A revisit to ruin probabilities in the presence of heavy-tailed insurance and financial risks

Insurance: Mathematics and Economics 73 (2017) 75–81

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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime

A revisit to ruin probabilities in the presence of heavy-tailed insurance and financial risks Yiqing Chen a , Zhongyi Yuan b,∗ a

College of Business and Public Administration, Drake University, 345 Aliber Hall, 2507 University Avenue, Des Moines, IA 50311, USA

b

Department of Risk Management, The Pennsylvania State University, 362 Business Building, University Park, PA 16802, USA

article

abstract

info

Article history: Received May 2016 Received in revised form November 2016 Accepted 9 January 2017 Available online 4 February 2017

Recently, Sun and Wei (2014) studied the finite-time ruin probability under a discrete-time insurance risk model, in which the one-period insurance and financial risks are assumed to be independent and identically distributed copies of a random pair (X , Y ). For the heavy-tailed case, under a restriction on the dependence structure of (X , Y ), they established an asymptotic formula for the finite-time ruin probability. In this paper we make an effort to remove this restriction as it excludes the cases with asymptotically dependent X and Y . We also extend the study to the infinite-time ruin probability. Employing a multivariate regular variation framework, we simplify the formula so that it shows in a transparent way how the ruin probabilities are affected by the tail dependence of (X , Y ). © 2017 Elsevier B.V. All rights reserved.

MSC: primary 91B30 secondary 62P05 62E20 62H20 Keywords: Ruin probability Heavy-tailed distributions Insurance and financial risks Asymptotics Tail dependence Regular variation

1. Introduction

where W0 = x. Iterating (1.1) yields

In this paper we consider an insolvency problem for an insurer who makes risky investments. We model the insurance business and the investments in a discrete-time model. Suppose that at time 0 the insurer holds an initial capital of amount x. At the beginning of each period k, the insurer invests its current wealth into a portfolio of assets that provides an overall return rate of Rk ∈ (−1, ∞). During each period k, the insurer’s realized net profit from the insurance business is denoted by a random variable Zk ∈ R, which is roughly equal to premiums collected minus insurance claims paid and other expenses incurred. Denote by Wk the wealth of the insurer at k, k ∈ N. The insurance business and investments together lead to an evolution of the wealth process as follows: Wk = (1 + Rk )Wk−1 + Zk ,

k ∈ N,

∗

(1.1)

Corresponding author. E-mail addresses: [email protected] (Y. Chen), [email protected] (Z. Yuan). http://dx.doi.org/10.1016/j.insmatheco.2017.01.005 0167-6687/© 2017 Elsevier B.V. All rights reserved.

Wk = x

k 

(1 + Rj ) +

j =1

k  i=1

k 

Zi

(1 + Rj ),

j=i+1

where, and throughout this paper, multiplication over an empty index set produces value 1 by convention. We consider the probabilities of ruin in both finite time and infinite time, which are given by

ψ(x; n) = P



 



min Wk < 0 W0 = x , 0≤k≤n

n ∈ N ∪ {∞},

where for n = ∞ we understand 0 ≤ k ≤ n as 0 ≤ k < ∞. The discrete-time risk model and the ruin probabilities described above were first introduced by Tang and Tsitsiashvili (2003, 2004), and were thereafter studied extensively in insurance mathematics and applied probability; see, e.g., Chen and Ng (2007), Chen (2011), Fougères and Mercadier (2012), Yang and Wang (2013), and Li and Tang (2015). Continuous-time versions of the study have also appeared in the literature; see Norberg (1999), Klüppelberg and Kostadinova (2008), Paulsen (2008), Heyde and Wang (2009), Bankovsky et al. (2011) and Hult and Lindskog (2011), among

76

Y. Chen, Z. Yuan / Insurance: Mathematics and Economics 73 (2017) 75–81

others. Generally, the importance of ruin theory lies in its guidance to insurers and regulators for risk capital calculation, as well as its application to the pricing of related insurance products. For example, to comply with regulation frameworks such as EU Solvency II, insurers have to hold enough risk capital so that the probability of ruin is sufficiently low. In addition, some insurancelinked securities such as contingent convertibles may use insurers’ probability of ruin as a trigger. Both cases will require an assessment of the probability of ruin, making the notion of ruin theory relevant. For ease of presentation, let us introduce

and g (·) are weakly equivalent, that is, 0 < lim inf f (x)/g (x) ≤ lim sup f (x)/g (x) < ∞. For two real numbers x and y, write x ∨ y = max{x, y}. For a nondecreasing function h, denote by h← (y) = inf {x ∈ R : h(x) ≥ y} its generalized inverse function. For a distribution function H and every 0 ≤ y ≤ 1, by relations (0.6b) and (0.6c) of Resnick (1987) we have H (H ← (y)) ≥ y,

For a random variable ξ distributed by H, write bξ (x) = 1/H



Xi = −Zi

and Yi =

1 1 + Ri

,

i ∈ N,

where Xi ∈ R quantifies the insurance risk and Yi ∈ [0, ∞) quantifies the financial risk. Then, for each n ∈ N ∪ {∞}, we can rewrite the probability of ruin as

 ψ(x; n) = P  =P

 min

0≤k≤n k

max 1≤k≤n



x

k 

(1 + Rj ) +

j =1

k  i =1

Zi

k 





(1 + Rj ) < 0

j =i +1

 Xi Πi > x ,

i =1

i

where Πi = j=1 Yj denotes the stochastic discount factor over the first i periods, i ∈ N. A recent trend of related studies is to describe the impact of the dependence between the insurance and financial risks on the asymptotic behavior of the ruin probability. In particular, Chen (2011) studied the finite-time ruin probability, assuming that the insurance and financial risks (Xi , Yi ), i ∈ N, are described by independent and identically distributed (i.i.d.) random pairs with common bivariate Farlie–Gumbel–Morgenstern distribution. Fougères and Mercadier (2012) carried on their study under a general multivariate regular variation (MRV) framework for the insurance and financial risks. Tang and Yuan (2012) introduced an algorithm for the computation of ruin probability, under autoregressive models for both one-period claim amounts and oneperiod log-return rates. In particular, Sun and Wei (2014) moved the study forward by considering a general class of heavy-tailed distributions for Xi and Yi , i ∈ N, and a weak within-period dependence structure between the two kinds of risks described by (3.1). However, as shown by Lemma 3.1, their assumption of dependence is quite restrictive and rules out the cases where Xi and Yi are asymptotically dependent. This paper is a significant continuation to this line of study, where we remove the assumption on the dependence structure needed by Sun and Wei (2014) for their result. The price we pay for this extension is a slightly more restrictive assumption on the distribution of Xi Yi . In addition, we also extend the study to the infinite-time horizon. The rest of the paper is organized as follows. In Section 2 we introduce some preliminaries needed for our study, in Section 3 we present our main results, and in Section 4 we provide some refinements of the results, which also show the impact of the tail dependence between the insurance and financial risks on the ruin probabilities. 2. Preliminaries

←

(x) = H ← (1 − 1/x).

(2.2)

Next, we recall some concepts of heavy-tailed distributions. A distribution function H on R is said to be long tailed, written as H ∈ L, if it has an ultimate right tail (that is, H (x) > 0 for all x ∈ R) and it holds that lim

H (x + y) H (x)

x→∞

= 1 for all y ∈ R.

This implies that there is some positive function a(·), with a(x) ↑ ∞ and a(x) = o(x), such that H (x + a(x)) ∼ H (x). A distribution function H on R is said to be dominatedly varying tailed, written as H ∈ D , if it has an ultimate right tail H satisfying lim sup x→∞

H (xy) H (x)

< ∞ for all 0 < y < 1.

The intersection L ∩ D is a class of heavy-tailed distributions with wide application; see, e.g., Geluk and Tang (2009) and Sun and Wei (2014) for related discussions. An important subclass of L ∩ D is the class C , which is large enough to contain almost all practically useful distribution functions in L ∩ D . A distribution function H on R is said to be in the class C , written as H ∈ C , if it has a consistently varying tail; that is, H (xy)

lim lim sup y↑1

H (x)

x→∞

lim lim inf y↓1

H (xy)

x→∞

H (x)

= 1,

or, equivalently,

= 1.

(2.3)

Clearly, the class C contains the famous class R as a proper subset. A distribution function H on R is said to be in the class R, written as H ∈ R, if it has a regularly varying tail; that is, if for some γ > 0, lim

x→∞

H (xy) H (x)

for all y > 0.

= y−γ

Furthermore, if the above holds, then we say that the tail H is regularly varying with index −γ , denoted as H ∈ RV−γ . We refer the reader to Bingham et al. (1987), Resnick (1987), and Embrechts et al. (1997) for comprehensive treatments of regular variation and heavy-tailed distributions. Finally, we introduce the lower and upper Matuszewska indices of distribution functions, which are very useful in describing their tail behavior. For a distribution function H on R with an ultimate right tail, its lower and upper Matuszewska indices are defined by ∗

 Let us first introduce some notational conventions. Throughout this paper, all limit relationships are according to x → ∞ unless otherwise stated. For two positive functions f (·) and g (·), we write f (x) . g (x) or g (x) & f (x) if lim sup f (x)/g (x) ≤ 1, write f (x) ∼ g (x) if lim f (x)/g (x) = 1, and write f (x) ≍ g (x) if f (·)

and H ← (y) ≤ x if and only if y ≤ H (x). (2.1)

M∗ = sup −

 ∗

M = inf −

log H (y) log y

log H ∗ (y) log y

 :y>1  :y>1 ,

and

Y. Chen, Z. Yuan / Insurance: Mathematics and Economics 73 (2017) 75–81

respectively, where ∗

H (y) = lim sup x→∞

H (xy) H ( x)

and

H ∗ (y) = lim inf x→∞

H (xy) H (x)

Lemma 3.1. For two non-degenerate non-negative random variables ξ and η that are asymptotically dependent, it holds that

.

H (x)

≤ c1 y−γ1

lim inf

(2.4)

holds for all xy ≥ x ≥ d1 ; if 0 ≤ M ∗ < ∞, then for every γ2 > M ∗ , there are some positive constants c2 and d2 such that the inequality H (x) H (xy)

≤ c2 yγ2

(2.5)

holds for all xy ≥ x ≥ d2 . By relation (2.5), it is easy to see that

  ∗ x−(M +ε) = o H (x)

(2.6)

holds for every ε > 0. 3. Main results From now on, we always assume that the risks across individual periods constitute a sequence of i.i.d. random pairs (Xi , Yi ), i ∈ N, with generic pair (X , Y ), and denote the distribution functions of random variables X , Y , and XY by F , G, and H, respectively.

Theorem 3.1. Assume that H ∈ L ∩ D with upper Matuszewska ∗ index M ∗ , that E [Y M +ε ] < ∞ for some ε > 0, and that lim lim sup

P (XY > x)

x→∞

= 0.

(3.1)

n 



> 0.

Proof. Note that, for every x > 0, P ξ η > bξ (2x)bη (2x) ≤ P ξ > bξ (2x) or η > bη (2x) ≤









1 x

,

where in the second step we used the first fact of relation (2.1). Moreover, using the second fact of (2.1) we see that bξ (2x)bη (2x) ≥ bξ η (x). Therefore, we have P ξ η > bξ η (x), η > bη (x)



lim inf



  x→∞ P ξ η > bξ η (x)   P ξ η > bξ (2x)bη (2x), η > bη (2x)   ≥ lim inf x→∞ P ξ η > bξ η (x)   P ξ > bξ (2x), η > bη (2x)   ≥ lim inf x→∞ P ξ η > bξ η ( x) λ (ξ , η)   = lim inf x→∞ 2xP ξ η > b (x) ξη > 0,   where in the last step we used the fact that P ξ η > bξ η (x) ≤ 1/x implied by relation (2.1). This completes the proof.



Apparently, Lemma 3.1 implies that asymptotic dependence between the insurance and financial risks in Theorem 3.1 would violate its condition (3.1). This greatly limits the usefulness of Theorem 3.1. In this paper, we show that the restriction (3.1) on the dependence structure of (X , Y ) can be removed, if we restrict the distribution of XY to the slightly smaller class C . 3.2. Ruin probability in finite time

P (Xi Πi > x) .

Theorem 3.2. Assume that H ∈ C with upper Matuszewska index ∗ M ∗ and that E [Y M +ε ] < ∞ for some ε > 0. Then relation (3.2) holds for each n ∈ N. We shall need the following lemma for the proof of Theorem 3.2: Lemma 3.2. Under the assumptions of Theorem 3.2, the products Xi Πi , i ∈ N, are pairwise asymptotically independent.

Then it holds for each n ∈ N that

ψ(x; n) ∼



Below is our main result on the finite-time ruin probability.

We start with restating the main result of Sun and Wei (2014).

A→∞



P ξ η > bξ η (x)

x→∞

3.1. On Sun and Wei (2014)

P (XY > x, Y > A)

P ξ η > bξ η (x), η > bη (x)



By Corollary 2.1.6 and Theorem 2.1.5 of Bingham et al. (1987), M∗ and M ∗ are the Matuszewska indices of 1/H, and we follow Tang and Tsitsiashvili (2003) to call them Matuszewska indices of H when no confusion could arise. It is known that 0 ≤ M∗ ≤ M ∗ ≤ ∞, and that two distribution functions with weakly equivalent tails have the same Matuszewska indices. In addition, one may verify that H ∈ D if and only if M ∗ < ∞, and H ∈ RV−γ if and only if M∗ = M ∗ = γ . The following two inequalities will be useful for what follows, and can be obtained by some obvious manipulations on Proposition 2.2.1 of Bingham et al. (1987). If 0 < M∗ ≤ ∞, then for every 0 < γ1 < M∗ , there are some positive constants c1 and d1 such that the inequality H (xy)

77

(3.2)

Proof. For arbitrarily chosen 1 ≤ i1 < i2 < ∞, by Lemma 7 of Tang and Yuan (2014), it holds that

i=1

P Xi1 Πi1 > x, Xi2 Πi2 > x



This result is consistent with and regarded as a significant generalization of Theorem 5.1 of Tang and Tsitsiashvili (2003). On the other hand, the result relies on the quite restrictive condition (3.1), which will generally be violated when X and Y are asymptotically dependent, as shown by Lemma 3.1. Recall the quantile function defined in relation (2.2). Also recall that two random variables ξ and η are said to be asymptotically dependent if the coefficient of upper tail dependence, defined by

  λ (ξ , η) = lim xP ξ > bξ (x), η > bη (x) , x→∞

is positive, and asymptotically independent if it is 0.



  = P Xi1 Πi1 > x, Xi2 Yi2 Πi2 −1 > x = o(1)H (x). Note that by Theorem 3.3(iv) of Cline and Samorodnitsky (1994), both Xi1 Πi1 and Xi2 Πi2 have a tail weakly equivalent to H (x). The claimed asymptotic independence between Xi1 Πi1 and Xi2 Πi2 follows straightforwardly.  For what follows, we denote by X + the positive part of a random variable X ; that is, X + = X ∨ 0. Proof of Theorem 3.2. Note that by Lemma 4.1 of Chen and Yuen (2009), the tail of Xi Πi is consistently varying for every i =

78

Y. Chen, Z. Yuan / Insurance: Mathematics and Economics 73 (2017) 75–81



1, . . . , n. For arbitrarily fixed 0 < ε < 1, we have

Xk Πk > x + a(x)

ψ(x; n)   n  + ≤P Xi Πi > x

≥ P (Xk Πk > x + a(x))   l  − P max Xi Πi > x, Xk Πk > x + a(x)

i=1

≤P

 n 

1≤l≤k−1

 Xi Πi > (1 − ε)x +



i=1

+P

Xi+ Πi > x,

i=1

≤

n 

n 

Xj+ Πj ≤ (1 − ε)x,

j =1

n  k=1

Xk+ Πk >

x



+

P

x

Xk Πk > +

≤

Xi Π i > ε x 3.3. Ruin probability in infinite time

P (Xi Πi > (1 − ε)x)

In the following result, we equip Theorem 3.2 with uniformity with respect to n ∈ N and extend the result to the infinite-time horizon:

i =1

+

n n  

 P

Xk Πk >

k=1 i=1,i̸=k

∼

n 

x n

, Xi Πi >

εx



n−1 Theorem 3.3. Assume that H ∈ C with lower and upper Matuszewska indices 0 < M∗ ≤ M ∗ < ∞, and that E [Y γ1 ∨ Y γ2 ] < 1 for some 0 < γ1 < M∗ and γ2 > M ∗ . Then relation (3.2) holds uniformly for all n ∈ N, and hence also for n = ∞.

P (Xi Πi > (1 − ε)x) ,

i=1

where in the last step we applied Lemma 3.2 and the dominated variation of the tails of X1 Π1 , . . . , Xn Πn to get rid of the second sum. It follows from the consistent variation of the tails of X1 Π1 , . . . , Xn Πn and the first relation in (2.3) that lim sup x→∞

ψ(x; n) n 

P (Xi Πi > x)

≤ lim lim sup ε↓0

n 

= 1. P (Xi Πi > x)

i =1

To derive the corresponding lower bound, introduce a stopping time

 τ (x) = inf k ∈ N :

k 

 Xi Π i > x

i=1

n 

Lemma 3.3. Suppose that the distribution function of a real-valued random variable ξ has lower and upper Matuszewska indices satisfying 0 < M∗ ≤ M ∗ < ∞. Then for every γ1 and γ2 with 0 < γ1 < M∗ ≤ M ∗ < γ2 < ∞, there exist some c > 0 and x0 = x0 (γ1 , γ2 ) > 0 such that, for all x > x0 and all nonnegative random variables η independent of ξ , P (ξ η > x)

and write

ψ(x; n) =

        ψ( x ; n)  lim sup n − 1 = 0. x→∞ n∈N    P (X Π > x)  i i   To prove Theorem 3.3, we need the following lemma, which can be obtained by following the proof of Lemma 3.2 of Hao and Tang (2012) using inequalities (2.4) and (2.5):

P (Xi Πi > (1 − ε)x)

i =1

x→∞

The uniformity in Theorem 3.3 is understood as

i =1

i=1 n 

∼ P (Xk Πk > x) ,

+

n i=1,i̸=k

k=1 n 

,



n 

P (ξ > x) P (τ (x) = k) .

(3.3)

k=1

By relation (2.6) and Theorem 2.2(iii) and 3.3(iv) of Cline and Samorodnitsky (1994), it is easy to verify that, for each k ∈ N, the distribution of Xk Πk is long tailed. Thus, we can find some positive function a(·), with a(x) ↑ ∞ and a(x) = o(x), such that

∞ 

lim lim sup

N →∞

Then

max 1≤l≤k−1

 ≥P

max 1≤l≤k−1

P Xi+ Πi > x



i =N +1

H (x)

x→∞

 =0

(3.4)

and that

P (τ (x) = k)

=P

≤ cE [ηγ1 ∨ ηγ2 ] .

Proof of Theorem 3.3. We largely follow the proof of Theorem 3.1 of Tang and Tsitsiashvili (2004) but also overcome several essential technical difficulties. First, let us prove that

P (Xk Πk > x + a(x)) ∼ P (Xk Πk > x) .



Xi Πi < −a(x), Xk Πk > x + a(x)

where in the last step we applied Lemma 7 of Tang and Yuan (2014) to get rid of the last two probabilities. Substituting this into (3.3) yields the lower bound. This completes the proof of Theorem 3.2. 

i =1





i=1

n

P (Xi Πi > (1 − ε)x)

n 



−P

 n 

i =1

k−1

l

k





X i Π i ≤ x,

i=1 l  i =1

 Xi Π i > x

i=1

Xi Πi ≤ x,

k−1  i =1

 P

∞ 

 Xi Πi > x +

= O(H (x));

(3.5)

i =1

Xi Πi ≥ −a(x),

∞

note that the almost sure convergence of i=1 Xi+ Πi can be easily verified by using relation (2.6) and Lemma 1.7 of Vervaat (1979).

Y. Chen, Z. Yuan / Insurance: Mathematics and Economics 73 (2017) 75–81

By Lemma 3.3, it holds for some c > 0, all large x, and all i ∈ N that P Xi Πi > x



+



 γ γ  ≤ cE Πi−11 ∨ Πi−21

H (x)

It follows that, for all n > N,

ψ(x; n) .

N  

P Xi+ Πi > x + ε H (x)



i =1

  ≤ c (E [Y γ1 ])i−1 + (E [Y γ2 ])i−1 . γ1

(3.6)

γ2

γ1

E [Y γ1 ]  E [Y γ2 ] qγ1

< 1.

qγ2

≤ (1 + ε)

γ2

Since both E [Y ] and E [Y ] are not greater than E [Y ∨ Y ] < 1, summing up (3.6) over i = N + 1, N + 2, . . . yields (3.4). To prove (3.5), choose some 0 < q < 1 close to 1 such that

n  

P Xi+ Πi > x .

P

∞ 

On the other hand, by Theorem 3.2 and relation (3.4), it holds for all n > N that

 ψ(x; n) ≥ ψ(x; N ) ∼

n 

Xi+ Πi > x

 =P

∞ 

i =1

Xi+ Πi > (1 − q)x

i =1

≤

∞ 



P Xi+ Πi > (1 − q)xqi−1

=



∞ 

+

P

Xi Yi

i=1

i−1  Yj

q

j =1

i −1  Yj

q

j =1

 ≤c

(1 − ε) 

n  

> (1 − q)x i−1

 +

qγ1

P Xi+ Πi > x .





. (1 + ε)

n  

P Xi+ Πi > x .



i =1

By Theorem 3.2, this two-sided asymptotic inequality holds uniformly for n = 1, . . . , N. Finally, since ε can be made arbitrarily small, the desired uniformity follows. 

E [Y γ2 ]

i−1 

qγ2

H ((1 − q)x).

(3.8)

and P

∞ 

We first introduce an important corollary of Theorem 3.3 for the case that H has a regularly varying tail. This will lead to a more explicit right-hand side of relation (3.2). Corollary 4.1. Suppose that H ∈ RV−γ for some γ > 0, and that   E [Y γ ] < 1 and E Y γ +ε < ∞ for some ε > 0. Then we have

ψ(x; n) ∼ H (x)

 Xi+ Πi > t

≤ P (ξ > t ) . (3.9)

ψ(x; ∞) ∼

For arbitrarily fixed large N ∈ N and all n > N, we have

 ψ(x; n) ≤ P

N 



i =1

 ≤P

n 

Xi+ Πi +

Xi+

i =N +1

i 

 Yj



ΠN > x

j =N +1

Xi Πi + ξ ΠN > x

∼

i =1



i =1

For arbitrarily fixed ε > 0, a similar relation to (3.6) shows that for N and x large and some c > 0, we have P (ξ ΠN > x)

∼l

Thus, there exists some δ > 0 such that E Y γ −δ ∨ Y γ +δ < 1. By Theorem 3.3, relation (3.2) holds uniformly for all n ∈ N. Furthermore, by Breiman’s theorem (see Breiman, 1965 and Cline and Samorodnitsky, 1994), it holds for each i ∈ N that



P Xi Πi > x + P (ξ ΠN > x) . +



s→0

Since the first relation in (3.9) implies the distribution function of ξ also belongs to L ∩ D , a similar idea to the proof of Lemma 3.2 can be used to show the pairwise asymptotic independence among ξ ΠN and Xi+ Πi , i = 1, . . . , N. Hence, following the proof of Theorem 3.2 we see that N  

.

Proof. By the monotone convergence theorem, it holds that



Xi+ Πi + ξ ΠN > x .



H (x) 1 − E [Y γ ]

lim E Y γ −s ∨ Y γ +s = E [Y γ ] < 1.

i =1

+

(4.1)

1 − E [Y γ ]





N

1 − (E [Y γ ])n

holds uniformly for all n ∈ N. In particular,

i=1

H (x)

n  

4. Further refinements

E [Y γ1 ]

P (ξ > x) ∼ lH (x)

P



P Xi+ Πi > x . ψ(x; n)

(3.7)





N 



i=1

> (1 − q)x .

Substituting (3.8) into (3.7) and noticing H ((1 − q)x) ≍ H (x), we obtain (3.5). Next, we prove the uniformity of relation (3.2) over n ∈ N. Note that by relation (3.5), there exists a nonnegative random variable ξ , independent of {Xi , Yi }i∈N , such that for some l > 0 and every t ∈ R,



P Xi+ Πi > x

To summarize, it holds uniformly for all n > N that



Similarly to (3.6), applying Lemma 3.3 again, it holds for some c > 0, all large x, and all i ∈ N that Xi+ Yi



i =1

i=1

∞  

n 

i=N +1

& (1 − ε)

qi−1

i=1

P

−

i=1







i=1

We derive



79

P (ξ ΠN > x) P (ξ > x)

  ≤ cl (E [Y γ1 ])N + (E [Y γ2 ])N ≤ ε.

P (Xi Πi > x) ∼ H (x) (E [Y γ ])

i −1



.

(4.2)

Substituting this into the right-hand side of (3.2) yields (4.1) for each n ∈ N. The uniformity of relation (4.1) can be verified as follows. The proof of Theorem 3.3 indicates that, for every 0 < ε < 1, one can find some large N ∈ N such that, for all n ≥ N,

(1 − ε)

N  

P Xi+ Πi > x . ψ(x; n)



i=1

. (1 + ε)

N  

P Xi+ Πi > x .

i=1



80

Y. Chen, Z. Yuan / Insurance: Mathematics and Economics 73 (2017) 75–81

(b) If 0 < yˆ < ∞, then

This together with relation (4.2) yields that

(1 − ε)H (x)

γ

1 − (E [Y ])

H (x) ∼ ν(B)F (x),

N

1 − E [Y γ ]

. ψ(x; n) . (1 + ε)H (x)

γ

1 − (E [Y ])

where B = {(t1 , t2 ) > 0 : t1 > yˆ −α }.

N

1 − E [Y γ ]

.

Thus, for some large N ∈ N and uniformly for all n ≥ N, it holds that

(1 − ε)2 H (x)

1 − (E [Y γ ])n 1 − E [Y γ ]

. ψ(x; n) . (1 + ε)H (x)

H (b(x)) = P 1 − (E [Y γ ])n 1 − E [Y γ ]

.

Due to Theorem 3.2 and relation (4.2), the last two-sided asymptotic inequality automatically holds uniformly for n = 1, . . . , N. Finally, by the arbitrariness of ε we conclude the proof.  The result for ψ(x; ∞) in Corollary 4.1 can also be obtained by simply applying Theorem 1 of Grey (1994). Under a mild condition on the dependence between X and Y , the corollary can be further refined to show the impact of tail dependence between X and Y on the ruin probabilities. We describe the dependence via an MRV structure, which we introduce below. Write E = [0, ∞]2 \ {0}, and denote by M+ (E) the space of non-negative Radon measures on E (i.e., measures that take finite values on compact subsets of E). We assume that the random  vector 1/F (X ), 1/G(Y ) satisfies

  xP

1

1 x

,

1

F (X ) G(Y )



Proof. (a) Since for every x ∈ R, the values of bX (1/F (x)) and x differ only on flat segments of F , which are assigned no probability mass, we have bX (1/F (X )) = X and bY (1/G(Y )) = Y almost surely (see also Proposition A.4 of McNeil et al. (2015)). It follows that

 v ∈ · → ν(·) in M+ (E),

(4.3)

ν(cB) = c

−1

ν(B),

(4.4)

where cB = {ct : t ∈ B}. Moreover, the measure ν completely characterizes the dependence of (X , Y ) in its tail area. In particular, a positive value of ν ((1, ∞]) implies that X and Y are asymptotically dependent, and hence by Lemma 3.1, relation (3.1) is violated. For more discussions on vague convergence and MRV, we refer the reader to Resnick (2007). Under condition (4.3), the transformed random vector (1/F (X ), 1/G(Y )) is said to have a Pareto copula, a concept introduced by Klüppelberg and Resnick (2008). Owing to their nonparametric nature, Pareto copulas are general enough to offer a wide range of dependence structures. Below we introduce Lemma 4.1, which characterizes the tail behavior of XY , and will be used to produce Corollary 4.2. Denote by yˆ the upper endpoint of G and write b(x) = bX (x)bY (x).

(4.5)

We consider both cases with yˆ = ∞ and 0 < yˆ < ∞. The latter case appears if a fixed positive proportion of the insurer’s assets is invested in a risk-free asset that guarantees a positive gross return rate. In the rest of this section, for 0 < α < ∞ and β = ∞ we understand 1/β = 0 and αβ/(α + β) = α as usual. Lemma 4.1. Suppose that F ∈ RV−α for some 0 < α < ∞ and that relation (4.3) holds with ν ((1, ∞]) > 0. (a) If G ∈ RV−β , 0 < β ≤ ∞, then H (x) ∼ ν(A) (b← (x))−1 , where A = {(t1 , t2 ) > 0 :

(4.6) 1/α 1/β t1 t2

> 1}.

1

1

1

,



F (X ) G(Y )

x



∈ Ax ,

where Ax = {(t1 , t2 ) > 0 : bX (xt1 ) bY (xt2 ) > b(x)}. By Proposition 0.8(v) of Resnick (1987), we have bX (·) ∈ RV1/α and bY (·) ∈ RV1/β , and hence, the set Ax can be written as



1/α 1/β

(t1 , t2 ) > 0 : (1 + o(1))t1 t2 bX (x)bY (x) > b(x)   1/α 1/β = (t1 , t2 ) > 0 : t1 t2 > 1 + o(1) .



This implies that for every small ε > 0 and large x,

(1 + ε)A ⊂ Ax ⊂ (1 − ε)A. It is easy to verify that all of the three sets (1 + ε)A, A, and (1 − ε)A are bounded away from 0, and by Lemma 7.1 of Shi et al. (2017), that all have a boundary with 0 mass assigned by ν . Hence, by relations (4.3) and (4.4), we have xH (b(x)) ≤ xP

v

with ν ((1, ∞]) > 0, where→ denotes vague convergence. This  means that 1/F (X ), 1/G(Y ) has a multivariate regularly varying tail with index −1. It is known that the limit measure ν is homogeneous, in that for every Borel subset B and every c > 0, we have

 

 

1

1 x

1

,

F (X ) G(Y )





∈ (1 − ε)A

→ (1 − ε)−1 ν(A). Proving the other asymptotic inequality in a similar way and noticing the arbitrariness of ε , we have H (b(x)) ∼ ν(A)x−1 .

(4.7)

Since b← (b(x)) ≤ x ≤ b← (b(x) + ε) holds for every x > 1 and ε > 0, it follows that x ∼ b← (b(x)). This together with relation (4.7) implies relation (4.6), and thus completes the proof of part (a). (b) Note that H (bX (x)) = P

  1 x

1

,

1

F (X ) G(Y )





∈ Bx ,

where Bx = {(t1 , t2 ) > 0 : bX (xt1 ) bY (xt2 ) > bX (x)}. Since bX (·) ∈ RV1/α and bY (·) ↑ yˆ , we can write the set Bx as



 1/α (t1 , t2 ) > 0 : (1 + o(1))t1 bX (x)ˆy > bX (x)   = (t1 , t2 ) > 0 : t1 > yˆ −α (1 + o(1)) .

The proof can then be completed in a similar fashion to that of part (a).  A special case of Lemma 4.1(a) with 0 < α = β < ∞ was discussed by Tang and Yuan (2015). It is easy to see that H is regularly varying with index −αβ/(α + β) in Lemma 4.1(a), and with index −α in Lemma 4.1(b). Obviously, in both cases the moment conditions required by Corollary 4.1 are satisfied, and hence, we have the following corollary: Corollary 4.2. Suppose that F ∈ RV−α , 0 < α < ∞, and that relation (4.3) is valid.

Y. Chen, Z. Yuan / Insurance: Mathematics and Economics 73 (2017) 75–81

(a) If G ∈ RV−β , 0 < β ≤ ∞, and E Y



 

ψ(x; n) ∼ ν(A)

1− E Y

αβ/(α+β)

 αβ/(α+β)

< 1, then

n

1 − E Y αβ/(α+β)



 (b← (x))−1 1/α 1/β

holds uniformly for all n ∈ N, where A = {(t1 , t2 ) > 0 : t1 t2 > 1} and the function b(·) is given by relation (4.5). In particular,

ψ(x; ∞) ∼

ν(A) 

1 − E Y αβ/(α+β)

 (b← (x))−1 .

(b) If 0 < yˆ < ∞ and E [Y α ] < 1, then

ψ(x; n) ∼ ν(B)

1 − (E [Y α ])n 1 − E [Y α ]

F ( x)

holds uniformly for all n ∈ N, where B = {(t1 , t2 ) > 0 : t1 > yˆ −α }. In particular,

ψ(x; ∞) ∼

ν(B) F (x). 1 − E [Y α ]

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