The construction of a quadratic predictor of the discounted renewal claims with dependence

Abstract

In this paper we propose a quadratic predictor of the compound discounted renewal aggregate claims when taking into account dependence within the inter-occurrence times. We compare the accuracy of the proposed quadratic predictor to the simulated value of that sum, by using specific mixture of exponential distributions to define the dependence structure between the inter-occurrence times.

Keywords

Discounted compound renewal aggregate sums moments Archimedean Copula random interest rate quadratic predictor

1. Introduction

In the insurance space, the discounted aggregate claims with the use of renewal processes has been a subject of study and the instantaneous interest rate used to discount the claims is either stochastic or constant. The discounted aggregate claims can be used to compute the discounted value of a surplus process, moments and as well as the moment generating functions. Most of the literature makes an assumption that the inter-arrival times and claim amounts are independent although this assumption might be too restrictive and therefore a generalization would be necessary.

The influence of inflation and interest on the present value of the claims process and on some of its functionalities, including the surplus process, has been studied by other authors such as Gerber [14], Taylor [30], Delbaen & Haezendonck [12], Waters [31], Willmot [32], Sundt & Teugels [26], Cai & Dickson [8], Léveillé & Garrido [21–23], Léveillé & et al. [24]. The latter formulated their results in a context of renewal, Jang [16] used martingales and jump diffusion processes to get the moments, and Kim & Kim [18] studied the problem in a Markov environment.

Léveillé and Garrido [21] determined the first two moments using the discounted aggregate claims in an economic environment. The work they stimulated on finding the distribution of the discounted aggregate claims with the use of renewal processes allow to calculate some functionals related to this distribution such as the popular risk measures Value-at-risk (VaR) and Tail Value-at-risk (TVaR). Léveillé and Garrido [22] extended their earlier work and derived the recursive moments of the compound renewal sums with discounted aggregate claims. This was improved on by, Léveillé and Adékambi [19,20] who found the simple and higher moments of the discounted compound renewal sums in the presence of a stochastic instantaneous interest rate.

The studies cite above make two key, common assumptions. They assume (i) that the claims inter arrival times and claim amount are independent, (ii) that the claims amounts are independent and identically distributed. The claim amounts are independent and identically distributed and the claims inter-arrival times are independent and identically distributed with an exponential distribution. Although these assumptions simplify our derivations they are too restrictive when it comes to the real world practical application of our models. For instance, in an ever changing economic environment the assumption that the inter-arrival times of dividends is independent may not necessarily be the case. Since the distribution of inter-dividends time in times of economic expansion and in times of economic crisis cannot be identically distributed. This consequently impacts on the company’s decision to reward shareholders with dividends payments or to invest profits to ride out any economy downturn, for example. It thus becomes necessary to generalize these assumptions so that the models so obtained are in tandem with real world peculiarities.

One approach in this quest for generalized models that has been particularly fruitful in research has been to dispense with independence and assume dependence; in particular, dependence between the claims inter-arrival times and claim amounts. Representative efforts in this regard are as follows. Bargès, Cossette, Loisel and Marceau [6] introduced the dependence of claims inter-arrival times and claims amount using a Farlie--Gumbel--Morgenstern (FGM) copula and then derived the first two moments of the discounted aggregate claims, Shendova and Zitikis [28] used a general point process to study the aggregate claims when there is dependence between claims inter-arrival times and claims amount. Most recently, Adékambi and Dziwa [2] found an explicit formula for the discounted compound renewal process when dependence is from a FGM copula and Adékambi [1] extended the work to find the second moment.

Due to noted weaknesses of the FGM copula, other copulas have been used. Representative works in this regard are Sarabia, Gómez-Déniz, Prieto and Jordá [27] who computed the discounted aggregate cash flows of dependent risk models using an Archimedean Copula in the presence of a mixing exponential distribution. Cossette, Marceau, Mtalai, Veilleux [11] also investigated the dependent risk models using Archimedean copula.

Albrecher, Constantinescu and Loisel [3] used mixing random variables to compute the ruin probability and they also allowed for the relaxation of the independence between claims inter-arrival times.

The aim of this study is to find explicit formula for the first two moments and the joint moment to be able to predict cashflows with renewal process when dependence is assumed between the dividends (the cashflows) and their inter-arrival times. The real-world motivation underlying the study is the desire for the ability to predict the discounted aggregate claims of an insurance. To the author best knowledge, there’s no published study that derives the moments where there is dependence among the inter-arrival times. It is hoped that this study makes a modest contribution to the burgeoning literature on dependence studies in classical risk theory.

The paper is organized as follows, in Section 2, we present the ordinary renewal process with dependence. Moments and higher joint moments of the aggregate discounted claims are derived in Sections 3 and 4. In Section 5, we apply the results of the preceding section when the subsequent inter-arrival times have a Pareto distribution with a Clayton copulas dependence. In Section 6, we examine the delayed renewal case with dependence and we derive its moments. The results are illustrated with numerical applications in Section 7. In Section 8, the conclusion follows.

2. The model

In this section, we introduce the ordinary renewal case with dependence

The process for the present value of the claim amounts is represented by $Z=\{Z(t),t\geq 0\}$ with Z (0) = 0 and $Z(t)=\sum _{i=1}^{N(t)}e^{-{\delta}T_{i}}X_{i}=\sum _{i=1}^{N(t)}D(T_{i})X_{i}$ , where Z(t) = 0 if N (t) = 0, D (T _k) = e ^{−I (T
_k)}, $I(T_{k})=\int _{0}^{T_{k}}{\delta}(x) \,dx$ . The claims counting processes $\{N(t),t\geq 0\}$ and $\{N_{d}(t),t\geq 0\}$ form, respectively, an ordinary and a delayed renewal process and, for $k\in N=\{1,2,3,\ldots \}$ . We define T _k to be the corresponding claim occurrence times and W _j = T _j − T _j−1, j = 1, 2, … to be the subsequent inter-claim times.

The claim severities $\{X_{k}\}_{k=1}^{\infty }$ are assumed to form an i.i.d. sequence and their higher moments exist over a subset $U\subset \mathbb{R}$ including a neighborhood of the origin.

In the usual ordinary renewal risk process, the sequences $\{W_{j}\}_{j=1}^{\infty }$ are assumed to be mutually independent. In this paper, we assume that W ₁, W ₂, … are dependent joined with Archimedean copulas.

Let 𝛩 be a random variable with pdf f _𝛩(𝜃) and we suppose that the Laplace transform of 𝛩 is given by $f_{{\Theta}}^{\ast }(s)=\int _{0}^{\infty }e^{-s{\theta}}f_{{\Theta}}({\theta})\,d{\theta}$ exist over a subset $K\subset \mathbb{R}$ including a neighborhood of the origin.

For a general set up, the results obtained above by using an exponential distribution for the conditional distribution of the time between successive claims, can be extended to other conditionally independent distributions.

For example, the conditional distribution of the inter-claims time can be written in the power form

$\Pr (W_{i}\geq x_{i}|{\Theta}={\theta})=(\bar{H}(x_{i}))^{{\theta}}$ for some distribution function H(x _i) and $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \Pr (W_{1}\geq x_{1},W_{2}\geq x_{2},\ldots ,w_{n}\geq x_{n}|{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\mathop{\prod }_{k=1}^{n}(\bar{H}(x_{i}))^{{\theta}}\end{eqnarray}.$ (1) For all n, i.e. is the common mixing parameter, then $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-23.0pt}\overline{F}_{W_{1},\ldots ,W_{n}(x_{1},\ldots ,x_{n})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}=\int _{0}^{\infty }\Pr (W_{1}\geq x_{1},\ldots ,W_{n}\geq x_{n}|{\Theta}={\theta})f_{{\Theta}}({\theta})d{\theta}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}=\displaystyle \overline{F}_{W_{1},\ldots ,W_{n}(x_{1},\ldots ,x_{n})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}=\int _{0}^{\infty }(\overline{H}(x_{1}))^{{\theta}}\ldots (\overline{H}(x_{n}))^{{\theta}}f_{{\Theta}}({\theta})d{\theta}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}=\displaystyle f_{{\Theta}}^{\ast }(-\log \overline{H}(x_{1})-\cdots -\overline{H}(x_{n}))\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}=\displaystyle f_{{\Theta}}^{\ast }(f_{{\Theta}}^{\ast -1}(\overline{F}_{W_{1}(x_{1})})+\cdots +f_{{\Theta}}^{\ast -1}(\overline{F}_{W_{n}(x_{n})}))\end{eqnarray}.$ (2) The form of dependence structure is again Archimedean with generator $∅(t)=f_{{\Theta}}^{\ast -1}(t)$ , where $\bar{F}_{W_{i}}(x)=f_{{\Theta}}^{\ast }(-\log \bar{H}(x_{i}))$ .

2.1. Remark: Specific mixture of exponential distributions

2.1.1. Ordinary renewal case

If the r.v.’s W ₁, W ₂, …, W _n are n dependent, positive and continuous r.v and that given 𝛩 = 𝜃, the r.v.’s W ₁, W ₂, …, W _n are conditionally independent and distributed as Exp (𝜃) and $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\displaystyle \Pr (W_{1}\geq x_{1},W_{2}\geq x_{2},\ldots ,w_{n}\geq x_{n}|{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle =\Pr (W_{1}\geq x_{1}|{\Theta}={\theta})\ldots \Pr (W_{n}\geq x_{n}|{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle =\displaystyle e^{-{\theta}x_{1}}\ldots e^{-{\theta}x_{n}}.\end{eqnarray}$ (3) It follows that $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\overline{F}_{W_{1}}(x_{1})=\Pr (W_{1}\geq x_{1})=\int _{0}^{\infty }e^{-{\theta}x}f_{{\Theta}}({\theta})d{\theta}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle =f_{{\Theta}}^{\ast }(x)=\int _{0}^{\infty }\Pr (W_{1}\geq x|{\Theta}={\theta})f_{{\Theta}}({\theta})d{\theta}.\end{eqnarray}$ (4) The joint distribution of the tail of W ₁, W ₂, …, W _n can be written as $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-23.0pt}\overline{F}_{(W_{1},\ldots ,W_{n}(x_{1},\ldots ,x_{n})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}=\int _{0}^{\infty }\Pr (W_{1}\geq x_{1},\ldots ,W_{n}\geq x_{n}∼|∼{\Theta}={\theta})f_{{\Theta}}({\theta})d{\theta}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}=\displaystyle \int _{0}^{\infty }e^{-{\theta}\mathop{\sum }_{i=1}^{n}x_{i}}f_{{\Theta}}({\theta})d{\theta}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}=\displaystyle f_{{\Theta}}^{\ast }\left(\mathop{\sum }_{i=1}^{n}x_{i}\right)=f_{{\Theta}}^{\ast }(f_{{\Theta}}^{\ast -1}(\overline{F}_{W_{1}(x_{1})})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\cdots +f_{{\Theta}}^{\ast -1}(\overline{F}_{W_{n}(x_{n})})).\end{eqnarray}$ (5) Otherwise from Sklar theorem, $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\Pr (W_{1}\geq x_{1,}\ldots ,w_{n}\geq x_{n})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\widetilde{C}(\overline{F}_{W_{1}(x_{1})},\ldots, \overline{F}_{W_{1}(x_{1})}).\end{eqnarray}$ (6) It follows that $\widetilde{C}(u_{1},\ldots ,u_{n})$ is an archimedean copula with $\begin{eqnarray}\displaystyle \displaystyle \widetilde{C}(u_{1},\ldots ,u_{n}) & = & \displaystyle f_{{\Theta}}^{\ast }(f_{{\Theta}}^{\ast -1}(u_{1})+\cdots +f_{{\Theta}}^{\ast -1}(u_{n}))\nonumber\\ \displaystyle & = & \displaystyle ∅^{-1}(∅(u_{1})+\cdots +∅(u_{n})),\end{eqnarray}$ (7) where $∅(t)=f_{{\Theta}}^{\ast -1}(t)$ is the generator of the Archimedean copula $\widetilde{C}$ .

2.1.2. The delayed renewal case dependence

Let (𝛬, 𝛩) be a positive random vector with cumulative distribution function F _{𝛬, 𝛩} and we suppose that its Laplace transform is given by $f_{{\Lambda},{\Theta}}^{\ast }(s)=\int _{0}^{\infty }\int _{0}^{\infty }e^{-{\theta}_{1}s-{\theta}t}dF_{{\Lambda},{\Theta}}({\theta}_{1},{\theta})$ exists over a subset $K\times K\subset \mathbb{R}^{2}$ including a neighborhood of the origin. We assume that W ₁, W ₂, … , W _n are n dependent random variables, conditionally independent given (𝛬, 𝛩) and distributed as Exponential distribution, W ₁|𝛬 = 𝜃₁ and W _i|𝛩 = 𝜃 ∼ Exp (𝜃), i = 2, … , n, such that the joint conditional survival function is given by $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\Pr (W_{1}\geq x_{1},W_{2}\geq x_{2},\ldots ,w_{n}\geq x_{n}|{\Lambda}={\theta}_{1},{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle =\Pr (W_{1}\geq x_{1}|{\Lambda}={\theta}_{1})\ldots \Pr (W_{n}\geq x_{n}|{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle =\displaystyle e^{-{\theta}x_{1}}\ldots e^{-{\theta}x_{n}}.\end{eqnarray}$ (8) The joint distribution of the tail of can be written as $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\overline{F}_{W_{1}}(x_{1})=\Pr (W_{1}\geq x_{1})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-5.0pt}=\int _{0}^{\infty }\Pr (W_{1}\geq x|{\Lambda}={\theta}_{1})f_{{\Lambda}}({\theta}_{1})d{\theta}_{1}=f_{{\Lambda}}^{\ast }(x).\nonumber\end{eqnarray}$ It follows that the survival function of the marginal distribution of W ₁ is $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-24.0pt}\overline{F}_{(W_{1},\ldots ,W_{n})}(x_{1},\ldots ,x_{1})=\int _{0}^{\infty }\int _{0}^{\infty }\Pr \big(W_{1}\geq x_{1},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-21.0pt}\times ∼W_{2}\geq x_{2},\ldots ,w_{n}\geq x_{n}|{\Lambda}={\theta}_{1},{\Theta}={\theta}\big)dF_{{\Lambda},{\Theta}}({\theta}_{1},{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-21.0pt}=\displaystyle \int _{0}^{\infty }\int _{0}^{\infty }e^{-{\theta}_{1}x_{1}}e^{-{\theta}\mathop{\sum }_{i=2}^{n}x_{1}}dF_{{\Lambda},{\Theta}}({\theta}_{1},{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-21.0pt}=\displaystyle f_{{\Lambda},{\Theta}}^{\ast }\left(x_{1},\mathop{\sum }_{i=2}^{n}x_{i}\right),\end{eqnarray}$ (9) and the survival function of the marginal distribution of W _i are, for i = 2, …, n, $\begin{eqnarray}\displaystyle \displaystyle \overline{F}_{W_{i}}(x_{1})=f_{{\Theta}}^{\ast }(x). & & \displaystyle \nonumber\end{eqnarray}$ Note that all the marginal distributions are not identically distributed. The first inter-arrival time then has distribution $\overline{F}_{W_{1}}(x_{1})=f_{{\Lambda}}^{\ast }(x)$ , while all subsequent inter-arrival times have cdf $\overline{F}_{W_{i}}(x_{1})=f_{{\Theta}}^{\ast }(x)$ , for i = 2, …, n. For the special case when $\bar{F}_{W_{1}}(x_{1})=\bar{F}_{W_{i}}(x_{1})$ , for i = 2, …, n, we get the ordinary renewal process.

From equation (9), we get $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\overline{F}_{W_{1},\ldots ,W_{n}}(x_{1},\ldots ,x_{1})=f_{{\Lambda},{\Theta}}^{\ast }(f_{{\Lambda}}^{\ast -1}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-12.0pt}\times ∼(\overline{F}_{W_{1}(x_{1})},f_{{\Theta}}^{\ast -1} (\overline{F}_{W_{2}(x_{2})},\ldots ,f_{{\Theta}}^{\ast -1}(\overline{F}_{W_{n}(x_{n})}).\nonumber\end{eqnarray}$ By applying Sklar’s theorem, see e,g. Sklar [29], the survival distribution function of W ₁, W ₂, …, W _n is $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\Pr (W_{1}\geq x_{1,}\ldots ,w_{n}\geq x_{n})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle =\widetilde{C}(\overline{F}_{W_{1}(x_{1})},\ldots \overline{F}_{W_{n}(x_{n})}).\nonumber\end{eqnarray}$ It follows that we have a factor copula of the form $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\widetilde{C}(u_{1},\ldots ,u_{n})=f_{{\Lambda},{\Theta}}^{\ast }\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼(f_{{\Lambda}}^{\ast -1}(u_{1})+f_{{\Theta}}^{\ast -1}(u_{2})+\cdots +f_{{\Theta}}^{\ast -1}(u_{n}))\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}={\psi}^{-1}({\psi}_{1}(u),∅(u_{2})+\cdots +∅(u_{n})),\end{eqnarray}$ (10) where $∅(t)=f_{{\Theta}}^{\ast -1}(t)$ , ${\psi}_{1}(t)=f_{{\Lambda}}^{\ast -1}(t)$ is the generator of the Archimedean copulas $\widetilde{C}$ .

(A) Pareto inter-arrival claims and Clayton Copula Dependence As in H. Albrecher et al. [3], if 𝛩 ∼ 𝛤(𝛼, 𝛽) with pdf $f_{{\Theta}}({\theta})=\frac{{\beta}^{{\alpha}}}{{\Gamma}({\alpha})}{\theta}^{{\alpha}-1}e^{-{\beta}{\theta}},{\theta}>0$ . It follows that W _i ∼ Pareto (𝛼, 𝛽) with survival function $\bar{F}_{W_{1}}(x)=\Pr (W_{1}\geq x)=L_{{\Theta}}(x)=(1+\frac{x}{{\beta}})^{-{\alpha}},x>0$ and $f_{{\Theta}}^{\ast -1}(t)={\beta}(t^{\frac{-1}{{\alpha}}}-1)$ .

The multivariate Pareto survival function of W ₁, W ₂, …, W _n can then be written as $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-25.0pt}\bar{F}_{(W_{1},W_{2},\ldots ,W_{n})}(x_{1},x_{2},\ldots ,x_{n})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-21.0pt}=\left(\!1+\mathop{\sum }_{i=1}^{n}\frac{x_{i}}{{\beta}}\right)^{-{\alpha}}\!x_{i}>0,\forall ∼i=1,\ldots n,\text{ and }{\alpha},{\beta}>0.\nonumber\end{eqnarray}$ The associated copula is the Clayton copula given by: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}C_{{\alpha}}(u_{1},\ldots ,u_{n})=(u_{1}^{-1/{\alpha}}+\cdots +u_{1}^{-1/{\alpha}}-n+1)^{-{\alpha}}.\nonumber\end{eqnarray}$ (B) Dependent Gamma inter-arrival Claims The gamma distribution with shape parameter 𝛼 ∈ (0,1] is completely monotone and then, it can be accommodated to the general model introduced in Section 1. We have the following result (Gleser, [15] and [5]).

Let W ∼ 𝛤(𝛼, 𝜆) be a gamma distribution with scale parameter 𝜆 and shape parameter 𝛼 ∈ (0,1] and pdf, $\begin{eqnarray}\displaystyle \displaystyle \hspace{-20.0pt}f_{W}(x)=\frac{{\lambda}^{{\alpha}}}{{\Gamma}({\alpha})}w^{{\alpha}-1}e^{-{\lambda}w},\quad w>0. & & \displaystyle \nonumber\end{eqnarray}$ Then, $\begin{eqnarray}\displaystyle \displaystyle f_{W}(x)=\int _{0}^{\infty }e^{-{\theta}w}f_{{\Theta}}({\theta})d{\theta}, & & \displaystyle \nonumber\end{eqnarray}$ where $\begin{eqnarray}\displaystyle \displaystyle f_{{\Theta}}({\theta})=\frac{({\theta}-{\lambda})^{-{\alpha}}{\lambda}^{{\alpha}}}{{\theta}{\Gamma}(1-{\alpha}){\Gamma}({\alpha})},\quad {\lambda}\leq {\theta}\lt \infty , & & \displaystyle\end{eqnarray}$ (11) and f _𝛩(𝜃) = 0 otherwise.

TheoremA .

The Laplace transform of the random variable 𝛩 with pdf equation (11) is, $\begin{eqnarray}\displaystyle \displaystyle L_{{\Theta}}(s)=\frac{{\Gamma}({\alpha},{\lambda}s)}{{\Gamma}({\alpha})},\quad s\geq 0, & & \displaystyle \nonumber\end{eqnarray}$ where ${\Gamma}(s,x)=\int _{x}^{\infty }t^{s-1}e^{-t}dt$ denotes the upper incomplete gamma function.

TheoremB .

From the Laplace transform of 𝛩, $f_{{\Theta}}^{\ast }(s)=\int _{0}^{\infty }e^{-s{\theta}}f_{{\Theta}}({\theta})d{\theta}$ , we take the derivative of $f_{{\Theta}}^{\ast }(s)$ with respect to s, and then have: $\begin{eqnarray}\displaystyle \displaystyle \frac{df_{{\Theta}}^{\ast }(s)}{ds} & = & \displaystyle \int _{0}^{\infty }-{\theta}e^{-s{\theta}}f_{{\Theta}}({\theta})d{\theta}\nonumber\\ \displaystyle & = & \displaystyle \displaystyle -f_{W}(s)\nonumber\\ \displaystyle & = & \displaystyle \displaystyle -\frac{{\lambda}^{{\alpha}}}{{\Gamma}({\alpha})}s^{{\alpha}-1}e^{-{\lambda}s}.\nonumber\end{eqnarray}$ Then, $\begin{eqnarray}\displaystyle \displaystyle f_{{\Theta}}^{\ast }(s)=\int _{s}^{\infty }\frac{{\lambda}^{{\alpha}}}{{\Gamma}({\alpha})}x^{{\alpha}-1}e^{-{\lambda}x}dx. & & \displaystyle \nonumber\end{eqnarray}$ With the change of variable 𝜆x = u, we have: $f_{{\Theta}}^{\ast }(s)=\int _{{\lambda}s}^{\infty }\frac{u^{{\alpha}-1}e^{-u}du}{{\Gamma}({\alpha})}=\frac{{\Gamma}({\alpha},{\lambda}s)}{{\Gamma}({\alpha})}$ .

Using Lemma 2.1, we get the generator function of the corresponding copula, which is given by, $\begin{eqnarray}\displaystyle \displaystyle ∅(t)=f_{{\Theta}}^{\ast -1}(t)=Q_{G_{{\alpha}}}(1-t), & & \displaystyle \nonumber\end{eqnarray}$ where Q _{G
_𝛼} represents the quantile function of a gamma distribution with mean 𝛼 and unit scale parameter. From Lemma 2.1, the joint survival function is, $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\Pr (W_{1}\geq x_{1},W_{2}\geq x_{2},\ldots ,w_{n}\geq x_{n})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\frac{{\Gamma}({\alpha},{\lambda}\mathop{\sum }_{j=1}^{n}x_{j})}{{\Gamma}({\alpha})},\quad \text{if }x_{1},\ldots ,x_{n}\geq 0,\nonumber\end{eqnarray}$ with marginal survival functions, $\Pr (X_{i}>x)=\frac{{\Gamma}({\alpha},{\lambda}x)}{{\Gamma}({\alpha})},x\geq 0,i=1,2,\ldots ,n$ .

The survival copula associated to the Exponential-Gamma dependent model is given by, $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\widetilde{C}(u_{1},\ldots ,u_{n})=1-F_{G_{{\alpha}}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼(G_{G_{{\alpha}}}(1-u_{1})+\cdots +G_{G_{{\alpha}}}(1-u_{n})),\nonumber\end{eqnarray}$ where F _{G
_𝛼} and Q _{G
_𝛼} represent the CDF and the quantile function, of a gamma distribution with shape parameter 𝛼.

(C) General Weibull inter-arrival claims with Gumbel Copula dependence

Let consider a positive stable random variable with pdf (see Feller [13]), $\begin{eqnarray}\displaystyle \displaystyle f_{{\Theta}}({\theta})=-\frac{1}{{\pi}{\theta}}\mathop{\sum }_{k=1}^{\infty }\frac{{\Gamma}(k{\alpha}+1)}{k!}(-{\theta}^{-{\alpha}})^{k}\sin ({\alpha}k{\pi}), & & \displaystyle \nonumber\end{eqnarray}$ and Laplace transform, $f_{{\Theta}}^{\ast }(s)=\exp (-s^{{\alpha}}),s\geq 0$ , and 𝛼 ∈ (0,1].

Using the expression of $f_{{\Theta}}^{\ast }(s)$ , we have $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\Pr (W_{1}\geq x_{1},W_{2}\geq x_{2},\ldots ,w_{n}\geq x_{n})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\exp (-(x_{1}+\cdots +x_{n})^{{\alpha}}),\nonumber\end{eqnarray}$ with marginal distribution, $\begin{eqnarray}\displaystyle \displaystyle \Pr (W_{i}>x)=\frac{{\Gamma}({\alpha},{\lambda}x)}{{\Gamma}({\alpha})},\quad x\geq 0,∼i=1,2,\ldots ,n, & & \displaystyle \nonumber\end{eqnarray}$ which are Weibull distributions with shape parameter 𝛼 ∈ (0, 1]. The generator of the Archimedean copula is $∅(t)=f_{{\Theta}}^{\ast -1}(t)=(-\log t)^{1/{\alpha}}$ and the survival copula is, $\begin{eqnarray}\displaystyle \displaystyle \widetilde{C}_{{\alpha}}(u_{1},\ldots ,u_{n})=\exp \left\{-\left[\mathop{\sum }_{k=1}^{n}(-\log u_{k})^{1/{\alpha}}\right]\right\}, & & \displaystyle \nonumber\end{eqnarray}$ which corresponds to the Gumbel copula.

(D) Inverse Gaussian Mixture of exponential inter-arrival claims. If 𝛩 ∼ IG (𝜆, 𝜇) has an inverse Gaussian distribution with parameters 𝜇 > 0 and 𝜆 > 0 and pdf, $\begin{eqnarray}\displaystyle \displaystyle f_{{\Theta}}({\theta})=\sqrt{\frac{{\lambda}}{2{\pi}}}{\theta}^{3/2}\exp \left(-\frac{{\lambda}({\theta}-{\mu})^{2}}{2{\mu}^{2}{\theta}}\right),\quad {\theta}>0. & & \displaystyle \nonumber\end{eqnarray}$ The corresponding mixing distribution for the marginal inter-occurrence times W _j is, $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\Pr (W_{i}>x)=\int _{0}^{\infty }e^{-x{\theta}}f_{{\Theta}}({\theta})d{\theta}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad =\exp \left\{-\frac{{\lambda}}{{\mu}}\left(\sqrt{1+\frac{2{\mu}^{2}x}{{\lambda}}}-1\right)\right\},\quad x\geq 0,\nonumber\end{eqnarray}$ for i = 1, 2, … , n and joint survival function $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\Pr (W_{1}\geq x_{1},W_{2}\geq x_{2},\ldots ,w_{n}\geq x_{n})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\exp \left\{-\frac{{\lambda}}{{\mu}}\left(\sqrt{1+\frac{2{\mu}^{2}\mathop{\sum }_{i=1}^{n}x_{i}}{{\lambda}}}-1\right)\right\},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \text{if }x_{1},\ldots ,x_{n}\geq 0.\nonumber\end{eqnarray}$ This model was introduced by Whitmore [33] and extended to the multivariable case by Whitmore and Lee [34]. Since the generator is, $\begin{eqnarray}\displaystyle \displaystyle ∅(t)=f_{{\Theta}}^{\ast -1}(t)=\frac{{\lambda}}{2{\mu}^{2}}\left\{\left(1-\frac{{\mu}}{{\lambda}}\log t\right)^{2}-1\right\}, & & \displaystyle \nonumber\end{eqnarray}$ the survival copula is given by: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-25.0pt}\widetilde{C}(u_{1},\ldots ,u_{n})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-24.0pt}=\exp \left\{-\frac{{\lambda}}{{\mu}}\left[\left(\mathop{\sum }_{i=1}^{n}\left(1-\frac{{\mu}}{{\lambda}}\log u_{i}\right)^{2}-n+1\right)^{1/2}-1\right]\right\},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \text{with if }0\leq u_{j}\leq 1,i=1,2,\ldots ,n.\nonumber\end{eqnarray}$ Otherwise from Sklar theorem, $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\Pr (W_{1}\geq x_{1},\ldots ,w_{n}\geq x_{n})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\widetilde{C}(\overline{F}_{W_{1}(x_{1})},\ldots \overline{F}_{W_{1}(x_{1})}).\end{eqnarray}$ (12) It follows that $\widetilde{C}(u_{1},\ldots ,u_{n})$ is an archimedean copula with $\begin{eqnarray}\displaystyle \displaystyle \widetilde{C}(u_{1},\ldots ,u_{n}) & = & \displaystyle f_{{\Theta}}^{\ast }(f_{{\Theta}}^{\ast -1}(u_{1})+\cdots +f_{{\Theta}}^{\ast -1}(u_{n}))\nonumber\\ \displaystyle & = & \displaystyle ∅^{-1}(∅(u)+\cdots +∅(u_{n})),\end{eqnarray}$ (13) where $∅(t)=f_{{\Theta}}^{\ast -1}(t)$ is the generator of the Archimedean copula $\widetilde{C}$ .

3. Joint moment with stochastic force of interest

If we now consider a stochastic net interest force, the previous method (the one that uses the renewal arguments) cannot be used and, what is more, it is not possible to obtain recursive formulas for the joint moments. By conditioning on the time of realization of the first claim, it is hardly the same process that is renewed, because we must take into consideration the time factor in the process that governs the force of interest.

Thus, we need a more general method that will help us find explicit formulas of the joint moment of our risk process for a stochastic force of interest. This method will be based essentially on the following lemma which gives the conditional joint distribution of the arrival times of claims knowing their number, for any renewal process.

TheoremC .
Consider an ordinary or a delayed renewal counting process, such as given in Section 1. Then, for $0=x_{0} \lt x_{1} \lt x_{2} \cdots\lt x_{k} \le t, i_{0} =0, 1=i_{1} \lt i_{2} \lt \cdots \lt i_{k} \le n$ and $1\le k\le n$ , the conditional joint density probability functions of $T_{i_{1} } ,T_{i_{2} } ,\ldots,T_{i_{k} } |N(t)=n, \Lambda =\lambda , \Theta =\theta$ and $T_{i_{1} } ,T_{i_{2} }, \ldots,T_{i_{k} } |N_{d} (t)=n, \Lambda =\lambda ,\Theta =\theta$ are given by:

For the ordinary renewal case: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}f_{T_{i_{1}},T_{i_{2}},\ldots ,T_{i_{k}}|N(t),{\Theta}={\theta}}(x_{1},x_{2},\ldots ,x_{k}|n,{\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}=\Bigg\{P(N(t-x_{k})=n-i_{k}|{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼\mathop{\prod }_{j=1}^{k}f_{T_{i_{j}}-T_{i_{j-1}}}(x_{j}-x_{j-1}|{\Theta}={\theta})\Bigg\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼\{P(N(t)=n|{\Theta}={\theta})\}^{-1}.\end{eqnarray}$ (14)

For the delay case: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-28.0pt}f_{T_{i_{1}},T_{i_{2}},\ldots ,T_{i_{k}}|N_{d}(t)}(x_{1},x_{2},\ldots ,x_{k}|n,{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-25.0pt}=\!\Bigg\{\!P(N_{d}(t-x_{k})=n-i_{k}|{\Lambda}={\lambda},{\Theta}={\theta})f_{T_{i_{1}}|{\Lambda}={\lambda}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-23.0pt}\times ∼(x_{j}|{\lambda})\mathop{\prod }_{j=2}^{k}f_{T_{i_{j}}-T_{i_{j-1}}|{\Theta}={\theta}}(x_{j}-x_{j-1}|{\theta})\Bigg\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-23.0pt}\times ∼\{P(N_{d}(t)=n|{\Lambda}={\lambda},{\Theta}={\theta})\}^{-1}.\end{eqnarray}$ (15)

TheoremD .
We only prove the delay renewal case since the proof adapts easily to the ordinary renewal case. Thus, we have $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-24.0pt}P\big(T_{i_{1}}\leq x_{1},T_{i_{2}}\leq x_{2},\ldots ,T_{i_{k}}\leq x_{k}|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad N(t)=n,{\Lambda}={\lambda},{\Theta}={\theta}\big)\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}=\big\{P\big(N(t)=n|T_{i_{1}}\leq x_{1},T_{i_{2}}\leq x_{2},\ldots ,\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad T_{i_{k}}\leq x_{k},{\Lambda}={\lambda},{\Theta}={\theta}\big)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼P\big(T_{i_{1}}\leq x_{1},T_{i_{2}}\leq x_{2},\ldots ,\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}T_{i_{k}}\leq x_{k},{\Lambda}={\lambda},{\Theta}={\theta}\big)\big\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼\{P(N(t)=n,{\Lambda}={\lambda},{\Theta}={\theta})\}^{-1}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}=\Bigg\{\int _{0}^{x_{1}}\int _{u_{1}}^{x_{2}}\ldots \int _{u_{k-1}}^{x_{k}}P\big(N(t)=n|T_{i_{1}}=u_{1},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad T_{i_{2}}=u_{2},\ldots ,T_{i_{k}}=u_{k},{\Lambda}={\lambda},{\Theta}={\theta}\big)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼f_{T_{i_{1}},T_{i_{2}},\ldots ,T_{i_{k}},{\Lambda},{\Theta}}(u_{1},u_{2},\ldots ,u_{k},{\lambda},{\theta})du_{k}\ldots u_{1}\Bigg\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼\{P(N(t)=n,{\Lambda}={\lambda},{\Theta}={\theta})\}^{-1}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}=\Bigg\{\int _{0}^{x_{1}}\int _{u_{1}}^{x_{2}}\ldots \int _{u_{k-1}}^{x_{k}}P\big(N(t-u_{k})=n-i_{k}|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad {\Lambda}={\lambda},{\Theta}={\theta}\big)f_{T_{i_{1}}|{\Lambda}={\lambda}}(x_{j}|{\lambda})\mathop{\prod }_{j=2}^{k}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼f_{T_{i_{j}-i_{j}-1}|{\Theta}={\theta}}(u_{j}-u_{j-1}|{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})du_{k}\ldots u_{1}\Bigg\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼\{P(N(t)=n|{\Lambda}={\lambda},{\Theta}={\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})\}^{-1}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}=\Bigg\{\int _{0}^{x_{1}}\int _{u_{1}}^{x_{2}}\ldots \int _{u_{k-1}}^{x_{k}}P(N(t-u_{k})=n-i_{k}|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼{\Lambda}={\lambda},{\Theta}={\theta})f_{T_{i_{1}}|{\Lambda}={\lambda}}(x_{j}|{\lambda})\mathop{\prod }_{j=2}^{k}f_{T_{i_{j}-i_{j}-1}|{\Theta}={\theta}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼(u_{j}-u_{j-1}|{\theta})du_{k}\ldots u_{1}\Bigg\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼\{P(N(t)=n|{\Lambda}={\lambda},{\Theta}={\theta})\}^{-1}.\nonumber\end{eqnarray}$ Since, $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-24.0pt}P(T_{i_{1}}\leq u_{1},T_{i_{2}}\leq u_{2},\ldots ,T_{i_{k}}\leq u_{k},{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=P(T_{i_{k}}\leq u_{k}|T_{i_{1}}\leq u_{1},T_{i_{2}}\leq u_{2},\ldots ,\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle T_{i_{k-1}}\leq u_{k-1},{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼P(T_{i_{1}}\leq u_{1},T_{i_{2}}\leq u_{2},\ldots ,\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle T_{i_{k}}\leq u_{k-1},{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\int _{0}^{u_{1}}\int _{v_{1}}^{u_{2}}\cdots \int _{v_{k-2}}^{u_{k-1}}P(T_{i_{k}}\leq u_{k}|T_{i_{1}}=v_{1},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle T_{i_{2}}=v_{2},\ldots ,T_{i_{k-1}}=v_{k-1},{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼f_{T_{i_{1}},\ldots ,T_{i_{k-1}},{\Theta}={\theta}} (v_{1},\ldots ,v_{k-1},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\Lambda}={\lambda},{\Theta}={\theta})dv_{1}\ldots dv_{k-1}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\int _{0}^{u}\int _{v_{1}}^{u_{2}}\cdots \int _{v_{k-2}}^{u_{k-1}}P(T_{i_{k}-i_{k-1}}\leq u_{k}-v_{k-1}|{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼f_{T_{i_{1}},\ldots ,T_{i_{k-1}}}(v_{1},\ldots ,v_{k-1}|{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad \times ∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})dv_{1}\ldots dv_{k-1}\nonumber\end{eqnarray}$ then we get $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}f_{T_{i_{1}},\ldots ,T_{i_{k}}}(u_{1},\ldots ,u_{k}|{\Lambda}={\lambda},{\Theta}={\theta})=f_{T_{i_{k-i_{k}-1}}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼(u_{k}-u_{k-1}|{\Theta}={\theta})f_{T_{i_{1}},\ldots ,T_{i_{k-1}}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼(u_{1},\ldots ,u_{k-1}|{\Lambda}={\lambda},{\Theta}={\theta}),\nonumber\end{eqnarray}$ which implies $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-23.0pt}f_{T_{i_{1}},\ldots ,T_{i_{k}}}(u_{1},\ldots ,u_{k}|{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle =f_{T_{i_{1}}|{\Lambda}={\lambda}}(u_{1}|{\Lambda}={\lambda})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼\mathop{\prod }_{j=2}^{k}f_{T_{i_{j}-i_{j}-1}|{\Theta}={\theta}}(u_{j}-u_{j-1}|{\Theta}={\theta}).\nonumber\end{eqnarray}$ Finally, $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-24.0pt}P\big(T_{i_{1}}\leq x_{1},T_{i_{2}}\leq x_{2},\ldots ,T_{i_{k}}\leq x_{k}|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad N(t)=n,{\Lambda}={\lambda},{\Theta}={\theta}\big)\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\Bigg\{\int _{0}^{x_{1}}\int _{u_{1}}^{x_{2}}\ldots \int _{u_{k-1}}^{x_{k}}P\big(N_{d}(t-u_{k})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad =n-i_{k}|{\Lambda}={\lambda},{\Theta}={\theta}\big)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-17.0pt}\quad \times ∼f_{T_{i_{1}}|{\Lambda}={\lambda}}(x_{j}|{\lambda})\mathop{\prod }_{j=2}^{k}f_{T_{i_{j}}-T_{i_{j-1}}|{\Theta}={\theta}}(x_{j}-x_{j-1}|{\theta})\Bigg\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-17.0pt}\quad \times ∼\{P(N(t)=n|{\Lambda}={\lambda},{\Theta}={\theta})\}^{-1},\nonumber\end{eqnarray}$ and equation (15) follows.

TheoremE .
(1) For k = n, we have for the ordinary renewal case (and similarly for the delayed renewal case) $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-24.0pt}f_{T_{1},\ldots ,T_{n}|N(t)=n,{\Theta}={\theta}}(x_{1},\ldots ,x_{n}|n,{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\bigg\{P(N(t-x_{n})=0|{\Theta}={\theta})\mathop{\prod }_{j=1}^{k}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad f_{{\tau}_{j}}(x_{j}-x_{j-1}|{\Theta}={\theta})\bigg\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼\{P(N(t)=n|{\Theta}={\theta})\}^{-1}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\bigg\{\bar{F}_{{\tau}_{1}}(t-x_{n}|{\Theta}={\theta})\mathop{\prod }_{j=1}^{k}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad f_{{\tau}_{j}}(x_{j}-x_{j-1}|{\Theta}={\theta})\bigg\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼\{P(N(t)=n|{\Theta}={\theta})\}^{-1}.\nonumber\end{eqnarray}$ The result follows by appropriately taking the k partial derivatives of the expression above.

TheoremF .
According to the assumptions of our risk model, and for a stochastic force of interest, the first three joint moments between Z (t) and Z (t + h) are given for and by: (1) $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}E[Z(t)Z(t+h)]=E[Z^{2}(t)]\displaystyle +E[X_{1}^{2}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t-u}^{t+h-u}E[D(u)D(u+v)]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼dm_{o}(v|{\theta})dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda},\nonumber\end{eqnarray}$ where $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}E[Z^{2}(t)]=E[X_{1}^{2}]\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}E[D^{2}(u)]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dm_{d}(u|{\lambda},{\theta})∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}+∼2E^{2}[X_{1}]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}E[D(u)D(u+v)]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dm_{o}(v|{\theta})∼dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda},\end{eqnarray}$ (16) (2) $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-23.0pt}E[Z^{2}(t)Z(t+h)]=E[Z^{3}(t)]+E[X_{1}^{2}]E[X_{1}]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}\int _{t-u}^{t+h-u}E[D^{2}(u)D(u+v)]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼dm_{o}(v|{\theta})dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}+∼2E^{3}[X_{1}]\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}\int _{t-u-v}^{t+h-u-v}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼E[D(u)D(u+v)D(u+v+w)]dm_{o}(w|{\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼dm_{o}(v|{\theta})dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\end{eqnarray}$ (17) where $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}E[Z^{3}(t)]=E[X_{1}^{3}]\int _{0}^{\infty }\int _{0}^{t}E[D^{3}(u)]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}+∼3E[X_{1}^{2}]E[X_{1}]\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}E\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼[D^{2}(u)D(u+v)]dm_{o}(v|{\theta})dm_{d}(u|{\lambda},{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}+∼3E[X_{1}^{2}]E[X_{1}]\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}E\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼[D^{2}(u)D(u+v)]dm_{o}(v|{\theta})dm_{d}(u|{\lambda},{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}+∼6E^{3}[X_{1}]\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}\int _{0}^{t-u-v}E\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼[D(u)D(u+v)D(u+v+w)]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼dm_{o}(w|{\theta})dm_{o}(v|{\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad \times ∼dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}E[Z(t)Z^{2}(t+h)]=E[Z^{3}(t)]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}+∼3E[X_{1}^{2}]E[X_{1}]\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t-u}^{t+h-u}E\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼[D(u)D^{2}(u+v)]dm_{o}(v|{\theta})∼dm_{d}(u|{\lambda},{\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}+∼3E[X_{1}^{2}]E[X_{1}]\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t-u}^{t+h-u}E\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼[D^{2}(u)D(u+v)]dm_{o}(v|{\theta})∼dm_{d}(u|{\lambda},{\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}+∼4E[X_{1}^{3}]\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}\int _{t-u-v}^{t+h-u-v}E\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼[D(u)D(u+v)D(u+v+w)]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dm_{o}(w|{\theta})dm_{o}(v|{\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}+∼2E[X_{1}^{2}]\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t-u}^{t+h-u}\int _{0}^{t+h-u-v}E\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼[D(u)D(u+v)D(u+v+w)]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dm_{o}(w|{\theta})dm_{o}(v|{\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\end{eqnarray}.$ (18)

TheoremG .
We illustrate the main idea of the proof by solving the first result of our theorem. Thus, let us obtain an expression of the generating function of the joint moments, for any sample path of 𝛿(x) the force of interest over the period [0, t + h]. An appropriate evaluation of the partial derivatives of E[e ^{xZ (t)+yZ (t+h)}|𝛿(z), z ∈ [0, t + h]]to (x, y) = (0,0) gives: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}E[Z^{2}(t)Z(t+h)|{\delta}(z),z\in [0,t+h]]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}=E[X_{1}^{3}]E\left[\mathop{\sum }_{k=1}^{N(t)}D^{3}(T_{k})|{\delta}(z),z\in [0,t+h]\right]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}+∼3E[X_{1}^{2}]E[X_{1}]E\Bigg[\mathop{\sum }_{k=1}^{N(t)-1}\mathop{\sum }_{j=k+1}^{N(t)}D^{2}(T_{k})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼D(T_{j})|{\delta}(z),z\in [0,t+h]\Bigg]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}+∼3E[X_{1}^{2}]E[X_{1}]E\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼\left[\mathop{\sum }_{k=1}^{N(t)-1}\mathop{\sum }_{j=k+1}^{N(t)}D(T_{k})D^{2}(T_{j})|{\delta}(z),z\in [0,t+h]\right]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼6E^{3}[X_{1}]E\Bigg[\mathop{\sum }_{k=1}^{N(t)-2}\mathop{\sum }_{j=k+1}^{N(t)-1}\mathop{\sum }_{l=j+1}^{N(t)}D(T_{k})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼D(T_{j})D(T_{l})|{\delta}(z),z\in [0,t+h]\Bigg]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}+∼E^{3}[X_{1}]E\Bigg[\mathop{\sum }_{k=1}^{N(t)}\mathop{\sum }_{j=N(t)+1}^{N(t+h)}D^{2}(T_{k})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼D(T_{j})|{\delta}(z),z\in [0,t+h]\Bigg]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}+∼2E[X_{1}^{2}]E[X_{1}]E\Bigg[\mathop{\sum }_{k=1}^{N(t)-1}\mathop{\sum }_{j=k+1}^{N(t)}\mathop{\sum }_{l=N(t)+1}^{N(t+h)}D(T_{k})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼D(T_{j})D(T_{l})|{\delta}(z),z\in [0,t+h]\Bigg].\nonumber\end{eqnarray}$ Now, if we first condition on N (t), 𝛬, and 𝛩 the first term of the previous summation gives: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}E\left[\mathop{\sum }_{k=1}^{N(t)}D^{3}(T_{k})|{\delta}(z),z\in [0,t+h]\right]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=E\left[E\left[\mathop{\sum }_{k=1}^{N(t)}D^{3}(T_{k})|N(t),{\Lambda},{\Theta},{\delta}(z),z\in [0,t]\right]\right]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=\mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{k=1}^{n}\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}D^{3}(u)f_{T_{k}|{\Lambda}={\lambda},{\Theta}={\theta}}(u|{\lambda},{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼P(N(t-u)=n-k|{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})dud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{n=k}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}D^{3}(u)f_{T_{k}|{\Lambda}={\lambda},{\Theta}={\theta}}(u|{\lambda},{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼P(N(t-u)=n-k|{\Theta}={\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})dud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=\mathop{\sum }_{k=1}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}D^{3}(u)dF_{W_{1}|{\Theta}={\theta}}^{\ast k}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=\int _{0}^{\infty }\!\int _{0}^{\infty }\!\int _{0}^{t}\!D^{3}(u)d\left\{\mathop{\sum }_{k=1}^{\infty }F_{W_{1}|{\Lambda}={\lambda},{\Theta}={\theta}}^{\ast k}(u|{\lambda},{\theta})\!\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}D^{3}(u)dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\end{eqnarray}$ For the second, we show that: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}E\left[\mathop{\sum }_{k=1}^{N(t)-1}\mathop{\sum }_{j=k+1}^{N(t)}D^{2}(T_{k})D(T_{j})|{\delta}(z),z\in [0,t+h]\right]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=E\Bigg[E\Bigg[\mathop{\sum }_{k=1}^{N(t)-1}\mathop{\sum }_{j=k+1}^{N(t)}D^{2}(T_{k})D(T_{j})|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼N(t),{\delta}(z),{\Lambda},{\Theta},z\in [0,t+h]\Bigg]\Bigg]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=\mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{k=1}^{n-1}\mathop{\sum }_{j=k+1}^{n}\int _{0}^{\infty }\int _{0}^{t}\int _{u}^{t}D^{2}(u)D(v)P\big(N(t-v)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad =n-j|{\Lambda}={\lambda},{\Theta}={\theta}\big)f_{T_{j-k}|{\Lambda}={\lambda}}(v-u|{\lambda})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{T_{k}|{\Theta}={\theta}}(u)f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=\mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{n=j}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{u}^{t}D^{2}(u)D(v)P\big(N(t-v)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\quad =n-j|{\Lambda}={\lambda},{\Theta}={\theta}\big)f_{T_{j-k}|{\Lambda}={\lambda}}(v-u|{\lambda})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{T_{k}|{\Lambda}={\lambda},{\Theta}={\theta}}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}D^{2}(u)D(u+v)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dF_{T_{j-k}|{\Lambda}={\lambda}}(v-u|{\lambda})DF_{T_{k}|{\Theta}={\theta}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u|{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}D(u)D(u+v)dm_{0}(v|{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\end{eqnarray}.$ And similarly, for the third term we have: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}E\left[\mathop{\sum }_{k=1}^{N(t)-1}\mathop{\sum }_{j=k+1}^{N(t)}D(T_{k})D^{2}(T_{j})|{\delta}(z),z\in [0,t+h]\right]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\int _{0}^{\infty }\int _{0}^{t-v}D(u)D^{2}(u+v)dm_{d}(v|{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\times ∼dm_{o}(u|{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}.\nonumber\end{eqnarray}$ For the last three terms, we obtain: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-135.0pt}E\Bigg[\mathop{\sum }_{k=1}^{N(t)-2}\mathop{\sum }_{j=k+1}^{N(t)-1}\mathop{\sum }_{l=j+1}^{N(t)}D(T_{k})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-120.0pt}\quad D(T_{j})D(T_{l})|{\delta}(z),z\in [0,t+h]\Bigg]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-130.0pt}=E\Bigg[E\Bigg[\mathop{\sum }_{k=1}^{N(t)-2}\mathop{\sum }_{j=k+1}^{N(t)-1}\mathop{\sum }_{l=j+1}^{N(t)}D(T_{k})D(T_{j})D(T_{l})|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-125.0pt}\times ∼N(t),{\delta}(z),{\Lambda},{\Theta},z\in [0,t+h]\Bigg]\Bigg]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-130.0pt}=\mathop{\sum }_{n=3}^{\infty }\mathop{\sum }_{k=1}^{n-2}\mathop{\sum }_{j=k+1}^{n-1}\mathop{\sum }_{n=l}^{n}\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{u}^{t}\int _{v}^{t}D(u)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-125.0pt}\times ∼D(v)D(w)P(N(t-w)=n-l|{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-125.0pt}\times ∼f_{T_{k},T_{j},T_{l},N(t)|{\Lambda}={\lambda},{\Theta}={\theta}}(u,v,w,n|{\lambda},{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-125.0pt}\times ∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})dwdvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-130.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{l=j+1}^{\infty }\mathop{\sum }_{n=l}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{u}^{t}\int _{v}^{t}D(u)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-125.0pt}\times ∼D(v)D(w)P(N(t-w)=n-l)f_{T_{l-j}|{\Lambda}={\lambda}}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-125.0pt}\times ∼(w-v|{\lambda})f_{T_{j-k}|{\Lambda}={\lambda}}(v-u|{\theta})f_{T_{k}|{\Theta}={\theta}}(u|{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-125.0pt}\times ∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})dwdvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-130.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{l=j+1}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}\int _{0}^{t-u-v}D(u)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-125.0pt}\times ∼D(u+v)D(u+v+w)P(N(t-w)=n-l)\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-125.0pt}\times ∼dF_{W_{2}|{\Lambda}={\lambda}}^{\ast (j-k)}(v-u|{\theta})d\{F_{W_{1}|{\Lambda}={\lambda}}\ast F_{W_{2}|{\Theta}={\theta}}^{\ast k-1}\}\nonumber\\ \displaystyle \displaystyle \text{}\text{}\times ∼(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}, & & \displaystyle \nonumber\end{eqnarray}$ which gives $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-25.0pt}E\Bigg[\mathop{\sum }_{k=1}^{N(t)-2}\mathop{\sum }_{j=k+1}^{N(t)-1}\mathop{\sum }_{l=j+1}^{N(t)}D(T_{k})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}D(T_{j})D(T_{l})|{\delta}(z),z\in [0,t+h]\Bigg]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle =\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}\int _{0}^{t-u-v}D(u)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼D(u+v)D(u+v+w)dm_{o}(w|{\theta})dm_{o}(v|{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-25.0pt}E\left[\mathop{\sum }_{k=1}^{N(t)}\mathop{\sum }_{j=N(t)+1}^{N(t+h)}D^{2}(T_{k})D(T_{j})|{\delta}(z),z\in [0,t+h]\right]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=E\Bigg[E\Bigg[\mathop{\sum }_{k=1}^{N(t)}\mathop{\sum }_{j=N(t)+1}^{N(t+h)}D^{2}(T_{k})D(T_{j})|N(t),\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad N(t+h),{\Lambda},{\Theta},{\delta}(z),z\in [0,t+h]\Bigg]\Bigg]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\mathop{\sum }_{n=1}^{\infty }\mathop{\sum }_{m=n+1}^{\infty }\mathop{\sum }_{k=1}^{n}\mathop{\sum }_{j=k+1}^{m}\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t}^{t+h}D^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u)D(v)f_{T_{k},T_{j},N(t),N(t+h)|{\Lambda}={\lambda},{\Theta}={\theta}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u,v,n,m|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{m=j}^{\infty }\mathop{\sum }_{n=k}^{j-1}\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t}^{t+h}D^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u)D(v)f_{T_{k},T_{j},N(t),N(t+h)|{\Lambda}={\lambda},{\Theta}={\theta}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u,v,n,m|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{m=j}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t}^{t+h}D^{2}(u)D(v)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{T_{k},T_{j},N(t),N(t+h)|{\Lambda}={\lambda},∼{\Theta}={\theta}}(u,v,m|{\lambda},{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{m=j}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t}^{t+h}D^{2}(u)D(v)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼P(N(t+h-v)=m-j|{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{T_{j-k}|{\Lambda}={\lambda}}(v-u|{\lambda})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{T_{k}|{\Theta}={\theta}}(u|{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{l=j+1}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t-u}^{t+h-u}D^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u)D(v)P(N(t+h-v)=m-j|{\Lambda}={\lambda},{\Theta}={\theta})\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dF_{W_{2}|{\Lambda}={\lambda}}^{\ast (j-k)}(v-u|{\theta})d\{F_{W_{1}|{\Lambda}={\lambda}}\ast F_{W_{2}|{\Theta}={\theta}}^{\ast k-1}\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t-u}^{t+h-u}D^{2}(u)D(v)dm_{o}(v|{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}E\Bigg[\mathop{\sum }_{k=1}^{N(t)-1}\mathop{\sum }_{j=k+1}^{N(t)}\mathop{\sum }_{l=N(t)+1}^{N(t+h)}D(T_{k})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼D(T_{j})D(T_{l})|{\delta}(z),z\in [0,t+h]\Bigg]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=E\Bigg[E\Bigg[\mathop{\sum }_{k=1}^{N(t)-1}\mathop{\sum }_{j=k+1}^{N(t)}\mathop{\sum }_{l=N(t)+1}^{N(t+h)}D(T_{k})D(T_{j})D(T_{l})|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼N(t),N(t+h),{\Lambda},{\Theta},{\delta}(z),z\in [0,t+h]\Bigg]\Bigg]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\mathop{\sum }_{n=2}^{\infty }\mathop{\sum }_{m=n+1}^{\infty }\mathop{\sum }_{k=1}^{n-1}\mathop{\sum }_{j=k+1}^{n}\mathop{\sum }_{l=n+1}^{m}\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{u}^{t}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼\int _{v}^{t+h}D(u)D(v)D(w)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{T_{k},T_{j},T_{j},N(t),N(t+h)|{\Lambda}={\lambda},{\Theta}={\theta}}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u,v,w,n,m|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{l=j+1}^{n-1}\mathop{\sum }_{m=l}^{n}\mathop{\sum }_{n=j}^{l-1}\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{u}^{t}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼\int _{v}^{t+h}D(u)D(v)D(w)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{T_{k},T_{j},T_{j},N(t),N(t+h)|{\Lambda}={\lambda},{\Theta}={\theta}}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u,v,w,n,m|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{l=j+1}^{\infty }\mathop{\sum }_{m=l}^{l-1}\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{u}^{t}\int _{t}^{t+h}D(u)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼D(v)D(w)f_{T_{k},T_{j},T_{j},N(t),N(t+h)|{\Lambda}={\lambda},{\Theta}={\theta}}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u,v,w,n,m|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{l=j+1}^{\infty }\mathop{\sum }_{m=l}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{u}^{t}\int _{t}^{t+h}D(u)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼D(v)D(w)P(N(t+h-w)=m-l|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼{\Lambda}={\lambda},{\Theta}={\theta})f_{T_{l-j}|{\Lambda}={\lambda}}(w-v)\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{T_{l-j}|{\Lambda}={\lambda}}(w-v)f_{T_{j-k}|{\Lambda}={\lambda}}(v-u)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼f_{T_{k}|{\Lambda}={\lambda},{\Theta}={\theta}}(u)f_{{\Theta}}({\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{j=k+1}^{\infty }\mathop{\sum }_{l=j+1}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{t}^{t-u}\int _{t-v}^{t+h-u-v}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼D(u)D(u+v)D(u+v+w)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼P(N(t+h-w)=m-l|{\Theta}={\theta})dF_{W_{2}|{\Theta}={\theta}}^{\ast (l-j)}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(w|{\Lambda}={\lambda})dF_{W_{2}|{\Theta}={\theta}}^{\ast (j-k)}(v|{\Theta}={\theta})dF_{W_{1}|{\Theta}={\theta}}^{\ast k}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼(u|{\Lambda}={\lambda},{\Theta}={\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})dvdud{\theta}d{\lambda}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}=\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{t}\int _{0}^{t-u}\int _{t-u-v}^{t+h-u-v}D(u)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼D(u+v)D(u+v+w)dm_{o}(w|{\theta})dm_{o}(v|{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼dm_{d}(u|{\lambda},{\theta})f_{{\Lambda},{\Theta}}({\lambda},{\theta})d{\theta}d{\lambda}\nonumber\end{eqnarray}.$ Finally, since each of the three previous integrals is a random variable, the result follows by taking the expectation of each of these expressions, a known property of the theory of stochastic integrals (see Karatzas [17], p. 3).

TheoremH .
Let $\{{\delta}(t),t\geq 0\}$ be an Itô process satisfying the stochastic differential equation of Ho--Lee--Merton d𝛿(t) = rdt + 𝜎 dB (t), with constant drift r and constant diffusion coefficient 𝜎, and where B (t) is a standard Brownian motion standard (see, Cairns [9], p. 87).

Thus, it is easy to prove that $\begin{eqnarray}\displaystyle \displaystyle \int _{0}^{t}{\delta}(x)dx\sim N({\delta}(0)t+r\frac{t^{2}}{2},{\sigma}^{2}\frac{t^{3}}{3}) & & \displaystyle \nonumber\end{eqnarray}.$ Using the property of Itô processes (see Oksendal [25], p. 14 ff), we can obtain $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}E[D^{3}(u)]=\exp \left\{-3{\delta}(0)u-\frac{3}{2}ru^{2}+\frac{3}{2}{\sigma}u^{3}\right\},\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}E[D^{2}(u)D(u+v)]=\exp \left\{-3{\delta}(0)(3u+v)-\frac{r}{2}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼\left.(v^{2}+2uv+3u^{2})+\frac{{\sigma}^{2}}{2}\left[\frac{(3u+v)^{3}+2v^{3}}{9}\right]\right\},\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}E[D(u)D^{2}(u+v)]=\exp \left\{-3{\delta}(0)(3u+2v)-\frac{r}{2}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\times ∼\left.(2v^{2}+4uv+3u^{2})+\frac{{\sigma}^{2}}{2}\left[\frac{(3u+2v)^{3}+4v^{3}}{9}\right]\right\},\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}E[D(u)D(u+v)D(u+v+w)]=\exp \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\times ∼\left\{\begin{array}{@{}l@{}}\displaystyle -3{\delta}(0)(3u+2v+v)\\ ∼-\frac{r}{2}[u^{2}+(u+v)^{2}+(u+v+w)^{2}]\\ \displaystyle ∼+\frac{{\sigma}^{2}}{2}\left[\frac{(3u+2v+w)^{3}+(2v+w)^{3}+3w^{3}}{18}\right]\end{array}\right\}.\nonumber\end{eqnarray}$

4. Quadratic predictor

In this session, our goal is to extend in different directions the results obtained by Léveillé & Adekambi [19,20]. Thus we present a quadratic predictor of the present value of the claims process.

4.1. Motivation

Having all the information on our risk process Z (t), until time t, we would like to be able to estimate the future behavior of this one, even in the case of a stochastic force of interest, so as to better reevaluate the premium for example. Since it is usually difficult to obtain the distribution of Z (t), one must then rely on methods of estimation, regression or simulation.

As we obtained the simple and joint moments of Z (t), we can for example construct predictors (linear, quadratic, polynomial) based on the minimization of the quadratic distance to predict the value of Z (t + h) if we know that of Z (t). These predictors will help us to better assess our risk, whether to estimate its variability over time or to readjust the premium, as and when information received by the insurer.

Let us construct a quadratic predictor, Q (t, h) = a + bZ (t) + cZ ²(t) where a and b are possibly dependent on t and h, minimizing the function B _{t, h} defined by: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}B_{t,h}(a,b)=E[[Z(t+h)-a-bZ(t)-cZ^{2}(t)]^{2}]\nonumber\end{eqnarray}.$ So we have: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\frac{\partial B_{t,h}(a,b)}{\partial a}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=-2E[Z(t+h)-a-bZ(t)-cZ^{2}(t)]=0\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\frac{\partial B_{t,h}(a,b)}{\partial b}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}=-2E[Z(t)[Z(t+h)-a-bZ(t)-cZ^{2}(t)]]=0\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\frac{\partial B_{t,h}(a,b)}{\partial c}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-17.0pt}=-2E[Z^{2}(t)[Z(t+h)-a-bZ(t)-cZ^{2}(t)]]=0.\nonumber\end{eqnarray}$ Hence, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\left(\begin{array}{@{}ccc@{}}\displaystyle 1 & E[Z(t)] & E[Z^{2}(t)]\\ E[Z(t)] & E[Z^{2}(t)] & E[Z^{3}(t)]\\ E[Z^{2}(t)] & E[Z^{3}(t)] & E[Z^{4}(t)]\end{array}\right)\left(\begin{array}{@{}c@{}}\displaystyle a\\ b\\ c\end{array}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}=\left(\begin{array}{@{}c@{}}\displaystyle E[Z(t+h)]\\ E[Z(t)Z(t+h)]\\ E[Z^{2}(t)Z(t+h)]\end{array}\right).\nonumber\end{eqnarray}$ With $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-25.0pt}\bullet ∼∼{\Delta}=E[Z^{2}(t)]E[Z^{4}(t)]-E^{2}[Z^{3}(t)]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}-E^{2}[Z(t)]E[Z^{4}(t)]-E[Z^{3}(t)]E[Z^{2}(t)]E[Z(t)]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-25.0pt}\bullet ∼∼{\Delta}_{a}=E[Z(t+h)]\{E[Z^{2}(t)]E[Z^{4}(t)]-E^{2}[Z^{3}(t)]\}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}-∼E[Z(t)Z(t+h)]\{E[Z(t)]E[Z^{4}(t)]-E[Z^{2}(t)]E[Z^{3}(t)]\}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}+∼E[Z(t)Z(t+h)]\{E[Z(t)]E[Z^{4}(t)]-E[Z^{2}(t)]E[Z^{3}(t)]\}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-25.0pt}\bullet ∼∼{\Delta}_{b}=E[Z(t)Z(t+h)]E[Z^{4}(t)]-E[Z^{2}(t)Z(t+h)]E[Z^{3}(t)]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}-∼E[Z(t)]\{E[Z(t+h)]E[Z^{4}(t)]-E[Z^{2}(t)Z(t+h)]E[Z^{2}(t)]\}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}+∼E[Z^{2}(t)]\{E[Z(t+h)]E[Z^{3}(t)]-E[Z^{2}(t)]E[Z(t)Z(t+h)]\}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-25.0pt}\bullet ∼∼{\Delta}_{c}=E[Z^{2}(t)Z(t+h)]E[Z^{2}(t)]-E[Z(t)Z(t+h)]E[Z^{3}(t)]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}-∼E[Z(t)]\{E[Z^{2}(t)Z(t+h)]E[Z(t)]-E[Z(t+h)]E[Z^{3}(t)]\}\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}+∼E[Z^{2}(t)]\{E\left[Z(t+h)Z(t)\right]E[Z(t)]-E[Z^{2}(t)]E[Z(t+h)]\}\nonumber\end{eqnarray}.$ We get from the previous equations, $\begin{eqnarray}\displaystyle \displaystyle a=\frac{{\Delta}_{a}}{{\Delta}},\quad b=\frac{{\Delta}_{b}}{{\Delta}},\quad c=\frac{{\Delta}_{c}}{{\Delta}} & & \displaystyle \nonumber\end{eqnarray}.$ We now consider the case where the claims amounts is 1, the number of claims follow a conditional Poisson distribution of parameter 𝛩 ∼ 𝛤(2,2) and 𝛿 = 0.005. Then from results of Theorem F , we have: $\begin{eqnarray}\displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}\bullet ∼∼E[X^{3}]=1,∼∼E[X]=1,∼∼E[X^{2}]=1,\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-16.0pt}E[Z(t)]=\left(\frac{1-e^{-{\delta}t}}{{\delta}}\right)E[{\Theta}],\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-16.0pt}Var[Z(t)]=\frac{1-e^{-2{\delta}t}}{{\delta}}E[{\Theta}],\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}\bullet ∼∼E[Z(t+h)]=\left(\frac{1-e^{-{\delta}(t+h)}}{{\delta}}\right)E[{\Theta}],\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-16.0pt}Cov(Z(t),Z(t+h))=\left(\frac{1-e^{-2{\delta}t}}{{\delta}}\right)E[{\Theta}]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}\bullet ∼∼E[Z^{3}(t)]=\frac{(1-e^{-3{\delta}t})}{3{\delta}}E[{\Theta}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +∼\frac{3(e^{-3{\delta}t}-e^{-2{\delta}t}-e^{-{\delta}t}+1)}{2{\delta}^{2}}E[{\Theta}^{2}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +∼\frac{(-e^{-3{\delta}t}+3e^{-2{\delta}t}-3e^{-{\delta}t}+1)}{{\delta}^{3}}E[{\Theta}^{3}]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}\bullet ∼∼E[Z^{4}(t)]=\frac{(1-e^{-4{\delta}t})}{4{\delta}}E[{\Theta}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +\frac{(3-4e^{-{\delta}t}+e^{-4{\delta}t})}{2{\delta}^{2}}E[{\Theta}^{2}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +\frac{3(1-2e^{-2{\delta}t}+e^{-4{\delta}t})}{4{\delta}^{2}}E[{\Theta}^{2}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +∼\frac{(3-4e^{-{\delta}t}+6e^{-2{\delta}t}-5e^{-4{\delta}t})}{2{\delta}^{3}}E[{\Theta}^{3}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +\frac{(1-4e^{-3{\delta}t}+3e^{-4{\delta}t})}{12{\delta}^{2}}E[{\Theta}^{2}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +∼\displaystyle \frac{(1-4e^{-{\delta}t}+6e^{-2{\delta}t}-3e^{-4{\delta}t})}{{\delta}^{4}}E[{\Theta}^{4}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +\frac{(3-4e^{-{\delta}t}-6e^{-2{\delta}t}+12e^{-3{\delta}t}-5e^{-4{\delta}t})}{2{\delta}^{3}}E[{\Theta}^{3}]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}\bullet ∼∼E[Z^{4}(t)]=\frac{(1-e^{-4{\delta}t})}{4{\delta}}E[{\Theta}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +\frac{(3-4e^{-{\delta}t}+e^{-4{\delta}t})}{2{\delta}^{2}}E[{\Theta}^{2}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +\frac{3(1-2e^{-2{\delta}t}+e^{-4{\delta}t})}{4{\delta}^{2}}E[{\Theta}^{2}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +∼\displaystyle \frac{(3-4e^{-{\delta}t}+6e^{-2{\delta}t}-5e^{-4{\delta}t})}{2{\delta}^{3}}E[{\Theta}^{3}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +\frac{(1-4e^{-3{\delta}t}+3e^{-4{\delta}t})}{12{\delta}^{2}}E[{\Theta}^{2}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +∼\displaystyle \frac{(1-4e^{-{\delta}t}+6e^{-2{\delta}t}-3e^{-4{\delta}t})}{{\delta}^{4}}E[{\Theta}^{4}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +\frac{(3-4e^{-{\delta}t}-6e^{-2{\delta}t}+12e^{-3{\delta}t}-5e^{-4{\delta}t})}{2{\delta}^{3}}E[{\Theta}^{3}]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}\bullet ∼∼E[Z(t)Z(t+h)]=\frac{(1-e^{-2{\delta}t})}{2{\delta}}E[{\Theta}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +\frac{(1-e^{-{\delta}t})(1-e^{-{\delta}(t+h)})}{{\delta}^{2}}E[{\Theta}^{2}]\nonumber\\ \displaystyle \displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}\bullet ∼∼E[Z^{2}(t)Z(t+h)]=\frac{1-e^{-3{\delta}t}}{3{\delta}}E[{\Theta}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +\frac{1-e^{-{\delta}t}}{2{\delta}^{2}}[2-e^{-{\delta}(t+h)}(1-e^{-2{\delta}t})]E[{\Theta}^{2}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +∼\frac{1}{2{\delta}^{3}}\big\{({\delta}+2)-e^{-{\delta}t}\big(4+3{\delta}e^{-{\delta}t}-2e^{-{\delta}t}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad -2{\delta}e^{-2{\delta}t}\big)-2e^{-{\delta}(t+h)}(1-e^{-{\delta}t})^{2}\big\}E[{\Theta}^{3}].\nonumber\end{eqnarray}$

We find that the difference between the estimated value and the simulated value is not very large for small h values relative to t. Similarly, when the value of h becomes large relative to t, the estimates are not very good. Obviously the price to pay with the estimated value is to calculate simple and joint moments, but with simulation we cannot perform sensitivity analysis on the discounted aggregate parameters. When the value of t is high, the discount factor cancels the effect of those moments.

Table 1
Comparison between Z _simul(t + h)|Z (t) and Q (t, h)

t h Z (t) Z _simul (t + h)|Z (t) Q (t, h )

1 0.001 0.967 1.007 1.01270

1 1 0.967 1.927 1.9957

1 10 0.967 11.011 10.7079

10 0.01 9.985 9.999 9.9945

10 1 9.985 11.113 10.9339

10 10 9.985 20.037 19.2634

100 0.01 100.1 100.241 100.1058

100 1 100.1 101.136 100.7050

100 10 100.1 110.077 106.0162

t	h	Z (t)	Z _simul (t + h)\|Z (t)	Q (t, h )
1	0.001	0.967	1.007	1.01270
1	1	0.967	1.927	1.9957
1	10	0.967	11.011	10.7079
10	0.01	9.985	9.999	9.9945
10	1	9.985	11.113	10.9339
10	10	9.985	20.037	19.2634
100	0.01	100.1	100.241	100.1058
100	1	100.1	101.136	100.7050
100	10	100.1	110.077	106.0162

5. Conclusion

We have constructed a quadratic predictor of the compound discounted renewal aggregate claims when taking into account dependence within the inter-occurrence times by giving explicit formulas for the joint and higher moments of that sum. To evaluate the accuracy of the proposed quadratic predictor, we compare its value to the simulated value of the compound discounted renewal aggregate. The techniques used are an extension of Léveillé and Adékambi [19,20].

Possible extensions to this research include the computation of the distribution of that sum with the same risk process.

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