**Remark 1**

If λ<sup>1</sup> ¼ λ<sup>2</sup> then Eq. (29) becomes

$$\begin{split} \tilde{\pi}\_{Z\_{4}}(r) &= \left( \Theta \lambda (\mu\_{1}^{\prime} - \mu\_{1}) \left( \frac{1}{\delta + 2\lambda} \right) + \frac{\lambda}{\delta} \mu\_{1} \right) \frac{1}{r} \\ &- \frac{1}{\delta + 2\lambda} \Theta \lambda (\mu\_{1}^{\prime} - \mu\_{1}) \frac{1}{r + \delta + 2\lambda} - \frac{\lambda}{\delta} \mu\_{1} \frac{1}{(r + \delta)} \\ &= \frac{\lambda}{\delta} \mu\_{1} \left[ \frac{1}{r} - \frac{1}{r + \delta} \right] + \Theta \lambda (\mu\_{1}^{\prime} - \mu\_{1}) \left( \frac{1}{\delta + 2\lambda} \right) \left[ \frac{1}{r} - \frac{1}{r + \delta + 2\lambda} \right] \\ &= \frac{\lambda \mu\_{1}}{r(r + \delta)} + \frac{\lambda \theta \left( \mu\_{1}^{\prime} - \mu\_{1} \right)}{r(r + \delta + 2\lambda)}, \end{split} \tag{30}$$

**Corollary 3.2**

~π2 *Zd* ð Þ¼ *r*

and

~π2 *Zo* ð Þ¼ *r*

~π2 *Zd* ð Þ¼ *r*

**281**

þ

From Bargès et al. [8], we have.

λ2μ<sup>2</sup> *r r*ð Þ <sup>þ</sup> <sup>2</sup><sup>δ</sup> <sup>þ</sup> <sup>θ</sup>

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

> > 2θ<sup>2</sup> λ2 <sup>2</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup>

2λ1ð Þ λ<sup>1</sup> � λ<sup>2</sup> *r r*ð Þ þ 2δ þ λ<sup>1</sup> ð Þ *r* þ 2δ þ 2λ<sup>1</sup>

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

> > 2θ<sup>2</sup> λ2 <sup>2</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup>

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

λ1 ð Þ *r* þ 2δ þ λ<sup>1</sup>

> λ2μ<sup>2</sup> *r r*ð Þ <sup>þ</sup> <sup>2</sup><sup>δ</sup> <sup>þ</sup> <sup>θ</sup>

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

Substituting Eqs. (39) and (40) into Eq. (38), yields:

þ

þ

þ

8

>>>>>>>>>>>>>><

�

þ

þ

>>>>>>>>>>>>>>:

The result in Theorem 3.1 when *n* ¼ 2 gives:

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

λ1 ð Þ *r* þ 2δ þ λ<sup>1</sup>

> ~π*Zo* ð Þ¼ *r*

2λ1ð Þ λ<sup>1</sup> � λ<sup>2</sup> *r r*ð Þ þ 2δ þ λ<sup>1</sup> ð Þ *r* þ 2δ þ 2λ<sup>1</sup>

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

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

> 2θλ<sup>2</sup> <sup>2</sup> μ<sup>1</sup> μ<sup>0</sup>

> > 1 *r*

ð Þ *r* þ 2δ ð Þ *r* þ 2δ þ 2λ<sup>2</sup> λ2ð Þ *r* þ 2δ þ 2λ<sup>1</sup>

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

λ<sup>2</sup> μ<sup>0</sup>

� �

2θλ<sup>2</sup> <sup>2</sup> μ<sup>1</sup> μ<sup>0</sup>

μ<sup>2</sup> þ 2μ<sup>1</sup>

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

þ 1

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

� � � �

λ<sup>2</sup> μ<sup>0</sup>

The second moment of f g *Zd*ð Þ*t ; t* ≥0 is given by the following development:

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

1 *r*

� �~π<sup>2</sup>

λ<sup>2</sup> μ<sup>0</sup>

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

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

ð Þ *r* þ 2δ ð Þ *r* þ 2δ þ 2λ<sup>2</sup> λ2ð Þ *r* þ 2δ þ 2λ<sup>1</sup>

μ<sup>2</sup> þ 2μ1~π*Zo*

� �

ð Þ*r*

*<sup>Z</sup>*<sup>0</sup> ð Þ*r ,*

2θλ<sup>2</sup> <sup>2</sup>μ<sup>1</sup> μ<sup>0</sup>

<sup>1</sup> � μ<sup>1</sup> � �

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

λ<sup>2</sup> μ<sup>0</sup>

2θλ<sup>2</sup> <sup>2</sup>μ<sup>1</sup> μ<sup>0</sup>

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

> <sup>1</sup> � μ<sup>1</sup> � �

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

(35)

(36)

(37)

9

>>>>>>>>>>>>>>=

*,*

>>>>>>>>>>>>>>;

(38)

þ 1

which is exactly the result of Bargès et al. [8]. The inverse of the Laplace transform in Eq. (29) will give

$$\begin{split} \pi\_{\mathbf{Z}\_{4}}(t) &= \left( \theta \lambda\_{1} (\mu\_{1}' - \mu\_{1}) \frac{\lambda\_{2} + \delta}{(\delta + \lambda\_{1})(\delta + 2\lambda\_{2})} + \frac{\lambda\_{1}\lambda\_{2} + \lambda\_{1}\delta}{8(\delta + \lambda\_{1})} \mu\_{1} \right) \\ &+ \left( \theta \lambda\_{1} (\mu\_{1}' - \mu\_{1}) \left( \frac{\lambda\_{2}}{\lambda\_{1} - 2\lambda\_{2}} + 1 \right) + (\lambda\_{2} - \lambda\_{1})\mu\_{1} \right) \frac{1}{\delta + \lambda\_{1}} e^{-(\delta + \lambda\_{1})t} \\ &- \theta \lambda\_{1} \frac{1}{\delta + 2\lambda\_{2}} \frac{\lambda\_{2}}{\lambda\_{1} - 2\lambda\_{2}} \left( \mu\_{1}' - \mu\_{1} \right) e^{-(\delta + 2\lambda\_{2})t} \\ &- 2\theta \lambda\_{1} \frac{1}{\delta + 2\lambda\_{1}} \left( \mu\_{1}' - \mu\_{1} \right) e^{-(\delta + 2\lambda\_{1})t} - \frac{\lambda\_{2}}{\delta} \mu\_{1} e^{-\delta t} \end{split} \tag{31}$$

#### **Remarks 2**

If θ ¼ 0 and λ<sup>1</sup> 6¼ λ<sup>2</sup> then

$$\begin{split} \pi\_{\mathsf{Z}\_{d}}(t) &= \left[ \frac{\lambda\_{1}}{\mathsf{S}} \left( \frac{\lambda\_{2} + \mathsf{\delta}}{\mathsf{S} + \lambda\_{1}} \right) - \frac{\lambda\_{2}}{\mathsf{S}} e^{-\mathsf{\delta}t} \right] \mathsf{\mu}\_{1} + \left( \frac{\lambda\_{2} - \lambda\_{1}}{\mathsf{S} + \lambda\_{1}} \right) \mathsf{\mu}\_{1} e^{-(\mathsf{\delta} + \lambda\_{1})t} \\ &= \lambda\_{2} \left( \frac{\mathsf{1} - e^{-\mathsf{\delta}t}}{\mathsf{\delta}} \right) \mathsf{\mu}\_{1} + (\lambda\_{1} - \lambda\_{2}) \left( \frac{\mathsf{1} - e^{-(\mathsf{\delta} + \lambda\_{1})t}}{\mathsf{\delta} + \lambda\_{1}} \right) \mathsf{\mu}\_{1} \\ &= \left\{ \lambda\_{2} \overline{a}\_{l|\mathsf{\delta}} + (\lambda\_{1} - \lambda\_{2}) \overline{a}\_{l|\mathsf{\delta} + \lambda\_{1}} \right\} \mathsf{\mu}\_{1} \end{split} \tag{32}$$

which is exactly the result of Léveillé et al. [15]. If λ<sup>1</sup> ¼ λ<sup>2</sup> and θ 6¼ 0 then

$$\pi\_{Z\_\*} (t) = \left( \Theta \lambda (\mu\_1' - \mu\_1) \frac{1}{(\delta + 2\lambda\_2)} + \frac{\lambda}{\delta} \mu\_1 \right)$$

$$- \Theta \lambda \frac{1}{\delta + 2\lambda} \left( \mu\_1' - \mu\_1 \right) e^{-(\delta + 2\lambda)t} - \frac{\lambda}{\delta} \mu\_1 e^{-\delta t} \tag{33}$$

$$= \frac{\lambda}{\delta} \left( 1 - e^{-\delta t} \right) \mu\_1 + \Theta \lambda \left( \frac{1 - e^{-(\delta + 2\lambda)t}}{\delta + 2\lambda} \right) \left( \mu\_1' - \mu\_1 \right)$$

which is exactly the result of Bargès et al. [8]. If λ<sup>1</sup> ¼ λ<sup>2</sup> and θ ¼ 0 then

$$
\pi\_{\mathcal{Z}\_\bullet}(t) = \frac{\lambda}{6} \left( \mathbf{1} - e^{-\delta t} \right) \mu\_1 = \lambda \overline{\mathfrak{a}}\_{t|\delta} \mu\_1,\tag{34}
$$

which is exactly the result of Léveillé et al. [15].

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

### **Corollary 3.2**

**Remark 1**

If λ<sup>1</sup> ¼ λ<sup>2</sup> then Eq. (29) becomes

*Probability, Combinatorics and Control*

� <sup>1</sup> δ þ 2λ

<sup>1</sup> � μ<sup>1</sup> 1

*r r*ð Þ <sup>þ</sup> <sup>δ</sup> <sup>þ</sup> λθ μ<sup>0</sup>

<sup>1</sup> � μ<sup>1</sup>

1 δ þ 2λ<sup>2</sup>

> 1 δ þ 2λ<sup>1</sup>

> > λ1 δ

¼ λ<sup>2</sup>

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

which is exactly the result of Bargès et al. [8].

λ<sup>2</sup> þ δ

λ2 λ<sup>1</sup> � 2λ<sup>2</sup>

> μ0 <sup>1</sup> � μ<sup>1</sup> *e*

λ<sup>2</sup> þ δ δ þ λ<sup>1</sup> 

<sup>1</sup> � *<sup>e</sup>*�*δ<sup>t</sup>* δ 

which is exactly the result of Léveillé et al. [15].

� θλ <sup>1</sup>

π*Zo* ðÞ¼ *t* θλ μ<sup>0</sup>

¼ λ <sup>δ</sup> <sup>1</sup> � *<sup>e</sup>*

which is exactly the result of Bargès et al. [8].

π*Zo* ðÞ¼ *t*

which is exactly the result of Léveillé et al. [15].

¼ λ2*at*j<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*jδþλ<sup>1</sup> μ<sup>1</sup>

> <sup>1</sup> � μ<sup>1</sup> 1

<sup>δ</sup> <sup>þ</sup> <sup>2</sup><sup>λ</sup> <sup>μ</sup><sup>0</sup>

λ δ 1 � *e*

θλ μ<sup>0</sup>

δ þ 2λ 

þ θλ μ<sup>0</sup>

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

The inverse of the Laplace transform in Eq. (29) will give

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

μ0 <sup>1</sup> � μ<sup>1</sup> *e*

> � λ2 δ *e* �*δt*

λ<sup>1</sup> � 2λ<sup>2</sup>

þ 1

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

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

ð Þ δ þ 2λ<sup>2</sup>

<sup>1</sup> � μ<sup>1</sup> *e*

�*δ<sup>t</sup>* <sup>μ</sup><sup>1</sup> <sup>þ</sup> θλ <sup>1</sup> � *<sup>e</sup>*�ð Þ <sup>δ</sup>þ2<sup>λ</sup> *<sup>t</sup>*

þ λ δ μ1

�ð Þ <sup>δ</sup>þ2<sup>λ</sup> *<sup>t</sup>* � <sup>λ</sup>

δ þ 2λ 

δ μ1*e* �*δt*

> μ0 <sup>1</sup> � μ<sup>1</sup>

�*δ<sup>t</sup>* <sup>μ</sup><sup>1</sup> <sup>¼</sup> <sup>λ</sup>*at*jδμ1*,* (34)

1

1

<sup>1</sup> � μ<sup>1</sup> 1

þ λ δ μ1

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

þ

<sup>1</sup> � μ<sup>1</sup> 1

*r*

δ μ1

δ þ 2λ 1

> λ1λ<sup>2</sup> þ λ1δ δ δð Þ þ λ<sup>1</sup>

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

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

�ð Þ δþ2λ<sup>2</sup> *t*

μ<sup>1</sup> þ

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

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

1 ð Þ *r* þ δ

*r*

μ1

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

μ1*e*

�ð Þ δþλ<sup>1</sup> *t*

μ1

� <sup>1</sup> *r* þ δ þ 2λ  (30)

(31)

(32)

(33)

~π*Zd* ð Þ¼ *r* θλ μ<sup>0</sup>

¼ λ δ μ1 1 *r* � <sup>1</sup> *r* þ δ 

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

<sup>¼</sup> λμ<sup>1</sup>

þ θλ<sup>1</sup> μ<sup>0</sup>

� θλ<sup>1</sup>

� 2θλ<sup>1</sup>

If θ ¼ 0 and λ<sup>1</sup> 6¼ λ<sup>2</sup> then

If λ<sup>1</sup> ¼ λ<sup>2</sup> and θ 6¼ 0 then

If λ<sup>1</sup> ¼ λ<sup>2</sup> and θ ¼ 0 then

**280**

π*Zd* ðÞ¼ *t*

**Remarks 2**

The second moment of f g *Zd*ð Þ*t ; t* ≥0 is given by the following development: The result in Theorem 3.1 when *n* ¼ 2 gives:

$$\begin{split} \bar{\tilde{\pi}}\_{Z\_4}^2(r) &= \frac{2\lambda\_1(\lambda\_1 - \lambda\_2)}{r(r + 2\delta + \lambda\_1)(r + 2\delta + 2\lambda\_1)} \left( \frac{1}{r}\mu\_2 + 2\mu\_1 \bar{\pi}\_{Z\_o}(r) \right) \\\\ &+ \frac{\lambda\_1}{(r + 2\delta + \lambda\_1)} \left( \frac{(r + 2\delta)(r + 2\delta + 2\lambda\_2)}{\lambda\_2(r + 2\delta + 2\lambda\_1)} + 1 \right) \bar{\pi}\_{Z\_0}^2(r), \end{split} \tag{35}$$

From Bargès et al. [8], we have.

$$
\tilde{\pi}\_{Z\_0}(r) = \frac{\lambda\_2 \mu\_1}{r(r+\delta)} \quad + \Theta \frac{\lambda\_2(\mu\_1' - \mu\_1)}{r(r+\delta+2\lambda\_2)} \tag{36}
$$

and

$$\begin{aligned} \tilde{\pi}\_{Z\_{\epsilon}}^{2}(r) &= \frac{\lambda\_{2}\mu\_{2}}{r(r+2\delta)} \quad + \theta \frac{\lambda\_{2}\left(\mu\_{1}^{\prime}-\mu\_{2}\right)}{r(r+2\delta+2\lambda\_{2})} \\\\ &+ \frac{2\lambda\_{2}^{2}\mu\_{1}^{2}}{r(r+\delta)(r+2\delta)} \quad + \frac{2\theta\lambda\_{2}^{2}\mu\_{1}(\mu\_{1}^{\prime}-\mu\_{1})}{r(r+\delta+2\lambda\_{2})(r+2\delta)} + \frac{2\theta\lambda\_{2}^{2}\mu\_{1}(\mu\_{1}^{\prime}-\mu\_{1})}{r(r+2\delta+2\lambda\_{2})(r+\delta)} \\\\ &+ \frac{2\theta^{2}\lambda\_{2}^{2}(\mu\_{1}^{\prime}-\mu\_{1})^{2}}{r(r+\delta+2\lambda\_{2})(r+2\delta+2\lambda\_{2})} \end{aligned} \tag{37}$$

Substituting Eqs. (39) and (40) into Eq. (38), yields:

$$
\begin{split}
\hat{\pi}\_{Z\_{4}}^{2}(r) &= \frac{2\lambda\_{1}(\lambda\_{1}-\lambda\_{2})}{r(r+2\delta+\lambda\_{1})(r+2\delta+2\lambda\_{4})} \left(\frac{1}{r}\mu\_{2}+2\mu\_{1}\left(\frac{\lambda\_{2}\mu\_{1}}{r(r+\delta)}+\theta\frac{\lambda\_{2}(\mu\_{1}^{\prime}-\mu\_{1})}{r(r+\delta+2\lambda\_{2})}\right)\right) \\ &+ \frac{\lambda\_{1}}{(r+2\delta+\lambda\_{1})} \left(\frac{(r+2\delta)(r+2\delta+2\lambda\_{2})}{\lambda\_{2}(r+2\delta+2\lambda\_{1})}+1\right) \\ \\ &\left\{\frac{\lambda\_{2}\mu\_{2}}{r(r+2\delta)}+\theta\frac{\lambda\_{2}(\mu\_{2}^{\prime}-\mu\_{2})}{r(r+2\delta+2\lambda\_{2})}\right. \\ &\times \left\{\begin{array}{c} \\ +\frac{2\lambda\_{2}^{2}\mu\_{1}^{2}}{r(r+\delta)(r+2\delta)}+\frac{2\theta\lambda\_{2}^{2}\mu\_{1}(\mu\_{1}^{\prime}-\mu\_{1})}{r(r+\delta+2\lambda\_{2})(r+2\delta)}+\frac{2\theta\lambda\_{2}^{2}\mu\_{1}(\mu\_{1}^{\prime}-\mu\_{1})}{r(r+2\delta+2\lambda\_{2})(r+\delta)} \\ \\ +\frac{2\theta^{2}\lambda\_{2}^{2}(\mu\_{1}^{\prime}-\mu\_{1})^{2}}{r(r+\delta+2\lambda\_{2})(r+2\delta+2\lambda\_{2})} \end{array} \right\}, \end{split} \tag{38}
$$

and rearranging Eq. (38), will give:

~π2 *Zd* ð Þ¼ *<sup>r</sup>* <sup>λ</sup>1μ<sup>2</sup> *r r*ð Þ þ 2δ þ λ<sup>1</sup> þ 2λ1λ2μ<sup>2</sup> 1 *r r*ð Þ þ δ ð Þ *r* þ 2δ þ λ<sup>1</sup> þ 2θλ1λ2μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � *r r*ð Þ þ δ þ 2λ<sup>2</sup> ð Þ *r* þ 2δ þ λ<sup>1</sup> þ θλ<sup>1</sup> μ<sup>0</sup> <sup>2</sup> � μ<sup>2</sup> � � ð Þ *r* þ 2δ *r r*ð Þ þ 2δ þ λ<sup>1</sup> ð Þ *r* þ 2δ þ 2λ<sup>1</sup> þ 2θλ1λ2μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � ð Þ *r* þ 2δ *r r*ð Þ þ δ ð Þ *r* þ 2δ þ λ<sup>1</sup> ð Þ *r* þ 2δ þ 2λ<sup>1</sup> <sup>þ</sup> <sup>2</sup>θ<sup>2</sup> λ1λ<sup>2</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup> ð Þ *r* þ 2δ *r r*ð Þ þ δ þ 2λ<sup>2</sup> ð Þ *r* þ 2δ þ λ<sup>1</sup> ð Þ *r* þ 2δ þ 2λ<sup>1</sup> þ λ1λ2μ<sup>2</sup> 1 *r r*ð Þ þ 2δ ð Þ *r* þ 2δ þ λ<sup>1</sup> þ θλ1λ<sup>2</sup> μ<sup>0</sup> <sup>2</sup> � μ<sup>2</sup> � � 1 *r r*ð Þ þ 2δ þ 2λ<sup>2</sup> ð Þ *r* þ 2δ þ λ<sup>1</sup> <sup>þ</sup> <sup>2</sup>λ1λ<sup>2</sup> 2μ2 1 1 *r r*ð Þ þ δ ð Þ *r* þ 2δ ð Þ *r* þ 2δ þ λ<sup>1</sup> <sup>þ</sup> <sup>2</sup>θ*λ*1λ<sup>2</sup> <sup>2</sup>μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � 1 *r r*ð Þ þ δ þ 2λ<sup>2</sup> ð Þ *r* þ 2δ ð Þ *r* þ 2δ þ λ<sup>1</sup> <sup>þ</sup> <sup>2</sup>θλ<sup>2</sup> <sup>2</sup>λ1μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � 1 *r r*ð Þ þ 2δ þ 2λ<sup>2</sup> ð Þ *r* þ δ ð Þ *r* þ 2δ þ λ<sup>1</sup> <sup>þ</sup> <sup>2</sup>θ<sup>2</sup> λ2 <sup>2</sup>λ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup> 1 *r r*ð Þ þ δ þ 2λ<sup>2</sup> ð Þ *r* þ 2δ þ 2λ<sup>2</sup> ð Þ *r* þ 2δ þ λ<sup>1</sup> *,* (39)

*<sup>γ</sup>*<sup>2</sup> ¼ � <sup>λ</sup>2μ<sup>2</sup>

2λ1λ2μ<sup>2</sup> 1 ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup>

*γ*<sup>3</sup> ¼

*γ*<sup>4</sup> ¼

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

8

>>>>>>>>>>>>>>>><

þ

>>>>>>>>>>>>>>>>:

8 >>>><

>>>>:

*<sup>γ</sup>*<sup>5</sup> ¼ � <sup>θ</sup>*λ*<sup>1</sup> <sup>μ</sup><sup>0</sup>

*<sup>γ</sup>*<sup>6</sup> ¼ � <sup>θ</sup>*λ*1λ<sup>2</sup> <sup>μ</sup><sup>0</sup>

**Remark 2** When

> ~π2 *Zd* ð Þ¼ *r*

**283**

þ

� <sup>2</sup>θ*λ*1λ2μ<sup>1</sup> <sup>μ</sup><sup>0</sup>

θ*λ*1λ<sup>2</sup> μ<sup>0</sup>

þ

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

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

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

2θ*λ*1λ<sup>2</sup>

�2θ*λ*1λ2μ<sup>1</sup> μ<sup>0</sup>

� <sup>2</sup>θ*λ*1λ<sup>2</sup>

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

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

> <sup>2</sup> � μ<sup>2</sup> � �

2λ<sup>2</sup> μ2 *r r*ð Þ <sup>þ</sup> <sup>δ</sup> ð Þ *<sup>r</sup>* <sup>þ</sup> <sup>2</sup><sup>δ</sup> <sup>þ</sup>

*r r*ð Þ <sup>þ</sup> <sup>2</sup><sup>δ</sup> <sup>þ</sup> θλ μ<sup>0</sup>

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

<sup>þ</sup> λμ<sup>2</sup>

2θλ<sup>2</sup> μ<sup>1</sup> μ<sup>0</sup>

which is exactly the result of Bargès et al. [8].

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

<sup>1</sup> � μ<sup>1</sup> � �

<sup>2</sup> � μ<sup>2</sup> � �

<sup>2</sup>μ<sup>1</sup> μ<sup>0</sup>

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

<sup>1</sup> � μ<sup>1</sup> � �

<sup>2</sup>μ<sup>1</sup> μ<sup>0</sup>

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

þ

<sup>1</sup> � μ<sup>1</sup> � �

2δ þ

� <sup>2</sup>θ<sup>2</sup>

þ

2θ*λ*1λ2μ<sup>1</sup> μ<sup>0</sup>

þ

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

<sup>1</sup> � μ<sup>1</sup> � �

� <sup>2</sup>λ<sup>2</sup>

þ

2θ<sup>2</sup>

<sup>1</sup> � μ<sup>1</sup> � �

( )

θ*λ*1λ<sup>2</sup> <sup>2</sup>μ<sup>1</sup> μ<sup>0</sup>

λ<sup>1</sup> ¼ λ<sup>2</sup>

2θλ<sup>2</sup> μ<sup>1</sup> μ<sup>0</sup>

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

λ1λ<sup>2</sup> μ<sup>0</sup>

� <sup>2</sup>θ<sup>2</sup>

� <sup>2</sup>θ<sup>2</sup>

<sup>1</sup> � μ<sup>1</sup> � �

<sup>1</sup> � μ<sup>1</sup> � �

*r r*ð Þ þ δ þ 2λ ð Þ *r* þ 2δ

2θ<sup>2</sup> λ<sup>2</sup> μ<sup>0</sup>

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

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

λ2 2μ2 1 <sup>δ</sup><sup>2</sup> <sup>þ</sup>

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

þ

λ1λ<sup>2</sup> μ<sup>0</sup>

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

2μ2 1 ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup>

θλ<sup>2</sup> <sup>2</sup>μ<sup>1</sup> μ<sup>0</sup>

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

( )

<sup>1</sup> � μ<sup>1</sup> � �

> <sup>1</sup> � μ<sup>1</sup> � �

> > þ

� <sup>2</sup>θλ<sup>2</sup>

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

λ2μ<sup>2</sup> 2δ þ λ<sup>1</sup>

<sup>2</sup>μ<sup>1</sup> μ<sup>0</sup>

ð Þ 2λ<sup>2</sup> � δ

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

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

þ

θ*λ*<sup>1</sup> μ<sup>0</sup>

<sup>1</sup> � μ<sup>1</sup> � �

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

(43)

9

>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>;

(44)

(45)

(46)

(47)

(48)

9 >>>>=

>>>>;

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

δ δð Þ � 2λ<sup>2</sup>

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

2θ*λ*1λ2μ<sup>1</sup> μ<sup>0</sup>

2θ<sup>2</sup> λ1λ<sup>2</sup> <sup>2</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup>

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

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

λ1λ<sup>2</sup> <sup>2</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup>

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

λ1λ<sup>2</sup> μ<sup>0</sup>

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

þ θ2 λ1λ<sup>2</sup> <sup>2</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup>

which can be simplified to

$$\tilde{\pi}\_{Z\_d}^2(r) = \frac{\chi\_0}{r} + \frac{\chi\_1}{r+8} + \frac{\chi\_2}{r+28} + \frac{\chi\_3}{r+28+\lambda\_1} + \frac{\chi\_4}{r+8+2\lambda\_2} + \frac{\chi\_5}{r+28+2\lambda\_1} + \frac{\chi\_6}{r+28+2\lambda\_2},\tag{40}$$

with,

*γ*<sup>0</sup> ¼ λ1μ<sup>2</sup> ð Þ 2δ þ λ<sup>1</sup> þ 2λ1λ2μ<sup>2</sup> 1 δð Þ 2δ þ λ<sup>1</sup> þ 2θ*λ*1λ2μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ 2λ<sup>2</sup> þ *θδλ*<sup>1</sup> μ<sup>0</sup> <sup>2</sup> � μ<sup>2</sup> � � ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup> þ 2θ*λ*1λ2μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup> þ 2*δθ*<sup>2</sup> λ1λ<sup>2</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup> ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup> ð Þ δ þ 2λ<sup>2</sup> þ λ1λ2μ<sup>2</sup> 2δð Þ 2δ þ λ<sup>1</sup> þ θ*λ*1λ<sup>2</sup> μ<sup>0</sup> <sup>2</sup> � μ<sup>2</sup> � � 2ð Þ δ þ λ<sup>2</sup> ð Þ 2δ þ λ<sup>1</sup> þ λ1λ<sup>2</sup> 2μ2 1 δ2 ð Þ 2δ þ λ<sup>1</sup> þ θ*λ*1λ<sup>2</sup> <sup>2</sup>μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � δ δð Þ þ 2λ<sup>2</sup> ð Þ 2δ þ λ<sup>1</sup> þ θ*λ*1λ<sup>2</sup> <sup>2</sup>μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � δ δð Þ þ λ<sup>2</sup> ð Þ 2δ þ λ<sup>1</sup> þ θ2 λ1λ<sup>2</sup> <sup>2</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup> 2ð Þ δ þ 2λ<sup>2</sup> ð Þ δ þ λ<sup>2</sup> ð Þ 2δ þ λ<sup>1</sup> 8 >>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>: 9 >>>>>>>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>>>>>>; (41)

$$\gamma\_1 = \left\{ -\frac{2\lambda\_1\lambda\_2\mu\_1^2}{\delta(\mathsf{S}+\lambda\_1)} - \frac{2\mathsf{R}\lambda\_1\lambda\_2\mu\_1(\mu\_1'-\mu\_1)}{(\mathsf{S}+\lambda\_1)(\mathsf{S}+2\lambda\_1)} - \frac{2\lambda\_1\lambda\_2^2\mu\_1^2}{\mathsf{S}^2(\mathsf{S}+\lambda\_1)} - \frac{2\mathsf{R}\lambda\_1\lambda\_2^2\mu\_1(\mu\_1'-\mu\_1)}{\mathsf{S}(\mathsf{S}+2\lambda\_2)(\mathsf{S}+\lambda\_1)} \right\} \tag{42}$$

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

$$\gamma\_2 = \left\{-\frac{\lambda\_2 \mu\_2}{2\delta} + \frac{\lambda\_2^2 \mu\_1^2}{8^2} + \frac{\Theta \lambda\_2^2 \mu\_1 (\mu\_1' - \mu\_1)}{\delta (\delta - 2\lambda\_2)}\right\} \tag{43}$$

*γ*<sup>3</sup> ¼ � <sup>λ</sup>1μ<sup>2</sup> 2δ þ λ<sup>1</sup> þ 2λ1λ2μ<sup>2</sup> 1 ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup> þ 2θ*λ*1λ2μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � ð Þ 2δ þ λ<sup>1</sup> ð Þ λ<sup>1</sup> þ δ � 2λ<sup>2</sup> þ θ*λ*<sup>1</sup> μ<sup>0</sup> <sup>2</sup> � μ<sup>2</sup> � � 2δ þ λ<sup>1</sup> � <sup>2</sup>θ*λ*1λ2μ<sup>1</sup> <sup>μ</sup><sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup> � <sup>2</sup>θ<sup>2</sup> λ1λ<sup>2</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup> ð Þ 2δ þ λ<sup>1</sup> ð Þ λ<sup>1</sup> þ δ � 2λ<sup>2</sup> þ λ2μ<sup>2</sup> 2δ þ λ<sup>1</sup> þ θ*λ*1λ<sup>2</sup> μ<sup>0</sup> <sup>2</sup> � μ<sup>2</sup> � � ð Þ 2δ þ λ<sup>1</sup> ð Þ λ<sup>1</sup> � 2λ<sup>2</sup> � <sup>2</sup>λ<sup>2</sup> 2μ2 1 ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup> � <sup>2</sup>θλ<sup>2</sup> <sup>2</sup>μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � ð Þ 2δ þ λ<sup>1</sup> ð Þ λ<sup>1</sup> þ δ � 2λ<sup>2</sup> þ 2θ*λ*1λ<sup>2</sup> <sup>2</sup>μ<sup>1</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � � ð Þ 2λ<sup>2</sup> � λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup> ð Þ 2δ þ λ<sup>1</sup> þ 2θ<sup>2</sup> λ1λ<sup>2</sup> <sup>2</sup> μ<sup>0</sup> <sup>1</sup> � μ<sup>1</sup> � �<sup>2</sup> ð Þ 2δ þ λ<sup>1</sup> ð Þ λ<sup>1</sup> þ δ � 2λ<sup>2</sup> ð Þ 2λ<sup>2</sup> � λ<sup>1</sup> 8 >>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>: 9 >>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>;

(44)

$$\chi\_{4} = \begin{cases} \frac{-2\mathfrak{H}\lambda\_{1}\lambda\_{2}\mathfrak{u}\_{1}\left(\mathfrak{h}\_{1}^{\prime}-\mathfrak{h}\_{1}\right)}{\left(\mathfrak{S}+2\lambda\_{2}\right)\left(\lambda\_{1}+\mathfrak{S}-2\lambda\_{2}\right)} + \frac{2\mathfrak{G}^{2}\lambda\_{1}\lambda\_{2}\left(\mathfrak{h}\_{1}^{\prime}-\mathfrak{h}\_{1}\right)^{2}\left(2\lambda\_{2}-\mathfrak{S}\right)}{\left(\mathfrak{S}+2\lambda\_{2}\right)\left(\lambda\_{1}+\mathfrak{S}-2\lambda\_{2}\right)\left(2\lambda\_{1}-2\lambda\_{2}+\mathfrak{S}\right)} \\ \quad - \frac{2\mathfrak{G}\lambda\_{1}\lambda\_{2}^{2}\mathfrak{u}\_{1}\left(\mathfrak{h}\_{1}^{\prime}-\mathfrak{h}\_{1}\right)}{\left(\mathfrak{S}+2\lambda\_{2}\right)\left(\mathfrak{S}-2\lambda\_{2}\right)\left(\lambda\_{1}+\mathfrak{S}-2\lambda\_{2}\right)} - \frac{2\mathfrak{G}^{2}\lambda\_{1}\lambda\_{2}^{2}\left(\mathfrak{h}\_{1}^{\prime}-\mathfrak{h}\_{1}\right)^{2}}{\mathfrak{S}\left(\mathfrak{S}+2\lambda\_{2}\right)\left(\lambda\_{1}+\mathfrak{S}-2\lambda\_{2}\right)} \end{cases} \tag{45}$$

$$\gamma\_5 = \left\{ -\frac{\Theta \lambda\_1 (\mu\_2' - \mu\_2)}{\delta + \lambda\_1} + \frac{2\Theta \lambda\_1 \lambda\_2 \mu\_1 (\mu\_1' - \mu\_1)}{(\delta + \lambda\_1)(\delta + 2\lambda\_1)} - \frac{2\Theta^2 \lambda\_1 \lambda\_2 \left(\mu\_1' - \mu\_1\right)^2}{(\delta + \lambda\_1)(2\lambda\_1 - \delta - 2\lambda\_2)} \right\} \tag{46}$$

$$\gamma\_6 = -\frac{\theta \lambda\_1 \lambda\_2 \left(\mu\_2' - \mu\_2\right)}{2(\ $ + \lambda\_2)(\lambda\_1 - 2\lambda\_2)} + \frac{\theta \lambda\_1 \lambda\_2^2 \mu\_1 \left(\mu\_1' - \mu\_1\right)}{(\$  + \lambda\_2)(\ $ + 2\lambda\_2)(\lambda\_1 - 2\lambda\_2)} + \frac{\theta^2 \lambda\_1 \lambda\_2^2 \left(\mu\_1' - \mu\_1\right)^2}{8(\$  + \lambda\_2)(\lambda\_1 - 2\lambda\_2)}\tag{47}$$
