The Asymptotic Tail Behavior of Discounted Total Claims for a Bivariate Risk Model with Constant Rate of Interest

2016 International Conference on Mathematical, Computational and Statistical Sciences and Engineering (MCSSE 2016) ISBN: 978-1-60595-396-0

The Asymptotic Tail Behavior of Discounted Total Claims for a Bivariate

Risk Model with Constant Rate of Interest

Ying-hua DONG

*

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China

*Corresponding author

Keywords: Asymptotic formula, Renewal model, Discounted total claims, Constant rate of interest, Subexponential distributions.

Abstract. In this paper,we consider a nonstandard bivariate risk model with constant rate of interest, in which the two kinds of claim sizes constitute a sequence of independent and identically distributed random vectors following a bivariate FGM distribution. When the two marginal distributions of the claim-size vector are subexponential, we present an asymptotic formula for the tail of discounted total claims.

Introduction

We study a nonstandard bivariate renewal risk model with constant force of interest, in which an insurer simultaneously operate two different lines of businesses. For any t 0, the two kinds of surplus processes of the insurer is represented by









 _



 t M t

i

t r i s

t r

rt _{e dC s X e} i

xe t U

0

) (

1

) ( 1

) (

1( ) ( )

 ,



 _



_  

 t N t

j

t r j s

t r

rt j

e Y s

dC e

ye t U

0

) (

1

) ( 2

) (

2( ) ( )

 ,

respectively, where the claim sizes {(X_i,Y_i),i1} form a sequence of independence, identically distributed random vector with generic vector (X,Y), whose marginal distributions are denoted by

F and G, respectively. (x,y) is the initial capital vector, and (C₁(t),C₂(t)) denotes the vector of total premium accumulated up to t with increasing components satisfying (C₁(0),C₂(0))(0,0).

} 0 ), (

{M t t  and {N(t),t 0} denotes the arrival processes with renewal functions ₁(t) and ₂(t), respectively. We assume that {(X,Y),(X_i,Y_i),i1}, {M(t),t0} and {N(t),t0} are mutually independent. We can prove that



 

 

1

1( ) ( ) i

i t

P

t 

 and







 

1

2( ) ( )

j

j t P

t 

 .

Throughout this paper, we suppose that (X,Y) follows a FGM distribution of the following form

), ( ) ( ) 1 ( ) ,

(X du Y dv F duG dv

P    



₍₁₎

Where (1,1] is a real number.

In the recent years, the one-dimensional renewal risk model has been widely investigated. The reader is referred to Klüppelberg and Stadtmüller [1] and Hao and Tang [2]. In addition, few articles considered a multi-risk model with constant force of interest. For example, Yang and Li [3] considered a bidimensional risk model with subexponential claims, in which the two kinds of businesses share a common claim-number process.

number processes, we study the asymptotic behavior of the tail probability of the discounted total claims.

Preliminaries and MainResults

This paper is concerned with heavy-tailed distributions, so we first introduce some related subclasses of heavy-tailed distributions. The reader can find them in Embrechts et al. [4] . Let X _{be a random} variable with distribution F _{and write} F(x)1F(x). _{We assume that} F(x)0 _{holds for all}

0



x . It is well known that the subexponential class is an important subset of the heavy-tailed distributions set.

A distribution F on [0,) _{is said to belong to the subexponential class} S, if the relation

2 ) (

) ( lim

* 2

 

 _{F x}

x F

x

holds, where F2* is 2-convolution of F _{with itself.}

A larger class than the class S _{is the class of long-tailed distribution, denoted by} L. By definition, L

F , if for any v0,

1 ) (

) (

lim  

  _{F x}

v x F

x .

A distribution F _{is said to belong to the dominated variation class} D, if for all 0v1,

. ) (

) (

lim 

  _{F x}

vx F x

As we know,

,

S L D 

and the inclusion is proper.

Hereafter, all limit relationships are for min(x,y) unless stated otherwise. For two positive functions a(x,y) _and b(x,y), we write a(x,y)~b(x,y), if

1 ) , (

) , ( lim

) ,

min( _{b x y}  y x a

y

x .

In this paper, we suppose that (X,Y) follows a bivariate Farlie-Gumbel-Morgenstern (FGM) distribution. Recall that a bivariate FGM distribution with marginal distributions F and G is given by

)) ( ) ( 1 )( ( ) ( ) ,

(x y F x G y F x G y

F   _{, (2)}

Where [1,1] is a constant. Clearly, if  0, then the equality reduces to a joint distribution function of two independent random variables.

Now we are in a position to state the main result.

Theorem 1 Consider the above renewal risk model. We suppose that {(X,Y),(X_i,Y_i),i1} is a sequence of independent, identically distributed random vectors following a common bivariate FGM distribution of the form (2) with [1,1]. Assume that the distributions of X and Ysatisfy FS and GS. Then for any fixed T 0 satisfying ₁(T)0 _and ₂(T)0,

(1)if [1,0], then

 

  

 







 



 



 NT T T _{ru rv}

j r j T

M

i

r

ie x Y e y F xe G ye d u v

X

P i j

0 0 1 2

) (

1 )

(

1

) ( ) ( ) ( ) ( ~

,   



) ( ) ( ) ( ) ( 1 0 0

v dP u dP ye G xe

F i i

rv ru

i T T

 

 

_  

(2) if



(0,1], M(T) 

E and EN(T)  for some  1, the the above relation still holds.

Some Lemmas

The following lemma is from Tang and Tsitsiashvili [5].

Lemma 1 Let X₁,,X_{n be} n real-valued independent random variables with common distribution F. If FS, then for any fixed n1,





 

 

  



 _ n

i

i i n

i i

iX x P c X x

c P

1 1

) (

~ .

The lemma below is due to Yang and Li [4].

Lemma 2 Suppose that {(X,Y),(X_i,Y_i),i1} is a sequence of independent, identically distributed and nonnegative random vectors following a bivariate FGM distribution of the form (2). If the distributions of X _and Ysatisfy FS and GS, then for any fixed n1 _and 0ab, it holds uniformly for all (c1,,cn) that







  



 

   

 



 n

i n

j

j j i

i n

j j j n

i i

iX x c Y y P c X x c Y y

c P

1 1 1

1

) ,

( ~

, .

Lemma 3 Let F and G be two distributions. If FL and GL, then there exists a positive slowly varying function l(x) at infinity satisfying 

  ( ) liml x

x and 0

) ( lim 

  _x

x l

x such that

1 )

( )) ( (

lim  



 _{F cx}

x dl cx F

x and _{( )} 1

)) ( (

lim  



 _{G cy}

y dl cy G

y

hold uniformly for all (c,d)[a,b]2, where 0ab.

The following lemma plays an important role in proving the main result.

Lemma 4 Suppose that {(X,Y),(X_i,Y_i),i1} is a sequence of independent, identically distributed and nonnegative random vectors following a bivariate FGM distribution of the form (2). If the distributions of X _and Y satisfy FS and GS , then for any fixed m1, n1 and

   a b

0 , it holds uniformly for all (c₁,,c_n) that







  



 

   

 



 m

i n

j

j j i

i n

j j j m

i i

iX x c Y y P c X x c Y y

c P

1 1 1

1

) ,

( ~

,

. (4)

Proof. Since FS, we can choose some positive slowly varying function l(x) at infinity satisfying l(x)0.5x, 

  ( ) liml x

x and 0

) ( lim 

  _x

x l

x such that

1 )

( )) ( (

lim  



 _{F cx}

x dl cx F

x and _{( )} 1

)) ( (

lim  



 _{G cy}

y dl cy G

y

hold uniformly for all 2

] , [ ) ,

(c d  a b , where 0ab.

With loss of generality, we suppose that mn. For this case, we split the probability on the left-hand side of (4) into three parts as follows:

   

 

 





 

y Y c x X c P

n

j j j m

i i i

1 1

,

   

 

 











 

 

n

j j j m

n i

i i m

i i

iX x cX l x cY y

c P

1 1

1

), (

, _

  

 

 











 



n

j j j n

i i i m

i i

iX x c X l x c Y y

c P

1 1

1

          









     n j j j m n i i i n i i i m i i

iX x c X l x c X l x c Y y

c P 1 1 1 1 ), ( ), ( , ). , ( ) , ( ) ,

( _{2 3}

1 x y J x y J x y

J  

 ₍₅₎

For J₁(x,y), it is clear that

      _          









    u X c dP y Y c u x X c P y x J m n i i i x l n j j j n i i i 1 ) ( 0 1 1

1( , ) , .

). , ( ) , ( ) , ( ), ( ), ( , ), ( , ), ( , , 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 y x J y x J y x J y Y c x l X c x l X c x X c P y Y c x l X c x X c P y Y c x l X c x X c P y Y c x X c P n j j j m n i i i n i i i m i i i n j j j n i i i m i i i n j j j m n i i i m i i i n j j j m i i i                                          

























              (5) ForJ₁(x,y), it is clear that

      _          









    u X c dP y Y c u x X c P y x J m n i i i x l n j j j n i i i 1 ) ( 0 1 1

1( , ) , .

Since FS and GS, by Lemma 2 and Lemma 3, we get



/



(1 ) ( / ) ( / ) ) / ( ~ ) , ( 1 1

1 i i

n i j i n j i c y G c x F c y G c x F y x

J





      



    n i n j j j i

iX x c Y y

c P 1 1 ) , ( ~

. (6) Following the proof of (6) with a slight modification, we obtain

) , ( 2 x y

J



  

  m n i n j j j i

iX x c Y y

c P 1 1 ) , ( ~

. (7) For J3(x,y), it follows that

      _           









     u X c dP y Y c x l u x X c P y x J m n i i i x l n j j j n i i i 1 ) ( 1 1

3( , ) ( ) ( ), .

Next we prove that J₃(x,y) is asymptotically negligible compared with the right-hand side of (4). By

Lemma 3 and Lemma 2,

( , )



( ) ( )



( )

1 1 ( )

3 x y Pc X x u l x dP c X u P c Y y

J j j

m n i i i i i n j i l x

       _    

 



     

(1 )



( ) ( )



( )

1 1 y Y c P u X c dP x l u x X c

P _{i i}

m n i i i i i n i        _     





    _.

From the above equality, we get for any , when x and y _{are sufficiently large,}

) , ( 3 x y

). ( ) ( ), ( , ) 1 )( 1 ( 1 1 1 1 y Y c P x l X c x l X c x X c

P _{j j}

n j m n i i i n i i i i m i

i  

     _{  }   









       (8)

According to Lemma 1, we have





       

 _ m

i i i m i i

iX x P c X x

c P 1 1 ) ( ~ .

Similarly as in (5) and (6), by Lemma 1, it follows that

      _{  }







    m n i i i n i i i i m i

iX x c X l x c X l x

c P 1 1 1 ) ( ), ( ,       _ 



 n i i

iX x

c P o

1

( .

By the above equality and (8), we have

        



  m i n j j j i

iX x c Y y

c P o y x J 1 1

3( , ) ( ,

. (9) Plugging (6), (7) and (9) into (5), we obtain (4).

The following lemma is a generalization of Lemma 3.3 in Yang and Li [4].

Lemma 5 Under the conditions of Lemma 4, for any  0, there are some constants K 0,

0 1 

b and b2 0 such that for all x0 and y0, m1 and n1,

) ( ) ( ) 1 ( ) 1 ( , 1 1 y G x F K y Y c x X c

P m n m n

n j j j m i i i               





  .

Proof of main results

For any fixed positive integer M , it holds for any fixed T that

_       





    ( ) 1 ) ( 1 , T N j r j T M i r

ie x Y e y

X

P i j 

     _{  } 

      

                M m N n M

m n N m M

N

n m M n M

1 1 1 1 1 1 1 1

         





   n T N m T M y e Y x e X P n j r j m i r i j

i _{, , ( ) , ( )}

1 1   ) ; , ( ) ; , ( ) ; , ( ) ; ,

( _{2 3 4}

1 x y T I x y T I x y T I x y T

I   

 _{. (10)}

We first consider I₁(x,y;T). For m1 _and n1, by Lemma 4, we have



      _{   } M m N n m i n j r j r

ie x Y e y M T m N T n

X P T

y x

I i j

1 1 1 1

1( , ; )~ ( , , ( ) , ( ) )

 

Following the corresponding proof in Lemma 3.4 of Yang and Li (2014), we have



 

( ) ( ) ( ) ( ) ~ ) ; ,

( _{1 2}

0 0

1 x y T F xe G ye d u d v

I T T ru rv  

) ( ) ( ) ( ) ( 1 0 0

v dP u dP ye G xe

F i i

rv ru i T T  

 

_   

. (11) Next we deal with I₂(x,y;T). We can see that

 





                   M

m n M

n j r j m i r

ie x Y e y M T m N T n

X P T

y x

I i j

1 1 1 1

2( , ; ) , , ( ) , ( )

  ) 1 ) ( ( ) 1 ) ( ( ,

1 1 0 0 1 1

              

 

 





        n v T N P m u T M P y e Y x e X P M

m n M

n j rv j m i ru i T T

dP(1 u)dP(1 v)_.

( , ; ) (1 ) (1 ) 1 ( ) ( ) ₁( ) ₂( )

0 0 ) ) ( ( ) ( )

(

2 x y T KE E F xe G ye d u d v

I   M T  NT _{N TM}

 

T T ru rv   .

Since {N(t),t0} is a renewal process, by the conditions of the theorem, we get

0 ) ( ) ( ) ( ) (

) ; , ( lim

0 0 1 2

2 

 



 T T _{ru rv} M

v d u d ye G xe F

T y x I

 

. (12) In the same way,

0 ) ( ) ( ) ( ) (

) , , ( ) ; , ( lim

0 0 1 2

4

3  

 



 T T _{ru rv} M

v d u d ye G xe F

T y x I T y x I

 

. (13) Putting (11), (12) and (13) into (10), we obtain the conclusion. This ends the proof of Theorem 1.

Acknowledgement

This research was suppoted by Project Funded by China Postdoctoral (No.2016M591885), Jiangsu Planned Projects for Postdoctoral Research Funds (No. 1501053A), the Statistical Research Program of National Bureau of Statistics of China (No. 2015LY83) and the foundation of Nanjing University of Information Science and Technology (No. 2014x026).

References

[1] C. Klüppelberg and U. Stadtmüller, Ruin probability in the presence of heavy-tails and interest rates, Scand. Actuar. J. 1 (1998), 49-58.

[2] X. Hao and Q. Tang, A uniform asymptotic estimate for discounted aggregate claims with subexponential tails, Insurance: Math. Econom. 43 (2008), 116-120.

[3] P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin, 1997.

[4] H. Yang and J. Li, Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims, Insurance: Math. Econom. 58 (2014), 185-192.

[5] Q. Tang and G. Tsitsiashvili, Randomly weighted sums of subexponential random variables with