sup> *δ δ*ð Þ þ *λ*<sup>1</sup> *:* 1 *r* þ

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

<sup>¼</sup> <sup>λ</sup><sup>2</sup>

þ

ð Þ δ þ λ<sup>1</sup> ð Þ δ þ 2λ<sup>2</sup>

λ2 λ<sup>1</sup> � 2λ<sup>2</sup>

<sup>¼</sup> <sup>δ</sup>

1 δ þ λ<sup>1</sup>

þ

� �

λ2 λ<sup>1</sup> � 2λ<sup>2</sup>

> 1 δ þ λ<sup>1</sup>

> > λ2 ð Þ δ þ λ<sup>1</sup>

ð Þ δ þ λ<sup>1</sup> ð Þ δ þ 2λ<sup>2</sup>

λ<sup>1</sup> � 2λ<sup>2</sup> � �

� � 1

� � 1

μ0 <sup>1</sup> � μ<sup>1</sup> � � 1

*r* þ δ þ 2λ<sup>1</sup>

� <sup>λ</sup><sup>1</sup> ð Þ δ þ λ<sup>1</sup>

þ

8 >>>><

>>>>:

*:* 1 *r* þ

� � λ<sup>2</sup> þ δ

λ2 λ<sup>1</sup> � 2λ<sup>2</sup>

> μ0 <sup>1</sup> � μ<sup>1</sup> � � 1

þ

Substituting Eqs. (24), (25), (26) and (27) into Eq. (23), yields:

ð Þ δ þ λ<sup>1</sup> ð Þ δ þ 2λ<sup>1</sup>

*:* <sup>1</sup> ð Þ *r* þ δ þ λ<sup>1</sup>

1 δ þ λ<sup>1</sup>

1 *r* þ δ þ λ<sup>1</sup>

� �

*:* <sup>1</sup> ð Þ *r* þ δ þ λ<sup>1</sup>

þ

� ð Þ λ<sup>1</sup> � λ<sup>2</sup> μ<sup>1</sup>

*r* þ δ þ 2λ<sup>2</sup>

� λ2 <sup>δ</sup> <sup>μ</sup><sup>1</sup>

*λ*2 ð Þ *δ* þ *λ*<sup>1</sup> *:* <sup>1</sup> ð Þ *r* þ *δ* þ *λ*<sup>1</sup>

1 *r* þ δ þ λ<sup>1</sup>

> � <sup>2</sup> δ þ 2λ<sup>1</sup>

1 *r*

λ2 ð Þ δ þ λ<sup>1</sup> ð Þ δ þ 2λ<sup>2</sup>

> 1 *r* þ δ þ λ<sup>1</sup>

> > � <sup>2</sup> δ þ 2λ<sup>1</sup>

λ1ð Þ λ<sup>2</sup> þ δ δ δð Þ þ λ<sup>1</sup>

δ ð Þ δ þ λ<sup>1</sup> ð Þ δ þ 2λ<sup>1</sup>

1 *r*

1 δ þ λ<sup>1</sup>

1 *r* þ δ þ λ<sup>1</sup> � *λ*2 *<sup>δ</sup> :* <sup>1</sup>

� <sup>1</sup> δ þ 2λ<sup>2</sup>

> 1 *r* þ δ þ 2λ<sup>1</sup>

> > 1 *r* þ δ þ 2λ<sup>2</sup>

> > > 9 >>>>=

> > > >>>>;

� �

1 *r*

� <sup>1</sup> δ þ 2λ<sup>2</sup>

� �

1 *r*

� λ2 <sup>δ</sup> *:* <sup>1</sup> ð Þ *r* þ δ

μ1

δ þ λ<sup>1</sup>

1 *r* þ δ *r*

1 *r* þ δ þ λ<sup>1</sup>

1 *r* þ δ þ 2λ<sup>1</sup>

ð Þ *<sup>r</sup>* <sup>þ</sup> *<sup>δ</sup>* (25)

1 *r* þ δ þ 2λ<sup>2</sup>

(26)

(27)

9 >>>>>=

>>>>>;

(28)

(29)

**Corollary 3.1**

The first moment of f g *Zd*ð Þ*t ; t*≥0 is given by:

$$\begin{split} \pi\_{Z\_4}(t) &= \left( \theta \dot{\lambda}\_1 (\mu\_1' - \mu\_1) \frac{\lambda\_2 + \delta}{(\delta + \lambda\_1)(\delta + 2\lambda\_2)} + \frac{\lambda\_1(\lambda\_2 + \delta)}{\delta(\delta + \lambda\_1)} \mu\_1 \right) \\ &+ \left( \theta \dot{\lambda}\_1 (\mu\_1' - \mu\_1) \left( \frac{\lambda\_1 - \lambda\_2}{\lambda\_1 - 2\lambda\_2} \right) - (\lambda\_1 - \lambda\_2)\mu\_1 \right) \frac{1}{\delta + \lambda\_1} e^{-(\delta + \lambda\_1)t} \\ &- \theta \dot{\lambda}\_1 \frac{1}{\delta + 2\lambda\_2} \frac{\lambda\_2}{\lambda\_1 - 2\lambda\_2} \left( \mu\_1' - \mu\_1 \right) e^{-(\delta + 2\lambda\_2)t} \\ &- 2\theta \lambda\_1 \frac{1}{\delta + 2\lambda\_1} \left( \mu\_1' - \mu\_1 \right) e^{-(\delta + 2\lambda\_1)t} - \frac{\lambda\_2}{\delta} \mu\_1 e^{-\delta t} \end{split} \tag{20}$$

*Proof*:

From Theorem 3.1, we have:

$$\begin{split} \tilde{\pi}\_{Z\_4}(r) &= \frac{\lambda\_1 \mu\_1}{r(r+\delta+\lambda\_1)} + \frac{\lambda\_1}{(r+\delta+\lambda\_1)} \tilde{\pi}\_{Z\_0}(r) \\ &+ \Theta(\mu\_1'-\mu\_1) \frac{\lambda\_1(r+\delta)}{r(r+\delta+\lambda\_1)(r+\delta+2\lambda\_1)} \end{split} \tag{21}$$

From Bargès et al. [8], we have

$$\bar{\pi}\_{Z\_{\circ}}(r) = \frac{\lambda\_2 \mu\_1}{r(r+\delta)} + \Theta \frac{\lambda\_2 \left(\mu\_1' - \mu\_1\right)}{r(r+\delta+2\lambda\_2)}\tag{22}$$

Substituting Eq. (22) into Eq. (21), yields

$$\begin{split} \tilde{\pi}\_{Z\_{2}}(r) &= \frac{\lambda\_{1}\mu\_{1}}{r(r+\delta+\lambda\_{1})} + \frac{\lambda\_{1}}{(r+\delta+\lambda\_{1})} \left\{ \frac{\lambda\_{2}\mu\_{1}}{r(r+\delta)} + \theta \frac{\lambda\_{2}(\mu\_{1}^{\prime}-\mu\_{1})}{r(r+\delta+2\lambda\_{2})} \right\} \\ &+ \theta\lambda\_{1}(\mu\_{1}^{\prime}-\mu\_{1}) \frac{(r+\delta)}{r(r+\delta+\lambda\_{1})(r+\delta+2\lambda\_{1})} \\ &= \left\{ \frac{\lambda\_{1}\lambda\_{2}}{r(r+\delta)(r+\delta+\lambda\_{1})} + \frac{\lambda\_{1}}{r(r+\delta+\lambda\_{1})} \right\}\mu\_{1} \\ &+ \theta\lambda\_{1}(\mu\_{1}^{\prime}-\mu\_{1}) \left\{ \frac{\lambda\_{2}}{r(r+\delta+\lambda\_{1})(r+\delta+2\lambda\_{2})} + \frac{r+\delta}{r(r+\delta+\lambda\_{1})(r+\delta+2\lambda\_{1})} \right\} \end{split} \tag{23}$$

with

$$\frac{\lambda\_1}{r(r+\delta+\lambda\_1)} = \frac{\lambda\_1}{(\delta+\lambda\_1)} \cdot \frac{1}{r} - \frac{\lambda\_1}{(\delta+\lambda\_1)} \cdot \frac{1}{(r+\delta+\lambda\_1)}\tag{24}$$

*Moments of the Discounted Aggregate Claims with Delay Inter-Occurrence Distribution… DOI: http://dx.doi.org/10.5772/intechopen.88699*

$$\frac{\lambda\_1 \lambda\_2}{\mathbf{r}(r+\delta)(r+\delta+\lambda\_1)} = \frac{\lambda\_1 \lambda\_2}{\delta(\delta+\lambda\_1)} \cdot \frac{1}{r} + \frac{\lambda\_2}{(\delta+\lambda\_1)} \cdot \frac{1}{(r+\delta+\lambda\_1)} - \frac{\lambda\_2}{\delta} \cdot \frac{1}{(r+\delta)} \tag{25}$$

$$\frac{\lambda\_2}{r(r+\delta+\lambda\_1)(r+\delta+2\lambda\_2)} = \frac{\lambda\_2}{(\delta+\lambda\_1)(\delta+2\lambda\_2)} \frac{1}{r}$$

$$+ \frac{\lambda\_2}{\lambda\_1 - 2\lambda\_2} \left[ \frac{1}{\delta+\lambda\_1} \frac{1}{r+\delta+\lambda\_1} - \frac{1}{\delta+2\lambda\_2} \frac{1}{r+\delta+2\lambda\_2} \right] \tag{26}$$

$$\frac{r+\delta}{r(r+\delta+\lambda\_1)(r+\delta+2\lambda\_1)} = \frac{\delta}{(\delta+\lambda\_1)(\delta+2\lambda\_1)} \frac{1}{r}$$

$$\begin{aligned} (\delta + \lambda\_1)(r + \delta + 2\lambda\_1) & \quad (\delta + \lambda\_1)(\delta + 2\lambda\_1) \, r \\ &+ \frac{1}{\delta + \lambda\_1} \frac{1}{r + \delta + \lambda\_1} - \frac{2}{\delta + 2\lambda\_1} \frac{1}{r + \delta + 2\lambda\_1} \end{aligned} \tag{27}$$

Substituting Eqs. (24), (25), (26) and (27) into Eq. (23), yields:

$$\begin{aligned} \dot{\pi}\_{Z\_{4}}(r) &= \mu\_{1} \left\{ \frac{\lambda\_{1}}{(\delta + \lambda\_{1})} \cdot \frac{1}{r} - \frac{\lambda\_{1}}{(\delta + \lambda\_{1})} \cdot \frac{1}{(r + \delta + \lambda\_{1})} \right\} \\ &+ \Theta(\mu\_{1}' - \mu\_{1}) \left\{ \frac{\lambda\_{2}}{(\delta + \lambda\_{1})(\delta + 2\lambda\_{2})} \frac{1}{r} \right\} \\ &+ \frac{\lambda\_{2}}{\lambda\_{1} - 2\lambda\_{2}} \left[ \frac{1}{\delta + \lambda\_{1}} \frac{1}{r + \delta + \lambda\_{1}} - \frac{1}{\delta + 2\lambda\_{2}} \frac{1}{r + \delta + 2\lambda\_{2}} \right] \\ &+ \Theta\lambda\_{1} (\mu\_{1}' - \mu\_{1}) \left\{ \frac{\delta}{(\delta + \lambda\_{1})(\delta + 2\lambda\_{1})} \frac{1}{r} \right\} \\ &+ \frac{1}{\delta + \lambda\_{1}} \frac{1}{r + \delta + \lambda\_{1}} - \frac{2}{\delta + 2\lambda\_{1}} \frac{1}{r + \delta + 2\lambda\_{1}} \end{aligned} \tag{28}$$

Rearranging the above equation, will give

$$\begin{aligned} \pi\_{Z\_2}(t) &= \left( \theta \dot{\lambda}\_1 (\mu\_1' - \mu\_1) \frac{\lambda\_2 + \delta}{(\delta + \lambda\_1)(\delta + 2\lambda\_2)} + \frac{\lambda\_1(\lambda\_2 + \delta)}{\delta(\delta + \lambda\_1)} \mu\_1 \right) \frac{1}{r} \\ &+ \left( \theta \dot{\lambda}\_1 (\mu\_1' - \mu\_1) \left( \frac{\lambda\_1 - \lambda\_2}{\lambda\_1 - 2\lambda\_2} \right) - (\lambda\_1 - \lambda\_2)\mu\_1 \right) \frac{1}{\delta + \lambda\_1} \frac{1}{r + \delta + \lambda\_1} \\\\ &- \theta \dot{\lambda}\_1 \frac{1}{\delta + 2\lambda\_2} \frac{\lambda\_2}{\lambda\_1 - 2\lambda\_2} \left( \mu\_1' - \mu\_1 \right) \frac{1}{r + \delta + 2\lambda\_2} \\\\ &- 2\theta \dot{\lambda}\_1 \frac{1}{\delta + 2\lambda\_1} \left( \mu\_1' - \mu\_1 \right) \frac{1}{r + \delta + 2\lambda\_1} - \frac{\lambda\_2}{\delta} \mu\_1 \frac{1}{r + \delta} \end{aligned} \tag{29}$$

Rearranging the above equation, we will get

λ2 ð Þ *r* þ *δm* X*m*�1 *k*¼1 *Ck*

<sup>þ</sup> <sup>θ</sup> <sup>λ</sup><sup>2</sup>

ð Þ δ þ λ<sup>1</sup> ð Þ δ þ 2λ<sup>2</sup>

λ<sup>1</sup> � 2λ<sup>2</sup> � �

� �

� � 1

μ0 <sup>1</sup> � μ<sup>1</sup> � �*e*

*<sup>m</sup>* <sup>μ</sup>*k*~πð Þ *<sup>m</sup>*�*<sup>k</sup> Zo* ð Þ*r*

ð Þ *r* þ *δm* þ 2λ<sup>2</sup>

þ

� ð Þ λ<sup>1</sup> � λ<sup>2</sup> μ<sup>1</sup>

�ð Þ δþ2λ<sup>2</sup> *t*

λ1 ð Þ *r* þ δ þ λ<sup>1</sup>

λ<sup>2</sup> μ<sup>0</sup>

μ1

þ

*:* <sup>1</sup> ð Þ *r* þ *δ* þ *λ*<sup>1</sup>

� �

<sup>δ</sup> <sup>μ</sup>1*<sup>e</sup>* �*δt*

*r r*ð Þ þ δ þ λ<sup>1</sup> ð Þ *r* þ δ þ 2λ<sup>1</sup>

<sup>1</sup> � μ<sup>1</sup> � � *r r*ð Þ þ δ þ 2λ<sup>2</sup>

� �

λ<sup>2</sup> μ<sup>0</sup>

<sup>1</sup> � μ<sup>1</sup> � � *r r*ð Þ þ δ þ 2λ<sup>2</sup>

> *r* þ δ *r r*ð Þ þ δ þ λ<sup>1</sup> ð Þ *r* þ δ þ 2λ<sup>1</sup>

�ð Þ <sup>δ</sup>þ2λ<sup>1</sup> *<sup>t</sup>* � <sup>λ</sup><sup>2</sup>

þ

� � λ1ð Þ *r* þ δ

λ2μ<sup>1</sup> *r r*ð Þ <sup>þ</sup> <sup>δ</sup> <sup>þ</sup> <sup>θ</sup>

X*m*�1 *k*¼1 *Ck <sup>m</sup>* μ<sup>0</sup> *<sup>k</sup>* � μ*<sup>k</sup>* � �~πð Þ *<sup>m</sup>*�*<sup>k</sup>*

λ1ð Þ λ<sup>2</sup> þ δ δ δð Þ þ λ<sup>1</sup>

μ1

δ þ λ<sup>1</sup> *e* �ð Þ δþλ<sup>1</sup> *t*

~π*<sup>Z</sup>*<sup>0</sup> ð Þ*r*

*Zo* ð Þ*r*

(19)

(20)

(21)

(22)

(23)

(24)

þ

*<sup>m</sup>* � μ*<sup>m</sup>* � � *r r*ð Þ þ *δm* þ 2λ<sup>2</sup>

The first moment of f g *Zd*ð Þ*t ; t*≥0 is given by:

<sup>1</sup> � μ<sup>1</sup> � � λ<sup>1</sup> � λ<sup>2</sup>

� � λ<sup>2</sup> þ δ

λ2 λ<sup>1</sup> � 2λ<sup>2</sup>

> μ0 <sup>1</sup> � μ<sup>1</sup> � �*e*

þ θ μ<sup>0</sup>

~π*Zo* ð Þ¼ *r*

� � ð Þ *r* þ δ

� �

� � λ<sup>2</sup>

λ1 ð Þ *r* þ δ þ λ<sup>1</sup>

þ

<sup>¼</sup> *<sup>λ</sup>*<sup>1</sup> ð Þ *δ* þ *λ*<sup>1</sup>

Substituting Eq. (22) into Eq. (21), yields

þ

<sup>1</sup> � μ<sup>1</sup>

*r r*ð Þ þ δ ð Þ *r* þ δ þ λ<sup>1</sup>

<sup>1</sup> � μ<sup>1</sup>

*λ*1 *r r*ð Þ þ *δ* þ *λ*<sup>1</sup>

λ1μ<sup>1</sup> *r r*ð Þ þ δ þ λ<sup>1</sup>

<sup>1</sup> � μ<sup>1</sup>

λ2μ<sup>1</sup> *r r*ð Þ <sup>þ</sup> <sup>δ</sup> <sup>þ</sup> <sup>θ</sup>

*r r*ð Þ þ δ þ λ<sup>1</sup> ð Þ *r* þ δ þ 2λ<sup>1</sup>

λ1 *r r*ð Þ þ δ þ λ<sup>1</sup>

*r r*ð Þ þ δ þ λ<sup>1</sup> ð Þ *r* þ δ þ 2λ<sup>2</sup>

*:* 1 *r* � *<sup>λ</sup>*<sup>1</sup> ð Þ *δ* þ *λ*<sup>1</sup>

<sup>1</sup> � μ<sup>1</sup>

1 δ þ 2λ<sup>2</sup>

> 1 δ þ 2λ<sup>1</sup>

~π*Zd* ð Þ¼ *r*

λ2μ*<sup>m</sup> r r*ð Þ þ *δm*

*Probability, Combinatorics and Control*

λ<sup>2</sup> μ<sup>0</sup>

þ θ*λ*<sup>1</sup> μ<sup>0</sup>

� θ*λ*<sup>1</sup>

� 2θ*λ*<sup>1</sup>

From Theorem 3.1, we have:

From Bargès et al. [8], we have

λ1μ<sup>1</sup> *r r*ð Þ þ δ þ λ<sup>1</sup>

þ θ*λ*<sup>1</sup> μ<sup>0</sup>

þ θλ<sup>1</sup> μ<sup>0</sup>

<sup>¼</sup> <sup>λ</sup>1λ<sup>2</sup>

þ θ

π*Zd* ðÞ¼ *t* θ*λ*<sup>1</sup> μ<sup>0</sup>

**Corollary 3.1**

*Proof*:

~π*Zd* ð Þ¼ *r*

with

**278**

~π*m Zo* ð Þ¼ *r*
