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

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1 Type II Bivariate Generalized Power Series Poisson Distribution and its Applications in Ris Analysis KK Jose a, and Shalitha Jacob a,b a Department of Statistics, St Thomas College, Pala, Arunapuram, Kerala , India b Mar Thoma College for Women, Perumbavoor, Kerala , India Abstract 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, pmf 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 ris 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 Keywords: bivariate generalized power series distribution; ruin probability; aggregate claims distribution 1 Introduction Bivariate discrete random variables taing non-negative integer values, have received considerable attention in the literature The type II bivariate Polya - Aeppli distribution was introduced by Minova and Balarishnan214) Kostadinova and Minova214) applied bivariate Poisson negative binomial distribution to bivariate ris processes 5 Furthermore Kostadinova215) introduced a bivariate ris model in which distribution of claim counting process 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 ris modeling With this as motivation, different bivariate compound Poisson distributions have been constructed 1 The family of bivariate generalized power series distribution is basically used for counting paired events It contains many important families lie bivariate Poisson, bivariate binomial, bivariate negative binomial and bivariate Corresponding author address: jstc@gmailcom KK Jose) URL: shalithajacob@yahooin Shalitha Jacob)

2 logarithmic series distributions The PMF of bivariate generalized power series distribution is given by P i, j) = a i,jθ1θ i j 2, i, j) T, 1) where = i,j a i,jθ i 1θ j 2, a i,j and T is a subset of cartesian product of the set of nonnegative integers In this paper we consider compound Poisson distribution with bivariate generalized power series compounding 15 distribution The bivariate type II Polya-Aeppli distribution and the bivariate Poisson negative binomial distribution are the special cases of bivariate generalized power series Poisson distribution 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, denoted by N 1 P o ) Suppose that X 1, X 2, X 3 follow generalized power series distribution with PGF P s) = gθs) gθ), where gθ) = i a iθ i, a i Now consider the random sum N = X 1 + X X N1, The distribution of N is called generalized power series Poisson distribution The PGF of the random variable N is given by Ψ N s) = e 1 P s)) gθs) 1 = e gθ) ) Then the corresponding PMF is given by P N = m) = e, m = C = e θ m m j) gθ) )j, m = 1, 2,, where C m j) = a 1 ), a 2 )a j ), j=m If 1, 2,, j ) is the ordered j-tuple of positive integers in the range set of the random variable X which sum up to m This paper is organised as follows In section 2 the joint probability mass function, correlation coefficient 2 and covariance of type II bivariate generalized power series Poisson distribution are derived In section 3 marginal distribution and conditional distribution of type II bivariate generalized power series Poisson distribution are given 2

3 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 ris model and 25 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 ris 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 ris model and the ruin probabilities 3 for the defined ris model 2 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 N 1 = X 1 + X X N and N 2 = Y 1 + Y Y N, 35 where N is independent of the compounding random vector X, Y ) and has a Poisson distribution with parameter Suppose that X, Y ) has a bivariate generalized power series distribution with PGF P s 1, s 2 ) = gθ 1s 1, θ 2 s 2 ) Then, the joint PGF of the bivariate random vector N 1, N 2 ) is given by Ψs 1, s 2 ) = e 1 P s1,s2)) = e 1 gθ 1 s 1,θ 2 s 2 ) gθ 1,θ 2 ) ) 2) The PGF in 2) can be written as Ψs 1, s 2 ) = e =1 gθ 1,θ 2) C ij )θ 1 s 1 ) i θ 2 s 2, 3) i,j 4 where C ij ) = i 1+i 2+ +i =i a i1j 1 a i2j 2 a i j, if i 1, i 2,, i ) is the ordered tuple of elements in the range of X j 1+j 2+ +j =j which sum up to i and j 1, j 2, j ) is the ordered tuple of elements in the range of Y which sum up to j 3

4 Differentiation in 3) leads to the following derivatives Ψ i,j) s 1, s 2 ) = e =1 gθ 1,θ 2) l i,m j where l i) = l! l i)!, mj) = m! m j)! and Ψ i,j) s 1, s 2 ) = i+j Ψs s i 1, s 2 ) 1 sj 2 Setting s 1 = s 2 = 1 in 4),we obtain the i, j) th factorial moments of N 1, N 2 ) C l,m )l i) m j) θ l 1θ m 2 s l i 1 s m j 2, 4) EN 1 N 1 1) N 1 i + 1)N 2 N 2 1) N 2 j + 1) = e = gθ 1,θ 2) l i,m j C lm )l i) m j) θ l 1θ m 2 21 Covariance and Correlation 45 The means are given by µ 1 = E N 1 ) = θ 1 θ 1 log and µ 2 = E N 2 ) = θ 2 θ 2 log Similarly the variances are obtained as Var N 1 ) = θ 1 θ 1 µ µ2 1, Var N 2 ) = θ 2 θ 2 µ µ2 2 From 2), we obtain 2 s 1 s 2 Ψs 1, s 2 ) = Setting s 1 = s 2 = 1 in 5), we easily obtain 2 gθ 1 s 1, θ 2 s 2 ) + s 1 s 2 ) gθ 1 s 1, θ 2 s 2 ) Ψs 1, s 2 ), s 2 EN 1 N 2 = θ 2 µ 1 + θ ) µ 1 µ 2 ) 2 s 1 gθ 1 s 1, θ 2 s 2 ) 5) The covariance and correlation between N 1 and N 2 are respectively given by Cov N 1, N 2 ) =θ 2 µ θ 2 µ 1µ 2 Corr N 1, N 2 ) = θ 2 2 θ 2 µ µ 1µ 2 θ i θ i µ i + 1 µ2 i i=1 ) 5 22 Joint Probability Mass Function The joint probability mass function of N 1, N 2 ) is obtained by expanding the PGF, Ψs 1, s 2 ) in powers of s 1 and s 2 Let fi, j) = P N 1 = i, N 2 = j), i = j =, 1, 2 denote the joint PMF of N 1, N 2 ) On the other hand, from Johnson et al, it is nown that fi, j) = Ψi,j) s 1, s 2 ) / s 1 = s 2 = i! 4

5 Using the PGF in 2) and the derivatives in 4) we obtain the joint PMF of N 1, N 2 ) and is given by f, ) = e 1 a ), gθ 1,θ 2 ), fi, j) = e θ i 1θ j 2 C ij ) =1 gθ 1,θ 2), i, j =, 1,, i, j), ), where C ij ) = i 1+i 2+ +i =i a i1,j 1 a i2,j 2 a i,j, if i 1, i 2,, i ) is the ordered tuple of elements in the range j 1+j 2+ +j =j of X which sums up to i and j 1, j 2, j ) is the ordered tuple of elements in the range of Y which sums up to j 3 Marginal Distributions The PGFs of the marginal compounding distributions are given by P 1 s 1 ) = P s 1, 1) = gθ1s1,θ2) i gθ 1,θ 2) = biθ1s1)i, where b i = j a ijθ j 2, is independent of θ 1, i biθi 1 P 2 s 2 ) = P 1, s 2 ) = gθ1,θ2s2) j gθ 1,θ 2) = cjθ2s2)j j cjθj 2 Therefore the above marginal PGFs can be written in the form, where c j = i a ijθ i 1, is independent of θ 2 P 1 s 1 ) = h 1θ 1 s 1 ) h 1 θ 1 ), P 2 s 2 ) = h 2θ 2 s 2 ) h 2 θ 2 ) 6) From which it follows that the random variable X has a generalized power series distribution with series function h 1 θ 1 ) = i b iθ i 1, where b i = j a ijθ 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 jθ j 2, where c j = i a ijθ 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 Ψ N1 s 1 ) = Ψs 1, 1) = e 1 h1θ1)), Ψ N2 s 2 ) = Ψ1, s 2 ) = e 1 h2θ2)) 7) The corresponding marginal distributions of N 1 and N 2 are easily obtained from 7),respectively, to be P N 1 = m) = e, m = where C m j) = j=m = e θ m 1 b 1, b 2 b j, C m j) h 1θ 1) )j, m = 1, 2,, If 1, 2,, j ) is the ordered j-tuple of positive integers in the range set of the random variable X which sum up to m 5

6 and P N 2 = m) = e, m = = e θ m 2 D m j) h 2θ 2) )j, m = 1, 2,, where D m j) = c 1, c 2 c j, j=m If 1, 2,, j ) is the ordered j-tuple of positive integers in the range set of the random variable Y which sum up to m 55 Then from it follows that marginal distributions of N 1 and N 2 belongs to univariate generalized power series Poisson distribution 31 Conditional Distribution From Johnson et al1997), the conditional PGF of N 2 given N 1 written Ψ N2/N 1=s 2 ), is where Ψ,) s 1, s 2 ) = s 1 Ψs 1, s 2 ) Substituting i, j) =, ) and s 1 = in 4), we get Ψ,), s 2 ) = e Ψ N2/N 1=s) = Ψ,), s 2 ) Ψ,), 1), 8) m=1 j gθ 1,θ 2) m! ) m C j m)θ 1θ 2 s 2 9) 6 Using 8) and 9) we obtain For =, we get Ψ N2/N 1=s 2 ) = m= j m=1 j Ψ N2/N 1=s 2 ) = Ψ, s 2) Ψ, 1) gθ 1,θ 2 ) = e g,θ 2 ) gθ 1,θ 2 ) ) m m! C j m)θ1θ 2 s 2 1) gθ 1,θ 2 ) ) m m! C j m)θ 1 θj 2 [ 1 g, θ ] 2s 2 ) It follows immediately that the conditional mean is In particular ) m fθ 1,θ 2 ) m=1 m! C j m)jθ j 2 E[N 2 /N 1 = ] = ) m fθ 1,θ 2 ) m=1 m! C j m)θ j 2 EN 2 /N 1 = ) = tθ 2 θ 2 g, θ 2 ) 6

7 4 Multivariate Extension Let X = X 1, X ) be a -dimensional random vector of generalized power series random variables The PGF of X is given by P s 1, s 2,, s ) = gθ 1s 1, θ 2 s 2,, θ s ) gθ 1, θ 2,, θ ) Define N i = X i1 + X i2 + + X in, i = 1, 2,,, where N is independent of compounding random vector X and has a Poisson distribution with parameter Then, the joint PGF of N 1, N 2,, N ) is given by Ψs 1, s 2,, s ) = e 1 P s1,s2, s )) = e 1 gθ 1 s 1,θ 2 s 2,,θ s ) gθ 1,θ 2,,θ ) ) 11) 65 The PGFs of the marginal compounding distributions are given by P Xi s i ) = P 1,, s i,, 1) = gθ 1,, θ i s i, θ ) gθ 1,, θ ) = h iθ i s i ), i = 1, 2,, 12) h i θ i ) Where h i θ i ) = xi b ix i )θ xi i and b i x i ) = x 1 x i 1x i+1 x a x1x 2 x θ x1 1 θxi 1 θxi+1 θx Therefore from 12) it follows that the random variable X i has a generalized power series distribution with series function h i θ i ) = xi b ix i )θ xi i as expanded in powers of θ i, other θ s treated as constants The marginal PGFs of N i, i = 1, 2, are obtained from 11) and 12), and are given by Ψ Ni s i ) = Ψ1,, s i,, 1) = e 1 h i θ i s i ) h i θ i ) ), i = 1, 2,, Then from it follows that N i, i = 1, 2, belongs to univariate generalized power series Poisson distribution The corresponding marginal PMFs are given by e, m = P N i = m) = e θi m h iθ i) C im j), m = 1, 2,, j where C im j) = j=m b i 1 )b i 2 ) b i j ) If 1, 2,, j ) is ordered j-tuple of positive integers in the range set of X i which sum up to m 41 Joint Probability Mass Function The PGF in 11) can be written as Ψs 1, s 2,, s ) = e i 1,i 2,i gθ 1,θ 2,,θ ) C i1,i 2,i j)θ 1 s 1 ) i1 θ 2 s 2 ) i2 θ s ) i 13) 7

8 7 Differentiation in 13) leads to the following derivatives Ψ r1,r2,,r ) s 1,, s ) = e i 1 r 1,,i r gθ 1,θ 2,,θ ) C i1,i 2,,i j)i r1) 1 θ i1 1 si1 r1 1 i r ) θ i s i r, 14) where i rj) j = i i, j = 1, 2,, and j r j)! Ψr1,r2,,r) s 1,, s ) = r1+r 2+ +r s r 1 1 s r 2 2,s r et al1997), it is nown that for r 1, r =, 1, and r 1, r 2,, r ),, ), fr 1, r 2,, r ) = Ψr1,r2,,r) s 1,, s ) / s 1 = = s = r 1! r! Ψs 1, s 2,, s ) From Johnson Denote by fi 1,, i ) = P N 1 = i 1,, N = i ), i 1 i =, 1, 2, the joint PMF of N 1, N 2,, N ) and is given by f,, ) = e 1 a ),, gθ 1,,θ ) fi 1, i 2,, i ) = e, gθ 1,,θ ) C i1,i 2,,i j)θ i1 1 θi2 2 θi, i 1, i 2,, i =, 1,, i 1, i 2,, i ),, ), where C i1,i 2,,i j) ai 11, i 21,, i 1 )ai 12, i 22,, i 2 ) ai 1j, i 2j,, i j ) i 11+i 12,+,+i ij=i 1 i 1 +i 2 + +i j =i 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,, Setting s 1 = s 2 = = s = 1 in 14), we obtain the joint factorial moment of N 1, N 2,, N ) 42 Conditional Distribution E[N 1 N 1 1) N 1 r 1 + 1) N N 1 ) N r +1 )] = e gθ 1,θ ) C i1,i 2,,i j)i r1) 1 θ i1 1 ir ) i 1 r 1,,i r From Johnson et al1997),the conditional PGF of N 2 N ), given N 1 written Ψ N2N /N 1=i 1 s 2,, s ), is Ψ N2N /N 1=i 1 s 2,, s ) = Ψi1,,,), s 2,, s ) Ψ i1,,), 1,, 1), 15) θ i where Ψ i1,,,) s 1,, s ) = i1 Ψs 1,, s ) s i1 1 8

9 Substituting r 1, r 2,, r ) = i 1,,, ) and s 1 = in 14), we get Ψ i1,,,), s 2,, s ) = e Using 15) and 16) we obtain For i 1 =, we get Ψ N2N /N 1=i 1 s 2,, s ) = i 2,,i i 2,,i gθ 1,,θ ) i 2,,i gθ 1,,θ ) C i1,i 2,i j)i 1!θ i1 1 θ 2s 2 ) i2 θ s ) i 16) C i1,i 2,i j)i 1!θ i1 1 θ 2s 2 ) i2 θ s ) i gθ 1,,θ ) Ψ N2N /N 1=s 2,, s ) = Ψ, s 2,, s ) Ψ, 1,, 1) = e g,θ 2,,θ ) gθ 1,θ 2,,θ ) The conditional PGF of N given N 1, N 2,, N 1 ) is Ψ N /N 1=i 1,,N 1 =i 1 s ) = Ψi1,,i 1,),,, s ) Ψ i1,,i 1,),,, 1) For i 1 = i 2 = = i 1 =, we get = i i gθ 1,,θ ) C i1,i 2,i j)i 1!θ i1 1 θi2 1 g, θ ) 2s 2,, θ s ) gθ 1, θ 2,, θ ) C i1,i 2,i j)i 1!i 2! i!θ i1 gθ 1,,θ ) Ψ N /N 1=N 2= =N 1 = = Ψ,,, s ) Ψ,,, 1) It follows immediately that the conditional mean is = e g,,,θ ) gθ 1,,θ ) 1 θi2 C i1,i 2,i j)i 1!i 2! i 1!θ i1 1 g,,, θ s ) gθ 1,, θ ) 2 θi 2 θi 1 1 θ s ) i ) 1 θi2 2 θi EN /N 1 = i 1, N 2 = i 2,, N 1 = i 1 ) = i i gθ 1,,θ ) C i1,i 2,i j)i 1! i 1!θ i1 1 θi 1 1 θ s ) i gθ 1,,θ ) C i1,i 2,i j)i 1!i 2! i 1!θ i1 1 θ 2) i2 θ i In particular EN /N 1 = = N 1 = ) = tθ θ g,,, θ ) gθ 1,, θ ) 75 5 Type II bivariate Generalized Power Series Poisson Process Consider a compound Poisson process with bivariate generalized power series compounding distribution, given in 1) The resulting bivariate counting process N 1 t), N 2 t)) has type II bivariate generalized power series Poisson 9

10 distribution with parameters t, θ 1 and θ 2 ie, f, ) = e t 1 a ), gθ 1,θ 2 ), fi, j) = e t θ i 1θ j 2 C ij ) =1 gθ 1,θ 2), i, j =, 1,, i, j), ), where C ij ) = i 1+i 2+ +i =i a i1,j 1 a l2,j 2 a i,j, if i 1, i 2,, i ) is the ordered tuple of elements in the range j 1+j 2+ +j =j of X which sums up to i and j 1, j 2, j ) is the ordered tuple of elements in the range of Y which sums up to j To express {N 1 t), N 2 t)), t } is type II bivariate generalized power series Poisson process with parameters 8 85 t, θ 1 and θ 2 we use the notation N 1 t), N 2 t)) BGP S II t, θ 1, θ 2 ) Remar: 1 1In the case of = 1 θ 1 θ 2 ) r, θ 1 = α, θ 2 = β and ai, j) = i+j j ) r+i+j 1 i+j ), the type II bivariate generalized power series Poisson process coincides with bivariate Poisson negative binomial process; see Kostadinova and Minova214) 2In the case of = 1 θ 1 θ 2 ) 1, θ 1 = α, θ 2 = β and ai, j) = i+j j ) r+i+j 1 i+j ), the type II bivariate generalized power series Poisson process coincides with type II bivariate Polya-Aeppli process; see Kostadinova215) 6 Type II Bivariate Generalized Power Series Poisson Ris Model Let us assume that there are two inds of claims X 1i and X 2i belonging to two classes We will investigate a two dimensional model U 1 t) = u 1 + c 1 t S 1 t), U 2 t) = u 2 + c 2 t S 2 t), 17) where u i, i = 1, 2, is the initial capital, c i >, i = 1, 2, is the constant premium income per unit time, N i t) is the 9 number of claims up to time t, X i is the size of the th claim and S i t) = N it) X ij, i = 1, 2 is the aggregate claims amount for i th class For fixed i = 1, 2, {X i } 1 are independent and identically distributed iid) nonnegative random variable with distribution function F i X i ) such that F i ) = and finite mean µ i = EX i ) Assume that {N i t), t }, {X 1, X 2 )} 1 are mutually independent and {X 1, X 2 )} 1 is a sequence of iid bivariate random vectors with joint distribution 95 function F x 1, x 2 ) Here we assume that bivariate counting process {N 1 t), N 2 t)), t } has a type II bivariate generalized power series Poisson process and will call the process the bivariate generalized power series Poisson ris model Now we consider the sum of both ris process 17), the joint capital for the two classes is given by: Ut) = U 1 t) + U 2 t) = u + ct St), 1

11 where u = u 1 + u 2, c = c 1 + c 2 and St) = S 1 t) + S 2 t) Central problem in ris theory is the modeling of the probability distribution for the aggregate claims The aggregate claims distribution is mainly used to compute ruin probabilities Hesselarger1996) introduced recursive formulas for the joint distribution of the bivariate aggregate claims random variables Clar and Homer23) used Fast Forier TransformationFFT) to compute bivariate aggregate claims distribution Here we derive bivariate aggregate claims distribution from type II bivariate generalized power series Poisson ris model via convolution Let Hu 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 F n x) = The joint CDF of aggregate claims is given by x F n 1 x y)fy)dy H S1t),S 2t))x, y, t) = P S 1 t) x, S 2 t) y) = P N 1 t) = i, N 2 t) = j)f1 i x)f j 2 y) i,j= = e t 1 a ), + e t i,=1 + e t j,=1 + e t i,j,=1 C i, )θ i 1 C,j )θ j 2 C i,j )θ i 1θ j 2 t gθ 1,θ 2) t gθ 1,θ 2) t gθ 1,θ 2) F i 1 x) F j 2 y) F i 1 x)f j 2 y) Let Nt) = N 1 t) + N 2 t) denotes the total number of claims happened in both classes 18) Then the PMF of Nt) is given by e t 1 a, gθ ), = 1,θ 2) P Nt) = ) = e t t gθ 1,θ 2) i= C i, i j)θ i 1θ2 i, = 1, 2, Now we consider the sum of aggregate claims of two classes St) = S 1 t) + S 2 t) Case 1:two classes have different claim size distribution In this case St) = N 1t) N 2t) X 1j + X 2j 11

12 1 and the corresponding CDF Gu) is given by Gu) = P St) x) = P N 1 t) = i, N 2 t) = j)f1 i F j 2 x), i,j= = e t 1 a ), gθ 1,θ 2 ) + e t + e t j,=1 C,j )θ j 2 i,=1 t gθ 1,θ 2) C i, )θ i 1 F j 2 t gθ 1,θ 2) x) + e t F i 1 x) i,j,=1 C i,j )θ i 1θ j 2 t gθ 1,θ 2) F i 1 F j 2 x) 19) Case 2:two classes have the same claim size distribution In this case St) = X 1 + X X Nt), where Nt) = N 1 t) + N 2 t) Denote by Gx) the CDF of St) and is given by Gx) = P St) x) = P Nt) = i)f i x), i= = e t 1 a ), gθ 1,θ 2 ) + e t i i, r= t gθ 1,θ 2) C r,i r j)θ r 1θ i r 2 F i x) 2) 7 Ruin probabilities Ruin theory for the bivariate ris model has been extensively considered in the literature It has been found that ruin probabilities are often fundamental interest in ris management purpose Chan et al23) discussed various ruin concept for bivariate ris process The time of ruin for the i th class i = 1, 2) is defined by τ i = inf{t ; U i t) < }, and the corresponding probability of ruin is Ψ i u i ) = P τ i < /U i ) = u i ) If for each i, the process U i t) for all t no ruin occurs), we indicae this by writing τ i = 15 12

13 Here we consider three inds of ruin time as follows The first one is τ max u 1, u 2 ) = inf{t / max U 1 t), U 2 t)) < }, 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 / min U 1 t), U 2 t)) < }, representing the first time when either U 1 t) or U 2 t) became negative, and last one is τ sum = inf{t /Ut) < }, representing the time when the joint capital for the two classes Ut) 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 ) Ψ max u 1, u 2 ) = P τ max < /U 1 ) = u 1, U 2 ) = u 2 ) = P maxu 1 t), U 2 t)) < ) = P U 1 t) <, U 2 t) < ) = P S 1 t) > u 1 + c 1 t, S 2 t)) > u 2 + c 2 t) = Hu 1 + c 1 t, u 2 + c 2 t) where Hu 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 ) Ψ min u 1, u 2 ) = P τ min < /U 1 ) = u 1, U 2 ) = u 2 ) = P minu 1 t), U 2 t)) < ) = 1 P minu 1 t), U 2 t)) > ) = 1 P U 1 t) >, U 2 t)) > ) = 1 P S 1 t) < u 1 + c 1 t, S 2 t) < u 2 + c 2 t) = 1 Hu 1 + c 1 t, u 2 + c 2 t), where Hu 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 ) Ψ sum u 1, u 2 ) = P τ sum < /U 1 ) = u 1, U 2 ) = u 2 ) = P Ut) < ) = P St) > u + ct) = Gu + ct) where Gx) is the survival function of St) 13

14 71 Bounds for Ruin Probability Most of the papers in the literature of bivariate ris 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 11 al23), Cai and Li25) and Yuen et al26) derived bounds for the ultimate ruin probability Ψ min u 1, u 2 ) Simple bounds for Ψ max u 1, u 2 ) was given by Cai and Li25, 27) 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 Ψ 1 u 1 ) Ψ 2 u 2 ) Ψ max u 1, u 2 ) minψ 1 u 1 ), Ψ 2 u 2 )) maxψ 1 u 1 ), Ψ 2 u 2 )) Ψ min u 1, u 2 ) Ψ 1 u 1 ) + Ψ 2 u 2 ) Ψ 1 u 1 )Ψ 2 u 2 ), 21) 115 where the final expression in the second equation is exactly the ruin probability in the case where {U 1 t)} t and {U 2 t)} t are independent If there is no initial capitals u 1 = u 2 = ), then the above relations becomes Ψ 1 ) Ψ 2 ) Ψ max, ) minψ 1 ), Ψ 2 )) maxψ 1 ), Ψ 2 ) Ψ min, ) Ψ 1 ) + Ψ 2 ) Ψ 1 )Ψ 2 ) 22) In the case of univariate generalized power series Poisson ris model the ruin probabilities are given by Ψ i ) = µ ih i θ i) c i h i θ 1 ) i = 1, 2 23) Using the equations 22) and 23) we can obtain bounds for the ruin probabilities Ψ max, ) and Ψ min, ) for the type II bivariate generalized power series Poisson ris model and are given by max Moreover, we have µ1 h 1θ 1 ) c 1 h 1 θ 1 ), µ 2h 2θ 2 ) c 2 h 2 θ 2 ) µ1 h Ψ max, ) min 1θ 1 ) c 1 h 1 θ 1 ), µ 2h ) 2θ 2 ) c 2 h 2 θ 2 ) 2 µ 1 µ 2 h 1θ 1 )h 2θ 2 ) c 1 c 2 h 1 θ 1 )h 2 θ 2 ) ) Ψ min, ) µ 1h 1θ 1 ) c 1 h 1 θ 1 ) + µ 2h 2θ 2 ) c 2 h 2 θ 2 ) 2 µ 1 µ 2 h 1θ 1 )h 2θ 2 ) c 1 c 2 h 1 θ 1 )h 2 θ 2 ) In the case of no initial capital above relation is Ψ min u 1, u 2 ) = Ψ 1 u 1 ) + Ψ 2 u 2 ) Ψ max u 1, u 2 ) Ψ min, ) = Ψ 1 ) + Ψ 2 ) Ψ max, ) and hence,we obtain Ψ min, ) = µ 1h 1θ 1 ) c 1 h 1 θ 1 ) + µ 2h θ 2 ) c 2 h 2 θ 2 ) Ψ max, ) 14

15 8 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 Φu 1, u 2 ) = P U 1 t), U 2 t) ; for all t ) and infinite time joint ruin probability is Ψu 1, u 2 ) = 1 Φu 1, u 2 ) In a small time interval, 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 Φu 1, u 2 ) = 1 h 1 a ) ), + oh) Φu 1 + c 1 h, u 2 + c 2 h) ) a1 θ 1 h u1+c 1h + + oh) Φu 1 + c 1 h x, u 2 + c 2 h)df 1 x) ) a1 θ 2 h u2+c 2h + + oh) Φu 1 + c 1 h, u 2 + c 2 h y)df 2 y) + a i,j θ1θ i j 2 h u1+c 1h u2+c 2h + oh) Φu 1 + c 1 h x, u 2 + c 2 h y)df1 i x)df j 2 y), i, where F m i x), i = 1, 2, m = 1, 2, is the distribution function of X i1 + X i2 + + X im 12 Rearranging the terms leads to Φu 1 + c 1 h, u 2 + c 2 h) Φu 1, u 2 ) h = 1 a, + a 1θ 1 + a 1θ 2 u1+c 1h u2+c 2h u1+c 1h u2+c 2h As h tends to zero, we get c 1 Φu 1, u 2 ) + c 2 Φu 1, u 2 ) = 1 a, u 1 u 2 + a 1θ 1 + a 1θ 2 + i, ) Φu 1 + c 1 h, u 2 + c 2 h) Φu 1 + c 1 h x, u 2 + c 2 h)df 1 x) Φu 1 + c 1 h, u 2 + c 2 h y)df 2 y) + i, a i,j θ i 1θ j 2 Φu 1 + c 1 h x, u 2 + c 2 h y)df1 i x)df j y) + oh) u1 u2 a i,j θ i 1θ j 2 ) Φu 1, u 2 ) Φu 1 x, u 2 )df 1 x) Φu 1, u 2 y)df 2 y) u1 u2 Φu 1 x, u 2 y)df i 1 x)df j 2 y) 2 24) It is difficult to solve this two dimensional integro differential equation 15

16 81 Laplace Transforms of the Survival Probabilities Having obtained the partial integro differential equationspide) for the survival probabilities Φu 1, u 2 ) of the surplus process 17), in the following we will derive the Laplace transforms for the survival probabilities Firstly, we define the following Laplace transforms ˇΦs 1, u 2 ) = e s1u1 Φu 1, u 2 )du, fi s i ) = e six df i x), i = 1, 2 and ˇΦs 1, s 2 ) = e s1u1 s2u2 Φu 1, u 2 )du Taing Laplace transform on both sides of the PIDE 24) with respect to u 1, we get c 1 s 1 ˇΦs1, u 2 ) Φ, u 2 )) + c 2 ˇΦs1, u 2 ) = 1 a ), u 2 On simplification we get c 2 u2 ˇΦs 1, u 2 ) + a 1θ 1 f1 s 1 ) ˇΦs 1, u 2 ) + a 1θ 2 ˇΦs 1, u 2 y)df 2 y) a i,j θ i + 1θ j i 2 f 1 s 1 ) u2 ˇΦs 1, u 2 y)df j 2 y), i, ˇΦs1, u 2 ) = c 1 Φ, u 2 ) + 1 a ), c 1 s 1 + a 1θ 1 f1 s 1 ) )ˇΦs 1, u 2 ) u 2 + a u2 1θ 2 a i,j θ ˇΦs 1, u 2 y)df 2 y) + 1θ i j i 2 f 1 s 1 ) u2 25) ˇΦs 1, u 2 y)df j 2 y), i, Taing Laplace transform on both sides of the PIDE 25) with respect to u 2, we get Finally we get c 2 s 2 ˇΦs1, s 2 ) ˇΦs 1, )) = c 1 ˇΦ, s2 ) + 1 a ), c 1 s 1 + a 1θ 1 f1 s 1 ) )ˇΦs1, s 2 ) + a 1θ 2 f2 s 2 ) a i,j θ ˇΦs 1, s 2 ) + 1θ i j i j 2 f 1 s 1 ) f 2 s 2) ˇΦs 1, s 2 ) ˇΦs 1, s 2 ) = i, c 1 s 1 ˇΦ, s2 ) + c 2 s 2 ˇΦs1, ) ) c 1 s 1 + c 2 s 2 1 a, gθ 1,θ 2) a1θ1 f 1s 1) gθ 1,θ 2) a1θ2 f 2s 2) gθ 1,θ 2) i, a i,jθ1 i θj 2 f i j 1 s1) f 2 s2) gθ 1,θ 2) 9 Conclusion 125 In this paper we introduced the type II bivariate generalized power series Poisson distribution as a compound Poisson distribution with generalized power series compounding distribution We have considered the bivariate ris 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 ris model are investigated Also the bounds for ruin probabilities are developed We obtained PIDE for the survival probability 13 and derived an expression for bivariate Laplace transform of ruin probabilities 16

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18 Kostadinova K215):Bivariate Polya-Aeppli ris model, Tom 54, by the authors; licensee Preprints, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution CC-BY) license 18