**Remark 2** When

and rearranging Eq. (38), will give:

þ

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

<sup>2</sup>λ1μ<sup>1</sup> μ<sup>0</sup>

which can be simplified to

*<sup>r</sup>* <sup>þ</sup> <sup>δ</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup>

þ

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

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

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

1 δ δð Þ þ λ<sup>1</sup>

2θ*λ*1λ2μ<sup>1</sup> μ<sup>0</sup>

θ*λ*1λ<sup>2</sup> μ<sup>0</sup>

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

*<sup>r</sup>* <sup>þ</sup> *<sup>γ</sup>*<sup>1</sup>

λ1μ<sup>2</sup> ð Þ 2δ þ λ<sup>1</sup>

þ

þ

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

þ

*<sup>γ</sup>*<sup>1</sup> ¼ � <sup>2</sup>λ1λ2μ<sup>2</sup>

<sup>1</sup> � μ<sup>1</sup>

� � ð Þ *r* þ 2δ

� � 1

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

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

� � ð Þ *r* þ 2δ

� �<sup>2</sup> ð Þ *r* þ 2δ

� � 1

� � 1

� �<sup>2</sup> 1

*<sup>r</sup>* <sup>þ</sup> <sup>2</sup><sup>δ</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>3</sup>

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

<sup>1</sup> � μ<sup>1</sup> � �

<sup>2</sup> � μ<sup>2</sup> � �

> <sup>1</sup> � μ<sup>1</sup> � �

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

*r* þ 2δ þ λ<sup>1</sup>

þ

þ

þ

þ

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

δ2

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

þ

*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>

*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>

*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>

*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>

2θ*λ*1λ2μ<sup>1</sup> μ<sup>0</sup>

2*δθ*<sup>2</sup>

λ1λ<sup>2</sup> 2μ2 1

ð Þ 2δ þ λ<sup>1</sup>

θ2 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> � �

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

λ1λ<sup>2</sup> μ<sup>0</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

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

2μ2 1

ð Þ δ þ λ<sup>1</sup>

� <sup>2</sup>λ1λ<sup>2</sup>

δ2

( )

<sup>þ</sup> *<sup>γ</sup>*<sup>4</sup> *r* þ δ þ 2λ<sup>2</sup>

> <sup>1</sup> � μ<sup>1</sup> � �

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

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

þ

*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>

<sup>þ</sup> <sup>2</sup>λ1λ<sup>2</sup> 2μ2 1

2θλ1λ2μ<sup>1</sup> μ<sup>0</sup>

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

<sup>1</sup> � μ<sup>1</sup> � �

þ λ1λ2μ<sup>2</sup>

*,*

<sup>þ</sup> *<sup>γ</sup>*<sup>5</sup> *r* þ 2δ þ 2λ<sup>1</sup>

*θδλ*<sup>1</sup> μ<sup>0</sup>

þ

<sup>1</sup> � μ<sup>1</sup> � �

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

� <sup>2</sup>θ*λ*1λ<sup>2</sup>

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

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

<sup>1</sup> � μ<sup>1</sup> � �

<sup>2</sup> � μ<sup>2</sup> � �

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

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

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

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

(39)

(40)

(41)

(42)

*,*

<sup>þ</sup> *<sup>γ</sup>*<sup>6</sup> *r* þ 2δ þ 2λ<sup>2</sup>

9

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

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

~π2 *Zd* ð Þ¼ *<sup>r</sup>* <sup>λ</sup>1μ<sup>2</sup>

þ θλ<sup>1</sup> μ<sup>0</sup>

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

þ θλ1λ<sup>2</sup> μ<sup>0</sup>

<sup>þ</sup> <sup>2</sup>θ*λ*1λ<sup>2</sup>

<sup>þ</sup> <sup>2</sup>θλ<sup>2</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>

ð Þ¼ *<sup>r</sup> <sup>γ</sup>*<sup>0</sup>

with,

8

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

*γ*<sup>0</sup> ¼

**282**

~π2 *Zd* *r r*ð Þ þ 2δ þ λ<sup>1</sup>

þ 2θλ1λ2μ<sup>1</sup> μ<sup>0</sup>

λ1λ<sup>2</sup> μ<sup>0</sup>

<sup>2</sup> � μ<sup>2</sup>

*Probability, Combinatorics and Control*

$$\lambda\_1 = \lambda\_2$$

$$
\bar{\pi}\_{Z\_4}^2(r) = \frac{2\lambda^2 \mu\_2}{r(r+8)(r+28)} + \frac{26\lambda^2 \mu\_1(\mu\_1' - \mu\_1)}{r(r+8+2\lambda)(r+28)}
$$

$$
+ \frac{\lambda \mu\_2}{r(r+28)} \quad + \frac{6\lambda(\mu\_2' - \mu\_2)}{r(r+28+2\lambda)}\tag{48}
$$

$$\frac{2\Theta\lambda^2\mu\_1(\mu\_1'-\mu\_1)}{r(r+\delta)(r+2\delta+2\lambda)} + \frac{2\Theta^2\lambda^2\left(\mu\_1'-\mu\_1\right)^2}{r(r+\delta+2\lambda)(r+2\delta+2\lambda)},$$

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

The Laplace transform in Eq. (49), is a combination of terms of the form:

$$\tilde{\mathbf{g}}(r) = \frac{1}{r(a\_1 + r)(a\_2 + r)...(a\_n + r)},\tag{49}$$

Then,

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

<sup>¼</sup> <sup>λ</sup>1ð Þ <sup>2</sup><sup>δ</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> 2δð Þ 2δ þ λ<sup>1</sup>

<sup>¼</sup> <sup>λ</sup>1ð Þ <sup>2</sup><sup>δ</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> 2δð Þ 2δ þ λ<sup>1</sup>

λ2λ1ð Þ 2δ þ λ<sup>2</sup> δ2

λ2λ1ð Þ 2δ þ λ<sup>2</sup> δ2

2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup>

ð Þ 2δ þ λ<sup>1</sup>

¼ λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup>

ð Þ 2δ þ λ<sup>1</sup>

¼ λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> <sup>þ</sup>

ðÞ¼ *t* λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup>

¼ λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> <sup>þ</sup>

¼ λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> <sup>þ</sup>

¼ λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup>

ðÞ¼ *t* λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup>

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

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

*e* �ð Þ 2δþλ<sup>1</sup> *t* μ2

> 1 þ λ2 δ

μ<sup>2</sup> <sup>þ</sup>

1 þ λ2 δ 

1 þ λ2 δ 

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

<sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> � *<sup>e</sup>*

<sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> � *<sup>e</sup>*

λ2λ1ð Þ 2δ þ λ<sup>2</sup> δ2

2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup>

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

ð Þ 2δ þ λ<sup>1</sup>

þ λ2 2μ2 1 <sup>δ</sup><sup>2</sup> � <sup>λ</sup>2μ<sup>2</sup> 2δ *<sup>e</sup>*

þ

þ

þ

þ

þ

� *e*

� *e*

� *e*

� <sup>2</sup>λ<sup>2</sup> δ *e*

þ 2λ<sup>2</sup>

To finally have:

þ 2λ<sup>2</sup>

π2 *Zd*

**285**

π2 *Zd* μ<sup>2</sup> þ

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

� λ2 2δ *e*

� λ2

<sup>μ</sup><sup>2</sup>

2λ2μ<sup>2</sup> 1 δ 

�2*δ<sup>t</sup>* <sup>þ</sup>

� <sup>2</sup>λ1λ2μ<sup>2</sup> 1 δ δð Þ þ λ<sup>1</sup>

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

1 2δ þ λ<sup>1</sup>

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

<sup>μ</sup><sup>2</sup>

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

<sup>2</sup><sup>δ</sup> <sup>1</sup> � <sup>2</sup>δ*at*j2<sup>δ</sup>

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

� <sup>2</sup>λ2ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> δð Þ 2δ þ λ<sup>1</sup>

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

1 þ λ2 δ *<sup>e</sup>*

> �*δ<sup>t</sup>* <sup>þ</sup> λ2 2 δ2 *e* �2*δ<sup>t</sup>* <sup>þ</sup>

�*δ<sup>t</sup>* <sup>þ</sup> <sup>2</sup> λ2 2 δ2 *e*

�*δ<sup>t</sup>* <sup>þ</sup> <sup>2</sup> λ2 2 δ2 *e*

<sup>1</sup> � <sup>2</sup>λ2ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> <sup>1</sup> � ð Þ <sup>2</sup><sup>δ</sup> <sup>þ</sup> <sup>λ</sup><sup>1</sup> *at*j2δþλ<sup>1</sup>

δð Þ 2δ þ λ<sup>1</sup>

<sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup>

> 2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup>

<sup>μ</sup><sup>2</sup>

2δ þ λ<sup>1</sup>

*<sup>e</sup>*

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

*e* �ð Þ 2δþλ<sup>1</sup> *t*

1 þ λ2 δ *<sup>e</sup>*

1 þ λ2 δ *<sup>e</sup>*

1 þ λ2 δ *<sup>e</sup>*

2λ<sup>2</sup>

2λ<sup>2</sup>

2λ<sup>2</sup>

2λ<sup>2</sup>

μ<sup>2</sup>

μ<sup>2</sup>

� 2 λ2 2 δ2 *e* �*δ<sup>t</sup>* <sup>þ</sup>

> � 2 λ2 2 δ2 *e* �*δ<sup>t</sup>* <sup>þ</sup>

� 2 λ2 2 <sup>δ</sup><sup>2</sup> <sup>1</sup> � <sup>δ</sup>*at*j2<sup>δ</sup> <sup>þ</sup>

<sup>μ</sup><sup>2</sup>

<sup>μ</sup><sup>2</sup>

1*e* �ð Þ 2δþλ<sup>1</sup> *t*

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

�*δt*

<sup>2</sup>λ2ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> <sup>μ</sup><sup>2</sup>

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

1 � ð Þ 2δ þ λ<sup>1</sup> *at*j2δþλ<sup>1</sup>

�2*δ<sup>t</sup>* � <sup>λ</sup><sup>2</sup> 2 δ2 *e* �2*δt*

�2*δ<sup>t</sup>* � <sup>λ</sup><sup>2</sup> 2 <sup>δ</sup><sup>2</sup> <sup>1</sup> � <sup>2</sup>δ*at*j2<sup>δ</sup>

μ<sup>2</sup>

*e* �ð Þ δþλ<sup>1</sup> *t*

<sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup>

<sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup>

<sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup>

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

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

*e* �ð Þ δþλ<sup>1</sup> *t*

2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup>

2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup>

1

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

1

1

1

1

1

1

1 � ð Þ δ þ λ<sup>1</sup> *at*jδþλ<sup>1</sup>

1*:*

1*,*

�ð Þ 2δþλ<sup>1</sup> *t*

*e* �ð Þ 2δþλ<sup>1</sup> *t*

1

μ2 1

(56)

(57)

(58)

π2 *Zd* ðÞ¼ *t*

with *g* a function defined for all non-negative real numbers. As described in the proof of Theorem 1.1 in Baeumer [16], each of these terms can be expressed as a combinations of partial fraction such as ~*g r*ð Þ¼ *γ*<sup>0</sup> 1 *<sup>r</sup>* þ *γ*<sup>1</sup> 1 *<sup>α</sup>*1þ*<sup>r</sup>* <sup>þ</sup> … <sup>þ</sup> *<sup>γ</sup><sup>n</sup>* <sup>1</sup> *<sup>α</sup>n*þ*<sup>r</sup>* where.

 $\gamma\_0 = \frac{1}{a\_1 \dots a\_n}$ , for  $i = 1$ , ...,  $n$ ,  $\gamma\_i = -\frac{1}{a\_i} \prod\_{\substack{j = 1, \ j \neq i}} \frac{1}{a\_j - a\_i}$ .

Since the inverse Laplace transform of <sup>1</sup> *<sup>α</sup>i*þ*<sup>r</sup>* is *<sup>e</sup>*�*αit* , it is easy to invert ~*g* and obtain

$$\mathbf{g(t)} = \boldsymbol{\gamma}\_0 + \boldsymbol{\gamma}\_1 \boldsymbol{e}^{-a\_1 t} + \boldsymbol{\gamma}\_2 \boldsymbol{e}^{-a\_2 t} + \dots + \boldsymbol{\gamma}\_n \boldsymbol{e}^{-a\_n t}.\tag{50}$$

Using Eq. (49) in Eq. (53), it results that

$$\pi\_{\mathcal{Z}\_4}^2(t) = \chi\_0 + \chi\_1 e^{-\delta t} + \chi\_2 e^{-2\delta t} + \chi\_3 e^{-(2\delta + \lambda\_4)t} + \chi\_4 e^{-(\delta + 2\lambda\_4)t} + \chi\_5 e^{-(2\delta + 2\lambda\_4)t} + \chi\_6 e^{-(2\delta + 2\lambda\_5)t}, t \ge 0,\tag{51}$$

where *<sup>γ</sup><sup>i</sup>* ð Þ*<sup>i</sup>*∈f g <sup>0</sup>*;*1*;*2*;*…*;*<sup>6</sup> are given by equation Eq. (50). **Remarks** If θ ¼ 0 then

$$\begin{aligned} \mathcal{H}\_0^{0=0} &= \frac{\lambda\_1 \mu\_2}{2\delta + \lambda\_1} + \frac{2\lambda\_1 \lambda\_2 \mu\_1^2}{8(2\delta + \lambda\_1)} + \frac{\lambda\_1 \lambda\_2 \mu\_2}{28(2\delta + \lambda\_1)} + \frac{\lambda\_1 \lambda\_2^2 \mu\_1^2}{8^2(2\delta + \lambda\_1)} \\\\ &= \frac{\lambda\_1}{2\delta + \lambda\_1} \left(\frac{2\delta + \lambda\_2}{2\delta}\right) \mu\_2 + \frac{\lambda\_1 \lambda\_2}{8(2\delta + \lambda\_1)} \left(\frac{2\delta + \lambda\_2}{\delta}\right) \mu\_1^2 \\\\ &= \frac{\lambda\_1 (2\delta + \lambda\_2)}{2\delta (2\delta + \lambda\_1)} \left\{ \mu\_2 + \frac{2\lambda\_2 \mu\_1^2}{\delta} \right\}, \end{aligned} \tag{52}$$

$$\eta\_1^{\theta=0} = -\frac{2\lambda\_1\lambda\_2\mu\_1^2}{\delta(\delta+\lambda\_1)} - \frac{2\lambda\_1\lambda\_2^2\mu\_1^2}{\delta^2(\delta+\lambda\_1)} = -\frac{2\lambda\_1\lambda\_2\mu\_1^2}{\delta(\delta+\lambda\_1)}\left(1+\frac{\lambda\_2}{\delta}\right), \\ \eta\_2^{\theta=0} = \frac{\lambda\_2^2\mu\_1^2}{\delta^2} - \frac{\lambda\_2\mu\_2}{2\delta} \tag{53}$$

$$
\begin{split}
\gamma\_{3}^{\theta=0} &= -\frac{\lambda\_{1}\mu\_{2}}{2\delta + \lambda\_{1}} + \frac{2\lambda\_{1}\lambda\_{2}\mu\_{1}^{2}}{(2\delta + \lambda\_{1})(\delta + \lambda\_{1})} + \frac{\lambda\_{2}\mu\_{2}}{2\delta + \lambda\_{1}} - \frac{2\lambda\_{2}^{2}\mu\_{1}^{2}}{(2\delta + \lambda\_{1})(\delta + \lambda\_{1})} \\\\
&= \frac{(\lambda\_{2} - \lambda\_{1})\mu\_{2}}{2\delta + \lambda\_{1}} + \frac{2\lambda\_{2}(\lambda\_{1} - \lambda\_{2})\mu\_{1}^{2}}{(2\delta + \lambda\_{1})(\delta + \lambda\_{1})},
\end{split} \tag{54}
$$

$$
\gamma\_{4}^{\theta=0} = \gamma\_{5}^{\theta=0} = \gamma\_{6}^{\theta=0}. \tag{55}
$$

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

Then,

The Laplace transform in Eq. (49), is a combination of terms of the form:

with *g* a function defined for all non-negative real numbers. As described in the proof of Theorem 1.1 in Baeumer [16], each of these terms can be expressed as a

> 1 *<sup>r</sup>* þ *γ*<sup>1</sup>

*<sup>α</sup>i*þ*<sup>r</sup>* is *<sup>e</sup>*�*αit*

�*α*2*<sup>t</sup>* <sup>þ</sup> … <sup>þ</sup> *<sup>γ</sup>ne*

�ð Þ <sup>δ</sup>þ2λ<sup>2</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>γ</sup>*5*<sup>e</sup>*

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

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

*,*

1 δ δð Þ þ λ<sup>1</sup>

þ

*,*

<sup>5</sup> <sup>¼</sup> *<sup>γ</sup>*<sup>θ</sup>¼<sup>0</sup>

1

1 þ λ2 δ � �

λ2μ<sup>2</sup> 2δ þ λ<sup>1</sup> þ

δ2

2δ þ λ<sup>2</sup> δ � �

> *, γ*<sup>θ</sup>¼<sup>0</sup> <sup>2</sup> <sup>¼</sup> <sup>λ</sup><sup>2</sup>

� <sup>2</sup>λ<sup>2</sup>

1 *αj*�*α<sup>i</sup>* .

*<sup>r</sup>*ð Þ *<sup>α</sup>*<sup>1</sup> <sup>þ</sup> *<sup>r</sup>* ð Þ *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>r</sup>* …ð Þ *<sup>α</sup><sup>n</sup>* <sup>þ</sup> *<sup>r</sup> ,* (49)

1

�*αnt*

*<sup>α</sup>*1þ*<sup>r</sup>* <sup>þ</sup> … <sup>þ</sup> *<sup>γ</sup><sup>n</sup>* <sup>1</sup>

�ð Þ <sup>2</sup>δþ2λ<sup>1</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>γ</sup>*6*<sup>e</sup>*

λ1λ<sup>2</sup> 2μ2 1

ð Þ 2δ þ λ<sup>1</sup>

μ2 1

2μ2 1 <sup>δ</sup><sup>2</sup> � <sup>λ</sup>2μ<sup>2</sup>

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

<sup>6</sup> *:* (55)

*<sup>α</sup>n*þ*<sup>r</sup>* where.

, it is easy to invert ~*g* and

*:* (50)

�ð Þ 2δþ2λ<sup>2</sup> *t*

*, t*≥ 0*,* (51)

(52)

<sup>2</sup><sup>δ</sup> (53)

(54)

<sup>~</sup>*g r*ð Þ¼ <sup>1</sup>

*αi* Q *<sup>j</sup>*¼1*,j*6¼*<sup>i</sup>*

�*α*1*<sup>t</sup>* <sup>þ</sup> *<sup>γ</sup>*2*<sup>e</sup>*

�ð Þ <sup>2</sup>δþλ<sup>1</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>γ</sup>*4*<sup>e</sup>*

combinations of partial fraction such as ~*g r*ð Þ¼ *γ*<sup>0</sup>

*Probability, Combinatorics and Control*

, for *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*,* …*, n*, *<sup>γ</sup><sup>i</sup>* ¼ � <sup>1</sup>

Since the inverse Laplace transform of <sup>1</sup>

Using Eq. (49) in Eq. (53), it results that

�*δ<sup>t</sup>* <sup>þ</sup> *<sup>γ</sup>*2*<sup>e</sup>*

*γ*θ¼<sup>0</sup>

<sup>0</sup> <sup>¼</sup> <sup>λ</sup>1μ<sup>2</sup> 2δ þ λ<sup>1</sup>

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

<sup>¼</sup> <sup>λ</sup>1ð Þ <sup>2</sup><sup>δ</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> 2δð Þ 2δ þ λ<sup>1</sup>

� <sup>2</sup>λ1λ<sup>2</sup>

δ2

þ

þ

2μ2 1

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

<sup>2</sup>λ2ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> <sup>μ</sup><sup>2</sup>

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

*γ*<sup>θ</sup>¼<sup>0</sup> <sup>4</sup> <sup>¼</sup> *<sup>γ</sup>*<sup>θ</sup>¼<sup>0</sup>

ð Þ δ þ λ<sup>1</sup>

*g t*ðÞ¼ *γ*<sup>0</sup> þ *γ*1*e*

�2*δ<sup>t</sup>* <sup>þ</sup> *<sup>γ</sup>*3*<sup>e</sup>*

where *<sup>γ</sup><sup>i</sup>* ð Þ*<sup>i</sup>*∈f g <sup>0</sup>*;*1*;*2*;*…*;*<sup>6</sup> are given by equation Eq. (50).

þ

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

2δ þ λ<sup>2</sup> 2δ � �

μ<sup>2</sup> þ

¼ � <sup>2</sup>λ1λ2μ<sup>2</sup>

þ

μ<sup>2</sup> þ

2λ2μ<sup>2</sup> 1 δ � �

*<sup>γ</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> *α*1…*α<sup>n</sup>*

ðÞ¼ *t γ*<sup>0</sup> þ *γ*1*e*

**Remarks** If θ ¼ 0 then

*γ*<sup>θ</sup>¼<sup>0</sup>

**284**

<sup>1</sup> ¼ � <sup>2</sup>λ1λ2μ<sup>2</sup>

<sup>3</sup> ¼ � <sup>λ</sup>1μ<sup>2</sup>

*γ*<sup>θ</sup>¼<sup>0</sup>

1 δ δð Þ þ λ<sup>1</sup>

2δ þ λ<sup>1</sup>

<sup>¼</sup> ð Þ <sup>λ</sup><sup>2</sup> � <sup>λ</sup><sup>1</sup> <sup>μ</sup><sup>2</sup> 2δ þ λ<sup>1</sup>

obtain

π2 *Zd*

π2 *Zd* ðÞ¼ *t* λ1ð Þ 2δ þ λ<sup>2</sup> 2δð Þ 2δ þ λ<sup>1</sup> μ<sup>2</sup> þ 2λ2μ<sup>2</sup> 1 δ � <sup>2</sup>λ1λ2μ<sup>2</sup> 1 δ δð Þ þ λ<sup>1</sup> 1 þ λ2 δ *<sup>e</sup>* �*δt* þ λ2 2μ2 1 <sup>δ</sup><sup>2</sup> � <sup>λ</sup>2μ<sup>2</sup> 2δ *<sup>e</sup>* �2*δ<sup>t</sup>* <sup>þ</sup> 1 2δ þ λ<sup>1</sup> ð Þ λ<sup>2</sup> � λ<sup>1</sup> μ<sup>2</sup> þ <sup>2</sup>λ2ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> <sup>μ</sup><sup>2</sup> 1 ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup> *<sup>e</sup>* �ð Þ 2δþλ<sup>1</sup> *t* <sup>¼</sup> <sup>λ</sup>1ð Þ <sup>2</sup><sup>δ</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> 2δð Þ 2δ þ λ<sup>1</sup> � λ2 2δ *e* �2*δ<sup>t</sup>* <sup>þ</sup> ð Þ <sup>λ</sup><sup>2</sup> � <sup>λ</sup><sup>1</sup> 2δ þ λ<sup>1</sup> *e* �ð Þ 2δþλ<sup>1</sup> *t* <sup>μ</sup><sup>2</sup> þ λ2λ1ð Þ 2δ þ λ<sup>2</sup> δ2 ð Þ 2δ þ λ<sup>1</sup> � <sup>2</sup>λ1λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> 1 þ λ2 δ *<sup>e</sup>* �*δ<sup>t</sup>* <sup>þ</sup> λ2 2 δ2 *e* �2*δ<sup>t</sup>* <sup>þ</sup> 2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> ð Þ 2δ þ λ<sup>1</sup> ð Þ δ þ λ<sup>1</sup> *e* �ð Þ 2δþλ<sup>1</sup> *t* μ2 1 <sup>¼</sup> <sup>λ</sup>1ð Þ <sup>2</sup><sup>δ</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> 2δð Þ 2δ þ λ<sup>1</sup> � λ2 <sup>2</sup><sup>δ</sup> <sup>1</sup> � <sup>2</sup>δ*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>2</sup> � <sup>λ</sup><sup>1</sup> 2δ þ λ<sup>1</sup> 1 � ð Þ 2δ þ λ<sup>1</sup> *at*j2δþλ<sup>1</sup> <sup>μ</sup><sup>2</sup> þ λ2λ1ð Þ 2δ þ λ<sup>2</sup> δ2 ð Þ 2δ þ λ<sup>1</sup> � <sup>2</sup>λ1λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> 1 þ λ2 δ *<sup>e</sup>* �*δ<sup>t</sup>* <sup>þ</sup> <sup>2</sup> λ2 2 δ2 *e* �2*δ<sup>t</sup>* � <sup>λ</sup><sup>2</sup> 2 δ2 *e* �2*δt* <sup>μ</sup><sup>2</sup> 1 þ 2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> � <sup>2</sup>λ2ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> δð Þ 2δ þ λ<sup>1</sup> <sup>μ</sup><sup>2</sup> 1*e* �ð Þ 2δþλ<sup>1</sup> *t* ¼ λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> þ λ2λ1ð Þ 2δ þ λ<sup>2</sup> δ2 ð Þ 2δ þ λ<sup>1</sup> � <sup>2</sup>λ1λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> 1 þ λ2 δ *<sup>e</sup>* �*δ<sup>t</sup>* <sup>þ</sup> <sup>2</sup> λ2 2 δ2 *e* �2*δ<sup>t</sup>* � <sup>λ</sup><sup>2</sup> 2 <sup>δ</sup><sup>2</sup> <sup>1</sup> � <sup>2</sup>δ*at*j2<sup>δ</sup> <sup>μ</sup><sup>2</sup> 1 þ 2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> *e* �ð Þ 2δþλ<sup>1</sup> *t* μ2 <sup>1</sup> � <sup>2</sup>λ2ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> <sup>1</sup> � ð Þ <sup>2</sup><sup>δ</sup> <sup>þ</sup> <sup>λ</sup><sup>1</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> 1 δð Þ 2δ þ λ<sup>1</sup> ¼ λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> <sup>þ</sup> 2λ<sup>2</sup> <sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> 1 � *e <sup>δ</sup><sup>t</sup>* 2λ1λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> 1 þ λ2 δ � 2 λ2 2 δ2 *e* �*δ<sup>t</sup>* <sup>þ</sup> 2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> *e* �ð Þ δþλ<sup>1</sup> *t* (56)

π2 *Zd* ðÞ¼ *t* λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> <sup>þ</sup> 2λ<sup>2</sup> <sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> 1 � *e <sup>δ</sup><sup>t</sup>* 2λ1λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> 1 þ λ2 δ � 2 λ2 2 δ2 *e* �*δ<sup>t</sup>* <sup>þ</sup> 2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> *e* �ð Þ δþλ<sup>1</sup> *t* ¼ λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> <sup>þ</sup> 2λ<sup>2</sup> <sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> 1 � *e <sup>δ</sup><sup>t</sup>* 2λ1λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> 1 þ λ2 δ � 2 λ2 2 <sup>δ</sup><sup>2</sup> <sup>1</sup> � <sup>δ</sup>*at*j2<sup>δ</sup> <sup>þ</sup> 2λ2ð Þ λ<sup>1</sup> � λ<sup>2</sup> δ δð Þ þ λ<sup>1</sup> 1 � ð Þ δ þ λ<sup>1</sup> *at*jδþλ<sup>1</sup> ¼ λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> <sup>þ</sup> 2λ<sup>2</sup> <sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> 1 � <sup>2</sup>λ<sup>2</sup> δ *e <sup>δ</sup><sup>t</sup>* <sup>λ</sup>2*at*j<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*jδþλ<sup>1</sup> ¼ λ2*at*j2<sup>δ</sup> þ ð Þ λ<sup>1</sup> � λ<sup>2</sup> *at*j2δþλ<sup>1</sup> μ<sup>2</sup> þ 2λ<sup>2</sup> <sup>δ</sup> <sup>λ</sup>2*at*j2<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*j2δþλ<sup>1</sup> � *<sup>e</sup> <sup>δ</sup><sup>t</sup>* <sup>λ</sup>2*at*j<sup>δ</sup> <sup>þ</sup> ð Þ <sup>λ</sup><sup>1</sup> � <sup>λ</sup><sup>2</sup> *at*jδþλ<sup>1</sup> μ<sup>2</sup> 1*:* (57)

To finally have:

$$\begin{split} \pi\_{\Sigma\_{4}}^{2}(\mathbf{t}) &= \left(\lambda\_{2}\overline{\mathbf{a}}\_{\mathbf{t}|2\delta} + (\lambda\_{1} - \lambda\_{2})\overline{\mathbf{a}}\_{\mathbf{t}|2\delta + \lambda\_{1}}\right)\mu\_{2} \\ &+ \frac{\mathcal{D}\lambda\_{2}}{\delta} \left(\lambda\_{2}\overline{\mathbf{a}}\_{\mathbf{t}|2\delta} + (\lambda\_{1} - \lambda\_{2})\overline{\mathbf{a}}\_{\mathbf{t}|2\delta + \lambda\_{1}} - \mathbf{e}^{\delta t} \left\{ \left(\lambda\_{2}\overline{\mathbf{a}}\_{\mathbf{t}|\delta} + (\lambda\_{1} - \lambda\_{2})\overline{\mathbf{a}}\_{\mathbf{t}|\delta + \lambda\_{1}}\right) \right\} \right) \mu\_{\mathbf{t}}^{2} \end{split} \tag{58}$$

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

$$\begin{split} \mathbf{y}\_{0}^{\lambda\_{1}=\lambda\_{2}} &= \frac{\lambda\mu\_{2}}{2\mathsf{\hat{s}}} + \frac{\lambda^{2}\mu\_{1}^{2}}{\mathsf{\hat{s}}^{2}} + \left(\frac{\mathsf{q}\lambda^{2}}{\mathsf{\hat{s}}(\mathsf{\hat{s}}+\lambda)} + \frac{\mathsf{q}\lambda^{2}}{\mathsf{\hat{s}}(\mathsf{\hat{s}}+2\mathsf{\hat{s}})}\right)\mu\_{1}(\mathsf{\mu}\_{1}^{\prime}-\mathsf{\mu}\_{1}) + \frac{\mathsf{q}\lambda(\mathsf{\mu}\_{2}^{\prime}-\mathsf{\mu}\_{2})}{2(\mathsf{\hat{s}}+\lambda)} \\ &+ \frac{\mathsf{q}^{2}\lambda^{2}\left(\mathsf{\mu}\_{1}^{\prime}-\mathsf{\mu}\_{1}\right)^{2}}{(\mathsf{\hat{s}}+\lambda)(\mathsf{\hat{s}}+2\mathsf{\hat{s}})}, \end{split} \tag{59}$$

$$\chi\_1^{\lambda\_1=\lambda\_2} = -\frac{2^{\lambda\_1^2 \mu\_1^2}}{\delta^2} - \frac{2\theta \lambda^2 \mu\_1 (\mu\_1' - \mu\_1)}{\delta(\delta + 2\lambda)},\\\chi\_2^{\lambda\_1=\lambda\_2} = -\frac{\lambda \mu\_2}{2\delta} + \frac{\lambda^2 \mu\_1^2}{\delta^2} + \frac{\theta \lambda^2 \mu\_1 (\mu\_1' - \mu\_1)}{\delta(\delta - 2\lambda)},\\\chi\_3^{\lambda\_1=\lambda\_2} = 0 \tag{60}$$

$$\gamma\_4^{\lambda\_1=\lambda\_2} = -\frac{2\Theta\lambda^2\mu\_1(\mu\_1'-\mu\_1)}{(8+2\lambda)(8-2\lambda)} - \frac{2\Theta^2\lambda^2(\mu\_1'-\mu\_1)^2}{8(8+2\lambda)},\tag{61}$$

We can rewrite ~π*Zo*

¼ 1 *r*

The term ~π*<sup>m</sup>*

*j*

*Zo*

<sup>1</sup>*; p*<sup>1</sup> � �*,* …*, in; j*

þ

**4.1 First two moments**

ð Þ <sup>1</sup> � *<sup>p</sup> Exp <sup>β</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

moment of *X* is <sup>μ</sup>*<sup>m</sup>* <sup>¼</sup> *<sup>p</sup> <sup>m</sup>*! *βm* 1

**Table 1.**

**287**

*i*1*; j* <sup>1</sup>*; p*<sup>1</sup> � �… *in; j*

<sup>1</sup> ¼ *m* � 1 þ *n, j*<sup>1</sup> þ … þ *j*

λ1 ð Þ *r* þ *m*δ þ λ<sup>1</sup>

<sup>þ</sup> ð Þ <sup>1</sup> � *<sup>p</sup> <sup>m</sup>*!

ð Þ <sup>1</sup> � *<sup>p</sup> <sup>m</sup>*!

*E Z*½ � *<sup>d</sup>*ð Þ*t for* λ*<sup>1</sup>* ¼ *1*, λ*<sup>2</sup>* ¼ *10*, δ ¼ *3*%*.*

*βm* 2 and μ<sup>0</sup>

~π2 *Zo* ð Þ¼ *r* 1 *r*

~π*m Zo* ð Þ¼ *r* 1 *r* X*m i*¼1

where <sup>℘</sup>*m,n* <sup>¼</sup> *<sup>i</sup>*1*; <sup>j</sup>*

and

~π*m Zd* ð Þ¼ *r*

**4. Application**

ð Þ*<sup>r</sup>* and <sup>~</sup>π<sup>2</sup>

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

~π*Zo* ð Þ¼ *r* 1 *r*

*Zo* ð Þ*r* as

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

ð Þ*r* can also be expressed using

*<sup>n</sup>; pn*

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

*<sup>n</sup>; pn*

For the numerical illustration, suppose that *<sup>X</sup>* � *pExp <sup>β</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>

*<sup>m</sup>* <sup>¼</sup> <sup>Ð</sup> ∞ 0

� �<sup>∈</sup> <sup>℘</sup>*m,nη<sup>k</sup> in; <sup>j</sup>*

*<sup>n</sup>* ¼ *m, j*<sup>1</sup> . … . *j*

( )

1 *r*

� �~π*<sup>m</sup>*

<sup>200</sup> � �, the inter-claim time distribution parameters <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup>*;* 4 and

*mxm*�<sup>1</sup>ð Þ <sup>1</sup> � *FX*ð Þ *<sup>x</sup>*

<sup>2</sup>*β*<sup>2</sup> ð Þ*<sup>m</sup>*. (72)

λ<sup>2</sup> ¼ 1, the interest rate δ ¼ 3% (**Tables 1**–**4**). We use three different values for the copula parameter θ ¼ �1*;* 0*;* 1, *p* ¼ <sup>1</sup>*=*<sup>3</sup> and fix the time *t* ¼ 1*;* 10*;* 100. The *m*th

**θ** *t* ¼ **1** *t* ¼ **10** *t* ¼ **100** �1 482.3375 4450 16,428 0 438.1057 4407.1 16,385 1 393.874 4364.2 16,342

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

*ηk*ð Þþ 1*;* 1*;* 0 *η<sup>k</sup>* ½ � ð Þ 1*;* 1*;* 1 *,* (67)

(68)

, (70)

(71)

<sup>1</sup>*; p*<sup>1</sup> � �*,* (69)

*ηk*ð Þþ 2*;* 2*;* 0 *ηk*ð Þþ 2*;* 2*;* 1 ð Þ *ηk*ð Þþ 2*;* 1*;* 0 *ηk*ð Þ 2*;* 1*;* 1 *ηk*ð Þþ 1*;* 1*;* 0 *η<sup>k</sup>* ½ � ð Þ ð Þ 1*;* 1*;* 1

þ *ηk*ð Þ 2*;* 1*;* 1 *ηk*ð Þþ 1*;* 1*;* 0 *ηk*ð Þ 2*;* 1*;* 1 *ηk*ð Þ 1*;* 1*;* 1

� � : *<sup>i</sup>*<sup>1</sup> <sup>¼</sup> *m, i*<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *in* <sup>¼</sup> *<sup>m</sup>* � <sup>1</sup> <sup>þ</sup> *n, i*<sup>1</sup> . … . *in,*

<sup>μ</sup>*<sup>m</sup>* <sup>þ</sup>X*<sup>m</sup>*�<sup>1</sup> *k*¼1 *Ck mE X<sup>k</sup>* 1 � �~πð Þ *<sup>m</sup>*�*<sup>k</sup> Zo* ð Þ*r*

*<sup>n</sup>; pn*

*<sup>n</sup>, p* ∈f g 0*;* 1

� � � … � *<sup>η</sup><sup>k</sup> <sup>i</sup>*1*; <sup>j</sup>*

!

*<sup>Z</sup>*<sup>0</sup> ð Þ*r ,*

<sup>80</sup> � �<sup>þ</sup>

*dx* <sup>¼</sup> <sup>μ</sup>*<sup>m</sup>* <sup>¼</sup> *<sup>p</sup> <sup>m</sup>*!

<sup>2</sup>*β*<sup>1</sup> ð Þ*<sup>m</sup>* þ

þ 1

2

*ηk*ð Þþ 2*;* 2*;* 0 *ηk*ð Þþ 2*;* 2*;* 1 *ηk*ð Þ 2*;* 1*;* 0 *ηk*ð Þþ 1*;* 1*;* 0 *ηk*ð Þ 2*;* 1*;* 0 *ηk*ð Þ 1*;* 1*;* 1

" #

$$\gamma\_5^{\lambda\_1=\lambda\_2} = -\frac{\Theta\lambda(\mu\_2'-\mu\_2)}{\Theta+\lambda} + \frac{2\Theta\lambda^2\mu\_1(\mu\_1'-\mu\_1)}{(\Theta+\lambda)(\Theta+2\lambda)} + \frac{2\Theta^2\lambda^2(\mu\_1'-\mu\_1)^2}{\Theta(\Theta+\lambda)},\tag{62}$$

$$\gamma\_{6}^{\lambda\_{1}=\lambda\_{2}} = \frac{\Theta\lambda(\mu\_{2}^{\prime}-\mu\_{2})}{2(\delta+\lambda)} - \frac{\Theta\lambda^{2}\mu\_{1}(\mu\_{1}^{\prime}-\mu\_{1})}{(\delta+\lambda)(\delta+2\lambda)} - \frac{\Theta^{2}\lambda^{2}\left(\mu\_{1}^{\prime}-\mu\_{1}\right)^{2}}{\delta(\delta+\lambda)}.\tag{63}$$

Then,

$$\begin{split} \pi\_{Z\_{\omega}}^{2}(t) &= \lambda \mu\_{2} \Big( \frac{1}{8} - \frac{e^{-\delta t}}{2\delta} \Big) + 6\lambda \left( \mu\_{1}^{\prime} - \mu\_{2} \right) \Big( \frac{1}{2\lambda + 2\delta} - \frac{e^{-(2\lambda + 2\delta)t}}{2\lambda + 2\delta} \Big) \\ &+ 2\lambda^{2} \mu\_{1}^{2} \Big( \frac{1}{2\delta^{2}} - \frac{e^{-\delta t}}{\delta^{2}} + \frac{e^{-2\delta t}}{2\delta^{2}} \Big) \\ &+ 2\theta\lambda^{2} \mu\_{1} \big( \mu\_{1}^{\prime} - \mu\_{1} \big) \Big( \frac{1}{2\delta(2\lambda + \delta)} - \frac{e^{-(2\lambda + \delta)t}}{(2\lambda + \delta)(-2\lambda + \delta)} + \frac{e^{-2\delta t}}{2\delta(-2\lambda + \delta)} \Big) \\ &+ 2\theta\lambda^{2} \mu\_{1} \big( \mu\_{1}^{\prime} - \mu\_{1} \big) \Big( \frac{1}{8(2\lambda + 2\delta)} - \frac{e^{-\delta t}}{8(2\lambda + \delta)} + \frac{e^{-(2\lambda + 2\delta)t}}{(2\lambda + 2\delta)(2\lambda + \delta)} \Big) \\ &+ 2\theta^{2}\lambda^{2} \big( \mu\_{1}^{\prime} - \mu\_{1} \big)^{2} \Big( \frac{1}{(2\lambda + \delta)(2\lambda + 2\delta)} - \frac{e^{-(2\lambda + \delta)t}}{\delta(2\lambda + \delta)} + \frac{e^{-(2\lambda + 2\delta)t}}{\delta(2\lambda + 2\delta)} \Big) \end{split} \tag{64}$$

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

$$
\pi\_{Z\_o}^2(t) = \lambda \overline{\pi}\_{t|28} \mu\_2 + \left(\lambda \overline{\pi}\_{t|28} \mu\_1\right)^2,\tag{65}
$$

which is exactly the result of Léveillé et al. [15]. **Remark 3.1**

Noting for *i* ¼ 1*,* 2*,* …*, m*,*j* ¼ 1*,* 2*,* …*, m*, *p* ¼ 0*,* 1 and *k*∈ N � f g0

$$\eta\_{k}(i,j,p) = \frac{\binom{i}{j} \Theta^{p} \lambda^{k} \left(E\left[\mathbf{X}^{j}\right]\right)^{1-p} \left(E\left[\mathbf{X}^{j}\right] - E\left[\mathbf{X}^{j}\right]\right)^{p}}{\left(r+p\times2\lambda+i\delta\right)^{k}}.\tag{66}$$

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

We can rewrite ~π*Zo* ð Þ*<sup>r</sup>* and <sup>~</sup>π<sup>2</sup> *Zo* ð Þ*r* as

$$
\tilde{\pi}\_{Z\_0}(r) = \frac{1}{r} [\eta\_k(\mathbf{1}, \mathbf{1}, \mathbf{0}) + \eta\_k(\mathbf{1}, \mathbf{1}, \mathbf{1})],\tag{67}
$$

$$\begin{split} \bar{\pi}\_{\mathbf{Z}\_{\*}}^{2}(r) &= \frac{1}{r} [\eta\_{k}(\mathbf{2},\mathbf{2},\mathbf{0}) + \eta\_{k}(\mathbf{2},\mathbf{2},\mathbf{1}) + (\eta\_{k}(\mathbf{2},\mathbf{1},\mathbf{0}) + \eta\_{k}(\mathbf{2},\mathbf{1},\mathbf{1}))(\eta\_{k}(\mathbf{1},\mathbf{1},\mathbf{0}) + \eta\_{k}(\mathbf{1},\mathbf{1},\mathbf{1}))] \\ &= \frac{1}{r} \Biggl[ \begin{split} \eta\_{k}(\mathbf{2},\mathbf{2},\mathbf{0}) + \eta\_{k}(\mathbf{2},\mathbf{2},\mathbf{1}) + \eta\_{k}(\mathbf{2},\mathbf{1},\mathbf{0})\eta\_{k}(\mathbf{1},\mathbf{1},\mathbf{0}) + \eta\_{k}(\mathbf{2},\mathbf{1},\mathbf{0})\eta\_{k}(\mathbf{1},\mathbf{1},\mathbf{1}) \\ &+ \eta\_{k}(\mathbf{2},\mathbf{1},\mathbf{1})\eta\_{k}(\mathbf{1},\mathbf{1},\mathbf{0}) + \eta\_{k}(\mathbf{2},\mathbf{1},\mathbf{1})\eta\_{k}(\mathbf{1},\mathbf{1},\mathbf{1}) \end{split} \tag{68} \end{split} \tag{69} $$

The term ~π*<sup>m</sup> Zo* ð Þ*r* can also be expressed using

$$\tilde{\pi}\_{\mathbf{Z}\_{\mathbf{o}}}^{m}(r) = \frac{1}{r} \sum\_{i=1}^{m} \left( i\_1, j\_1, p\_1 \right) ... \left( i\_n, j\_n, p\_n \right) \in \wp\_{m,n} \eta\_k \left( i\_n, j\_n, p\_n \right) \times ... \times \eta\_k \left( i\_1, j\_1, p\_1 \right), \tag{69}$$

where

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

θλ<sup>2</sup> δ δð Þ <sup>þ</sup> <sup>λ</sup> <sup>þ</sup>

θλ<sup>2</sup> δ δð Þ þ 2λ

μ<sup>1</sup> μ<sup>0</sup>

ð Þ <sup>δ</sup> <sup>þ</sup> <sup>λ</sup> ð Þ <sup>δ</sup> <sup>þ</sup> <sup>2</sup><sup>λ</sup> *,* (59)

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

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

<sup>2</sup>δð Þ <sup>2</sup><sup>λ</sup> <sup>þ</sup> <sup>δ</sup> � *<sup>e</sup>*�ð Þ <sup>2</sup>λþ<sup>δ</sup> *<sup>t</sup>*

ð Þ <sup>2</sup><sup>λ</sup> <sup>þ</sup> <sup>δ</sup> ð Þ <sup>2</sup><sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>δ</sup> � *<sup>e</sup>*�ð Þ <sup>2</sup>λþ<sup>δ</sup> *<sup>t</sup>*

<sup>δ</sup>ð Þ <sup>2</sup><sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>δ</sup> � *<sup>e</sup>*�*δ<sup>t</sup>*

ðÞ¼ *t* λ*at*j<sup>2</sup>δμ<sup>2</sup> þ λ*at*j<sup>2</sup>δμ<sup>1</sup>

<sup>θ</sup>*<sup>p</sup>*λ*<sup>k</sup> E X<sup>j</sup>* � � � � <sup>1</sup>�*<sup>p</sup> E X*0*<sup>j</sup>* h i

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

μ<sup>1</sup> μ<sup>0</sup>

<sup>1</sup> � μ<sup>1</sup> � � <sup>þ</sup> θλ μ<sup>0</sup>

> θλ2μ<sup>1</sup> <sup>μ</sup><sup>0</sup> ð Þ <sup>1</sup>�μ<sup>1</sup> δ δð Þ �2<sup>λ</sup> ,*<sup>γ</sup>*

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

λ<sup>2</sup> μ<sup>0</sup>

2λ þ 2δ

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

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

� *E X<sup>j</sup>* � � � � *<sup>p</sup>*

ð Þ *<sup>r</sup>* <sup>þ</sup> *<sup>p</sup>* � <sup>2</sup><sup>λ</sup> <sup>þ</sup> *<sup>i</sup><sup>δ</sup> <sup>k</sup> :* (66)

� �

� �

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

� �<sup>2</sup>

� �

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

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

λ<sup>2</sup> μ<sup>0</sup>

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

<sup>2</sup><sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>δ</sup> � *<sup>e</sup>*�ð Þ <sup>2</sup>λþ2<sup>δ</sup> *<sup>t</sup>*

� �

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

λ1¼λ<sup>2</sup>

δ δð Þ <sup>þ</sup> <sup>2</sup><sup>λ</sup> *,* (61)

δ δð Þ <sup>þ</sup> <sup>λ</sup> *,* (62)

δ δð Þ <sup>þ</sup> <sup>λ</sup> *:* (63)

*e*�2*δ<sup>t</sup>* 2δð Þ �2λ þ δ

*e*�ð Þ <sup>2</sup>λþ2<sup>δ</sup> *<sup>t</sup>* ð Þ 2λ þ 2δ ð Þ 2λ þ δ

> *e*�ð Þ <sup>2</sup>λþ2<sup>δ</sup> *<sup>t</sup>* δð Þ 2λ þ 2δ

*,* (65)

(64)

<sup>3</sup> ¼ 0 (60)

� �

λ1¼λ<sup>2</sup> <sup>2</sup> ¼ � λμ<sup>2</sup>

μ<sup>1</sup> μ<sup>0</sup>

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

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

If λ<sup>1</sup> ¼ λ<sup>2</sup> then

2δ þ

þ θ2 λ<sup>2</sup> μ<sup>0</sup>

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

*Probability, Combinatorics and Control*

1

<sup>5</sup> ¼ � θλ μ<sup>0</sup>

1 <sup>δ</sup> � *<sup>e</sup>*�*δ<sup>t</sup>* 2δ � �

*γ* λ1¼λ<sup>2</sup> <sup>6</sup> <sup>¼</sup> θλ μ<sup>0</sup>

ðÞ¼ *t* λμ<sup>2</sup>

<sup>þ</sup> <sup>2</sup>λ<sup>2</sup> μ2 1 1 <sup>2</sup>δ<sup>2</sup> � *<sup>e</sup>*�*δ<sup>t</sup>*

<sup>þ</sup> <sup>2</sup>θλ<sup>2</sup>

<sup>þ</sup> <sup>2</sup>θλ<sup>2</sup>

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

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

**Remark 3.1**

**286**

μ<sup>1</sup> μ<sup>0</sup>

μ<sup>1</sup> μ<sup>0</sup>

λ<sup>2</sup> μ<sup>0</sup>

<sup>1</sup> � μ<sup>1</sup> � � 1

<sup>1</sup> � μ<sup>1</sup> � � 1

<sup>1</sup> � μ<sup>1</sup>

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

π2 *Zo*

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

*i j* � �

*ηk*ð Þ¼ *i; j; p*

Noting for *i* ¼ 1*,* 2*,* …*, m*,*j* ¼ 1*,* 2*,* …*, m*, *p* ¼ 0*,* 1 and *k*∈ N � f g0

*γ* λ1¼λ<sup>2</sup>

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

<sup>δ</sup><sup>2</sup> � <sup>2</sup>θλ2μ<sup>1</sup> <sup>μ</sup><sup>0</sup> ð Þ <sup>1</sup>�μ<sup>1</sup>

δ δð Þ <sup>þ</sup>2<sup>λ</sup> , *<sup>γ</sup>*

<sup>4</sup> ¼ � <sup>2</sup>θλ<sup>2</sup>

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

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

> > þ θλ μ<sup>0</sup>

<sup>δ</sup><sup>2</sup> <sup>þ</sup>

� �<sup>2</sup> 1

� �

<sup>2</sup> � μ<sup>2</sup> � � 1

> *e*�2*δ<sup>t</sup>* 2δ<sup>2</sup>

*γ* λ1¼λ<sup>2</sup> <sup>0</sup> <sup>¼</sup> λμ<sup>2</sup>

*γ* λ1¼λ<sup>2</sup> <sup>1</sup> ¼ � <sup>2</sup>λ2μ<sup>2</sup>

> *γ* λ1¼λ<sup>2</sup>

Then,

π2 *Zd*

$$\otimes\_{m,n} = \left\{ \begin{aligned} (i\_1, j\_1, p\_1), \dots, (i\_n, j\_n, p\_n) &: i\_1 = m, i\_1 + \dots + i\_n = m - 1 + n, i\_1 \ge \dots \ge i\_m \\ j\_1 = m - 1 + n, j\_1 + \dots + j\_n = m, j\_1 \ge \dots \ge j\_n, p \in \{0, 1\} \\ \text{and} \end{aligned} \right\}, \text{(70)}$$

$$\begin{split} \tilde{\pi}\_{Z\_d}^m(r) &= \frac{2\lambda\_1(\lambda\_1 - \lambda\_2)}{r(r + m\mathfrak{G} + \lambda\_1)(r + m\mathfrak{G} + 2\lambda\_1)} \left( \frac{1}{r}\mu\_m + \sum\_{k=1}^{m-1} C\_m^k E\left[X\_1^k\right] \tilde{\pi}\_{Z\_b}^{(m-k)}(r) \right) \\ &+ \frac{\lambda\_1}{(r + m\mathfrak{G} + \lambda\_1)} \left( \frac{(r + m\mathfrak{G})(r + m\mathfrak{G} + 2\lambda\_2)}{\lambda\_2(r + m\mathfrak{G} + 2\lambda\_1)} + 1 \right) \tilde{\pi}\_{Z\_b}^m(r), \end{split} \tag{71}$$
