On a Bivariate Poisson Negative Binomial Risk Process

—In this paper we deﬁne a bivariate counting process as a compound Poisson process with bivariate negative binomial compounding distribution. We investigate some of its basic properties, recursion formulas and probability mass function. Then we consider a risk model in which the claim counting process is the deﬁned bivariate Poisson negative binomial process. For the deﬁned risk model we derive the distribution of the time to ruin in two cases and the corresponding Laplace transforms. We discuss in detail the particular case of exponentially distributed claims.


I. Introduction
We consider the stochastic process N(t), t > 0 defined on a fixed probability space (Ω, F , P) and given by where X i , i = 1, 2, . . .are independent, identically distributed (iid) as X random variables, independent of N 1 (t).We suppose that the counting process N 1 (t) is a Poisson process with intensity λ > 0 (N 1 (t) ∼ Po(λt)).In this case N(t) is a compound Poisson process.The probability mass function (PMF) and probability generating function (PGF) of N 1 (t) are given by P(N 1 (t) = i) = (λt) i e −λt i! , i = 0, 1, . . .
and ψ N 1 (t) (s) = e −λt (1−s) . ( The compound Poisson distribution is analyzed by many authors; see Johnson et al. [3], Grandell, [2], Minkova [8].The corresponding compound Poisson process is commonly used as a counting process in risk models; see for example Klugman et al. [5], Minkova [9]. In this paper we suppose that the compounding random variable X has a bivariate negative binomial distribution, given in the next Section II.In Section III we define a counting process with the Bivariate Poisson Negative Binomial distribution (BPNB).We derive the moments and the joint PMF.Then, in Section IV, two types of ruin probability are considered for the risk model with BPNB distributed counting process.We derive the Laplace transforms and analyze the case of exponentially distributed claims.

II. Bivariate Negative Binomial Distribution
Let us consider the bivariate negative binomial distribution, defined by the following PGF, given in Kocherlakota and Kocherlakota [6] where γ = 1 − α − β and r ≥ 1 is a given integer number.We use the notation (X, Y) ∼ BNB(r, α, β).

III. The bivariate counting process
In this section we consider a compound Poisson process with bivariate negative binomial compounding distribution.The resulting process is a bivariate counting process (N 1 (t), N 2 (t)), defined by the PGF where ψ 1 (s 1 , s 2 ) is the PGF of the compounding distribution, given in (4).We say that the counting process defined by ( 9) has a bivariate Poisson Negative binomial distribution with parameters λt, α and β, and use the notation (N 1 (t), N 2 (t)) ∼ BPNB(λt, α, β).The marginal distributions are defined by the following PGFs where ψ 1 (s 1 ) and ψ 1 (s 2 ) are given by ( 6) and (7).
The means are given by E(N From ( 9) we obtain Upon setting s 1 = s 2 = 1 we obtain the product moment of N 1 (t) and N 2 (t) to be which yields the covariance between N 1 (t) and N 2 (t) as For the correlation coefficient we have In terms of ρ 1 and ρ 2 , the covariance and the correlation coefficient have the forms .

IV. Bivariate risk model
Consider the following bivariate surplus process for two lines of business.Here u 1 and u 2 are the initial capitals, c 1 , c 2 represent the premium incomes per unit time and Z 1 , Z 1 1 , Z 1 2 , . . ., and Z 2 , Z 2 1 , Z 2 2 , . . .are two independent sequences of independent random variables, independent of the counting processes N 1 (t) and N 2 (t), representing the corresponding claim sizes.The univariate case of this model was analyzed in Kostadinova [7].Let µ 1 = E(Z 1 ) and µ 2 = E(Z 2 ) be the means of the claims.Denote by S 1 (t) j=1 Z 2 j the corresponding accumulated claim processes.The model, analyzed in Chan et al., [1] is the case when N 1 (t) = N 2 (t) = N(t).Here we consider two possible times to ruin and the corresponding ruin probabilities Ψ max (u 1 , u 2 ) = P(τ max < ∞) and Ψ sum (u 1 , u 2 ) = P(τ sum < ∞).For the event of τ max we have the following: It follows that the ruin probability ψ max (u 1 , u 2 ) is the joint survival function of (S 1 (t), S 2 (t)).
In a similar way, we obtain the event for the τ sum : i.e., the ruin probability ψ sum (u 1 , u 2 ) is the survival function of the sum S 1 (t) + S 2 (t).
According to the definition of Ψ max (u 1 , u 2 ), for the no initial capitals, we have the following where Ψ 1 (0) and Ψ 2 (0) are the ruin probabilities of the models U 1 (t) and U 2 (t) with no initial capitals.The univariate Poisson Negative binomial risk model is analyzed in [7], where it is given that and It follows that the upper bound of the ruin probability is given by The lower bound has the form

A. Laplace transforms
Denote by LT Z 1 (s 1 ) and LT Z 2 (s 2 ) the Laplace transforms of the random variables Z 1 and Z 2 .Then, the Laplace transform of (S 1 (t), S 2 (t)) is given by According to the construction of the counting process, for the Laplace transform of (S 1 (t), S 2 (t)) we have of the marginal compounding distributions, we obtain the Laplace transforms of the marginal compound distributions to be We need the following result about Laplace transforms, given in Omey and Minkova ([10]) Lemma 1.For the joint survival function In our case we have ∞ 0 ∞ 0 e −s 1 x−s 2 y P(S 1 (t) > x, S 2 (t) > y)dxdy Lemma 2. For the survival function P(S 1 (t) + S 2 (t) > x) we have Then, for the ruin probability ψ sum we have:

B. Exponentially distributed claims
Let us consider the case of exponentially distributed claim sizes, i.e.
where Γ(n) is a Gamma function and Γ(a, x) = ∞ x t a−1 e −t dt is the incomplete Gamma function, the truncated exponential sum function.
For the ruin probability ψ max we have where In this case we have Substituting x = u 1 + c 1 t and y = u 2 + c 2 t in the last expression, we obtain the ruin probability ψ max (u 1 , u 2 ), as it was shown in the previous section.
The survival function of the sum S 1 (t) + S 2 (t) is the survival function of the sum of claims, i.e.V. Conclusion In this study we introduce a compound Poisson process with bivariate negative binomial compounding distribution.Also, we find the moments and the joint probability mass function.Then we define the bivariate risk model with bivariate Poisson negative binomial counting process.We find the Laplace transform of the ruin probability and investigate a special case of exponentially distributed claims.