Type II Bivariate Generalized Power Series Poisson Distribution and its Applications in Risk Analysis

In this paper we consider type II bivariate generalized power series Poisson distribution as a compound Poisson distribution with bivariate generalized power series compounding distribution. We obtain some properties, p.m.f and conditional distributions. In addition we also give a brief discussion about the multivariate extension of this case. Then we introduce type II bivariate generalized power series Poisson process and consider a bivariate risk model with type II bivariate generalized power series Poisson model as the counting process. For this model we derive distribution of the time to ruin and bounds for the probability of ruin. We obtain partial integro-differential equation for the ruin probabilities and express its bivariate transform through two univariate boundary transforms, where one of the initial capitals is fixed at zero.


Introduction
Bivariate discrete random variables taking non-negative integer values, have received considerable attention in the literature.The type II bivariate Polya -Aeppli distribution was introduced by Minkova and Balakrishnan(2014).Kostadinova and Minkova(2014) applied bivariate Poisson negative binomial distribution to bivariate risk processes.
Furthermore Kostadinova(2015) introduced a bivariate risk model in which distribution of claim counting process 5 is the bivariate Polya-Aeppli distribution.In the literature it has been found that bivariate compound Poisson distributions are very flexible and can be used efficiently in bivariate risk modeling.With this as motivation, different bivariate compound Poisson distributions have been constructed.
The family of bivariate generalized power series distribution is basically used for counting paired events.It contains many important families like bivariate Poisson, bivariate binomial, bivariate negative binomial and bivariate For the univariate case, where X 1 , X 2 , X 3 ... are independent and identically distributed random variables, independent of N 1 and N 1 has a Poisson distribution with parameter λ, denoted by N 1 ∼ P o (λ).Suppose that X 1 , X 2 , X 3 ... follow generalized power series distribution with PGF P (s) = g(θs) g(θ) , where g(θ) = i a i θ i , a i ≥ 0. Now consider the random sum The distribution of N is called generalized power series Poisson distribution.
Then the corresponding PMF is given by where C m (j) = k1+k2+...+kj =m a(k 1 ), a(k 2 )...a(k j ), If (k 1 , k 2 , ..., k j ) is the ordered j-tuple of positive integers in the range set of the random variable X which sum up to m.
A multivariate extension of the generalized power series Poisson distribution and its properties are discussed in section 4. Bivariate counting processes with type II bivariate generalized power series Poisson distribution is introduced in section 5.In section 6 we consider type II bivariate generalized power series Poisson risk model and derive the distributions of bivariate aggregate claims and sum of aggregate claims of two classes.Section 7 presents three types of ruin probabilities and an expression for ruin probabilities for a type II bivariate generalized power series Poisson risk model is derived.In addition, the bounds for the ruin probabilities are developed.In section 8 a system of partial integro differential equation for the ruin probabilities is developed and the Laplace transform is derived.Section 8 deals with multivariate generalized power series Poisson risk model and the ruin probabilities for the defined risk model.

Bivariate Generalized Power Series Poisson Distribution
Let us consider the sequence (X i , Y i ), i = 1, 2, ... of independent and identically distributed random variables, distributed as (X, Y ). Define where N is independent of the compounding random vector (X, Y ) and has a Poisson distribution with parameter λ.
Suppose that (X, Y ) has a bivariate generalized power series distribution with PGF Then, the joint PGF of the bivariate random vector (N 1 , N 2 ) is given by ) . (2) The PGF in (2) can be written as where is the ordered k tuple of elements in the range of X which sum up to i and (j 1 , j 2 , • • • j k ) is the ordered k tuple of elements in the range of Y which sum up to j .
Differentiation in (3) leads to the following derivatives where Setting s 1 = s 2 = 1 in (4),we obtain the (i, j) th factorial moments of (N 1 , N 2 )

Covariance and Correlation
The means are given by Similarly the variances are obtained as From (2), we obtain Setting s 1 = s 2 = 1 in (5), we easily obtain The covariance and correlation between N 1 and N 2 are respectively given by

Joint Probability Mass Function 50
The joint probability mass function of (N 1 , N 2 ) is obtained by expanding the PGF, Ψ(s 1 , s 2 ) in powers of s 1 and s 2 .
On the other hand, from Johnson et al., it is known that Using the PGF in (2) and the derivatives in (4) we obtain the joint PMF of (N 1 , N 2 ) and is given by is the ordered k tuple of elements in the range of X which sums up to i and (j 1 , j 2 , • • • j k ) is the ordered k tuple of elements in the range of Y which sums up to j .

Marginal Distributions
The PGFs of the marginal compounding distributions are given by , where b i = j a ij θ j 2 and θ 2 are treated as constants.
Analogously, Y has a generalized power series distribution with series function h 2 (θ 2 ) = j c j θ j 2 , where c j = i a ij θ i 1 and θ 1 are treated as constants.
Then , from (2) and ( 6), we obtain the corresponding marginal PGFs of N 1 and N 2 The corresponding marginal distributions of N 1 and N 2 are easily obtained from (7),respectively, to be where .., k j ) is the ordered j-tuple of positive integers in the range set of the random variable X which sum up to m. and where positive integers in the range set of the random variable Y which sum up to m.
Then from it follows that marginal distributions of N 1 and N 2 belongs to univariate generalized power series Poisson 55 distribution.

Conditional Distribution
From Johnson et al.(1997), the conditional P.G.F. of where Substituting (i, j) = (k, 0) and s 1 = 0 in (4), we get Using ( 8) and ( 9) we obtain 60 For k = 0, we get It follows immediately that the conditional mean is In particular .

Multivariate Extension
Let X = (X 1 , • • • X k ) be a k-dimensional random vector of generalized power series random variables.
The PGF of X is given by Define where N is independent of compounding random vector X and has a Poisson distribution with parameter λ.
Then, the joint PGF of (N The PGFs of the marginal compounding distributions are given by Where Therefore from ( 12) it follows that the random variable X i has a generalized power series distribution with series as expanded in powers of θ i , other θ's treated as constants.
The marginal PGFs of N i , i = 1, 2, . . .k are obtained from ( 11) and ( 12), and are given by Then from it follows that N i , i = 1, 2, . . .k belongs to univariate generalized power series Poisson distribution.
The corresponding marginal P.M.Fs are given by where ) is ordered j-tuple of positive integers in the range set of X i which sum up to m.
If (i l1 , i l2 , . . ., i lj ) is the ordered j tuple of elements in the range set of X l which sum up to i l , l = 1, 2, . . ., k.

Type II bivariate
is the ordered k tuple of elements in the range of X which sums up to i and (j 1 , j 2 , • • • j k ) is the ordered k tuple of elements in the range of Y which sums up to j .
Remark: 1. 1.In the case of g(θ , the type II bivariate generalized power series Poisson process coincides with bivariate Poisson negative binomial process; see Kostadinova and Minkova(2014).
2.In the case of g(θ , the type II bivariate generalized power series Poisson process coincides with type II bivariate Polya-Aeppli process; see Kostadinova(2015).

Type II Bivariate Generalized Power Series Poisson Risk Model
Let us assume that there are two kinds of claims X 1i and X 2i belonging to two classes.We will investigate a two dimensional model where u i , i = 1, 2, is the initial capital, c i > 0, i = 1, 2, is the constant premium income per unit time, N i (t) is the number of claims up to time t, X ik is the size of the k th claim and S i (t) = Ni(t) j=1 X ij , i = 1, 2 is the aggregate claims amount for i th class.
For fixed i = 1, 2, {X ik } k≥1 are independent and identically distributed (i.i.d) nonnegative random variable with distribution function F i (X i ) such that F i (0) = 0 and finite mean µ are mutually independent and {(X 1k , X 2k )} k≥1 is a sequence of i.i.d bivariate random vectors with joint distribution function F (x 1 , x 2 ).Here we assume that bivariate counting process {(N 1 (t), N 2 (t)), t ≥ 0} has a type II bivariate generalized power series Poisson process and will call the process the bivariate generalized power series Poisson risk model.
Now we consider the sum of both risk process (17), the joint capital for the two classes is given by: , Central problem in risk theory is the modeling of the probability distribution for the aggregate claims.The aggregate claims distribution is mainly used to compute ruin probabilities.Hesselarger(1996) The joint CDF of aggregate claims is given by Let N (t) = N 1 (t) + N 2 (t) denotes the total number of claims happened in both classes.
Then the PMF of N (t) is given by Now we consider the sum of aggregate claims of two classes Case 1:two classes have different claim size distribution In this case and the corresponding CDF G(u) is given by 100 Case 2:two classes have the same claim size distribution In this case where Denote by G(x) the CDF of S(t) and is given by (20)

Ruin probabilities
Ruin theory for the bivariate risk model has been extensively considered in the literature.It has been found that ruin probabilities are often fundamental interest in risk management purpose.Chan et al.(2003) discussed various ruin concept for bivariate risk process.
The time of ruin for the i th class (i = 1, 2) is defined by and the corresponding probability of ruin is If for each i, the process U i (t) ≥ 0 for all t ≥ 0 (no ruin occurs), we indicae this by writing τ i = ∞.

105
Here we consider three kinds of ruin time as follows.The first one is 0}, representing the first time when both U 1 (t) and U 2 (t) became negative, whereas the second one is τ min (u 1 , u 2 ) = inf{t ≥ 0/ min (U 1 (t), U 2 (t)) < 0}, representing the first time when either U 1 (t) or U 2 (t) became negative, and last one is τ sum = inf{t ≥ 0/U (t) < 0}, representing the time when the joint capital for the two classes U (t) became negative.The associated ruin probabilities will be respectively denoted by Ψ max (u 1 , u 2 ), Ψ min (u 1 , u 2 ) and Ψ sum (u 1 , u 2 ).
First we derive the expression for the ruin probability Ψ max (u 1 , u 2 ) where H(u 1 , u 2 ) is the joint survival function of (S 1 (t), S 2 (t)).
Next we consider the expression for the ruin probability Ψ min (u 1 , u 2 ) where H(u 1 , u 2 ) is the joint CDF of (S 1 (t), S 2 (t)) given by ( 18).
Finally we derive the expression for the ruin probability Ψ sum (u 1 , u 2 ) where G(x) is the survival function of S(t) .

Bounds for Ruin Probability
Most of the papers in the literature of bivariate risk theory are concerned with ruin probabilities.Exact solutions of these probabilities are rarely available, and existing result are mostly in the form of bounds.Chan et al.(2003) , Cai and Li(2005) and Yuen et al.(2006) derived bounds for the ultimate ruin probability Ψ min (u 1 , u 2 ). 110 Simple bounds for Ψ max (u 1 , u 2 ) was given by Cai andLi(2005, 2007).
The lower and upper bounds on Ψ max (u 1 , u 2 ) and Ψ min (u 1 , u 2 ) are respectively described by the following inequalities.
where the final expression in the second equation is exactly the ruin probability in the case where {U 1 (t)} t≥0 and 115 {U 2 (t)} t≥0 are independent.
If there is no initial capitals (u 1 = u 2 = 0), then the above relations becomes In the case of univariate generalized power series Poisson risk model the ruin probabilities are given by Using the equations ( 22) and ( 23) we can obtain bounds for the ruin probabilities Ψ max (0, 0) and Ψ min (0, 0) for the type II bivariate generalized power series Poisson risk model and are given by Moreover, we have In the case of no initial capital above relation is and hence,we obtain

Two Dimensional Integro Differential Equation
In this section we will derive partial integro differential equation for the bivariate survival probability for the bivariate surplus process (17) defined in section 6.
Define the infinite time joint survival probability Φ(u 1 , u 2 ) = P (U 1 (t) ≥ 0, U 2 (t) ≥ 0; for all t ≥ 0). and infinite time joint ruin probability is Ψ In a small time interval (0, h] , there are following possible cases: no claim, one claim from class 1 and no claim from class 2, no claim from class 1 and one claim from class 2, one or more than one claims from each class.It follows that where Rearranging the terms leads to As h tends to zero, we get It is difficult to solve this two dimensional integro differential equation.

Laplace Transforms of the Survival Probabilities
Having obtained the partial integro differential equations(PIDE) for the survival probabilities Φ(u 1 , u 2 ) of the surplus process (17), in the following we will derive the Laplace transforms for the survival probabilities.

Conclusion
In this paper we introduced the type II bivariate generalized power series Poisson distribution as a compound 125 Poisson distribution with generalized power series compounding distribution.We have considered the bivariate risk model with type II bivariate generalized power series Poisson distribution as claim number distribution.Three models of ruin and the probabilities of ruin for the type II bivariate generalized power series Poisson risk model are investigated.Also the bounds for ruin probabilities are developed.We obtained PIDE for the survival probability and derived an expression for bivariate Laplace transform of ruin probabilities. 130 introduced recursive formulas for the joint distribution of the bivariate aggregate claims random variables.Clark and Homer(2003) used Fast Forier Transformation(FFT) to compute bivariate aggregate claims distribution.Here we derive bivariate aggregate claims distribution from type II bivariate generalized power series Poisson risk model via convolution.Let H(u 1 , u 2 , t) denotes the joint cumulative distribution function of bivariate aggregate claims, (S 1 (t), S 2 (t)) and and F * n (x) is the n-fold convolution of claim amount distribution which can be calculated recursively as