ASYMPTOTIC MULTIVARIATE FINITE-TIME RUIN PROBABILITIES ...

BULLETIN FRANÇAIS D’ACTUARIAT, Vol. 13, n° 26, juin – décembre 2013, pp. 79 - 92

ASYMPTOTIC MULTIVARIATE FINITE-TIME RUIN PROBABILITIES WITH HEAVY-TAILED CLAIM AMOUNTS:

IMPACT OF DEPENDENCE AND OPTIMAL RESERVE ALLOCATION

Romain BIARD1

Laboratoire de mathématiques de Besançon2

Abstract

In ruin theory, the univariate model may be found too restrictive to describe

accurately the complex evolution of the reserves of an insurance company. In the case

where the company is composed of multiple lines of business, we compute asymptotics of

finite-time ruin probabilities. Capital transfers between lines are partially allowed. When

claim amounts are regularly varying distributed, several forms of dependence between the

lines are considered. We also study the optimal allocation of a large global initial reserve in

order to minimize the asymptotic ruin probability.

Keywords

Multivariate finite-time ruin probabilities; multivariate regular variation; capital transfer;

optimal allocation.

1 [email protected] 2 Laboratoire de mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030 Besançon, France

94 R. BIARD

1. INTRODUCTION

This paper deals with an insurance company with multiple lines of business. Each

line is assumed to be exposed to catastrophic risks like earthquakes, floods or terrorist

attacks. Such risks may affect several lines of the company at the same time, so dependence

between the lines is considered. Each line may correspond to a business in a specific

country or to a type of policy offered by the company. Capital transfers between the lines

are strictly regulated. Nevertheless, we assume here that a piece of the amount of each line

is allowed to recover losses of an another one. Our study is done in a finite-time framework. Actually, insurance regulation is based

on 1-year time horizon in the standard formula, and on finite-time horizon usually

comprised between 1 and 10 years in internal models. Suppose now that the company owns a global initial reserve to share between the

lines. Due to the specific risk exposition of each line, the choice of the allocation may have

a huge impact of its solvency. In Loisel (2005) and Biard et al. (2010), this optimal

allocation problem is concerned with the minimization of the expected time-integrated

negative part of a risk process. In this paper, we focus on the finite-time multivariate ruin

probability for our minimization problem. In risk theory, multivariate context has been studied scarcely compared to the

univariate one. For the univariate setting, the reader is referred e.g. to the comprehensive

books by Rolski et al. (1999), Asmussen and Albrecher (2010), Goovaerts et al. (2001) and

the references therein. Concerning the multivariate setting, light-tailed case is studied in

Collamore (1996, 2002) and a discrete approach is investigated in Picard et al., (2003). In

this paper, we are concerned with the heavy-tailed case, studied previously by Hult and

Lindskog (2006b, 2006a). The paper is organized as follows. In Section 2, we present the framework of the

paper and we define the quantities under interest. Section 3 deals with the computation of

the asymptotics of the multivariate finite-time ruin probability in context of dependence and

Section 4 investigates optimal allocation problems.

2. FRAMEWORK

Throughout the paper, vectors are denoted by bold letters. For example,

1 , , d dx x x . Moreover, we define 0, ,00 , 1, ,11 , ie the unit

ASYMPTOTIC MULTIVARIATE FINITE-TIME RUIN PROBABILITIES WITH HEAVY-TAILED CLAIM AMOUNTS

95

vector whose i th component is equal to 1 and for 1 1k d , 1

kk ii1 e .

2.1 Multivariate Regular Variation

In order to describe losses of catastrophic risks, we choose the heavy-tailed class of

regularly varying random variables. The typical example of these kinds of random variables

is the Pareto distribution. This random variable class well describes catastrophic risks in the

sense that, in the case of large initial reserve, the ruin of the company may be only caused

by one big loss. Definition 2.1 A function L on 0, is slowly varying at if

1, for every t 0.limu

L tuL u

We write 0L .

Definition 2.2 (Univariate Regular Variation) A -valued random variable X

is regularly varying if there exists 0 , such that

,limu

P X tut

P X u

for every t 0,

or equivalently if, ,P X u u L u for some 0L .

We write X .

Definition 2.3 (Multivariate Regular Variation) An d -valued random vector 1 , , dX XX

with unbounded support is regularly varying if there exists a nonzero Radon

measure defined on \ 0d such that

,lim| |u

P uAA

P u

X

X (1)

for every Borel set dA bounded away from 0 (i.e. 0 A ) with 0A .

We can also use an equivalent definition using the spectral measure. An d -valued random vector

96 R. BIARD

1 , , dX XX

with unbounded support is regularly varying if there exists an 0 and a

probability measure on the unit sphere 1 :| | 1dS x x such that

| | , /| |

,lim| |u

P xu Sx S

P u

X X X

X (2)

for every 0x and Borel sets 1dS S . The probability measure is called the

spectral measure of X . As a consequence, we have for every 0u and Borel set dA bounded away

from 0

.uA u A

We write , X .

For a general presentation of heavy-tailed theory, the reader is referred e.g. to the

book of Resnick (2007).

2.2 The model

To describe the reserve of an insurance company with d lines of business, we

consider a multivariate risk process 0t t R . Denote by 0u the global initial reserve

and by 0,1 da the vector which describes the part of u which is allocated to each

branch. As a consequence, we have 1 1da a . The premium rates are captured in

0, d c . The aggregate claim amount process 0t t S is assumed to be a multivariate

Poisson process, that is to say

1

,N t

t iiS X

where N t is a Poisson process with parameter 0 and 1i i X is a d -valued

independent and identically distributed sequence. We denote by X their common

distribution. Hence, we have, for 0t ,


97

1

.N t

t ii

u t R a c X (3)

Throughout this paper, X will be regularly varying for some 1 and measure .

2.3 Multivariate finite-time ruin probability

In the univariate setting, the finite-time ruin probability is defined as, for , 0u T ,

00,

, 0, , 0| .supt tT

u T P t T R R u P S ct u

In the multivariate case, there is not a unique definition. For example in Cai and Li,

2005, Cai and Li, 2007, we can find several definitions, depending on the interest. For

, 0u T let us define

- the probability that the sum of the line reserves becomes negative

before T ,

0, 1

, ;supd

j jsum t

T j

u T P S c t u

(4)

- the probability that all the line reserves become negative before T ,

0,1

, ;supd

j j jand t

Tj

u T P S c t a u (5)

- the probability that one of the line reserve becomes negative before T ,

0,1

, ;supd

j j jor t

Tj

u T P S c t a u (6)

- and the probability that all the line reserves are negative at a given time

before T ,

, 0, , 1, , 0 .jsim tu T P t T j d R

(7)

98 R. BIARD

Here, we investigate the definition proposed by Hult and Lindskog (2006a). For 0,1 , let

1 1

: 0 0 ,d d

k k

k k

F x x

x (8)

where min and max . For 0T , we define the multivariate finite-time ruin

probability , ,d u T as the probability that the risk reserve process tR hits F at some

time t before T . Explicitly, for , 0u T , we have

, , 0, , .d tu T P t T F

R (9)

Remark 2.4 The ruin set F corresponds to the possibility to transfer from positive

lines a fraction 0,1 to cover a negative position of another line. For 0 , no

transfer is allowed and ,0d or and for 1 , transfer is allowed without restrictions

and ,1d sum .

In Figure 1, the set F is represented for 0, 1/2 and 1 in the the two-

dimensional case.

Figure 1: F for =0, 1/2 and 1 in two dimensions

The following result, from Hult and Lindskog (2006a), gives the asymptotic of the

finite-time multivariate ruin probability for a large initial reserve. Proposition 2.5 (Hult and Lindskog (2006a)) For a risk process 0t t R given

by (3) with a common distribution , X for some 1 and measure , we

have, for 0T and large u ,


99

, , | | .d u T T F P u a X (10)

This result is the base of our computations. Actually, after giving the assumptions on the dependence structure between claim amount of each line, we can exhibit and then

get the asymptotic ruin probability. The following lemma gives F a for some basic

forms of X . Lemma 2.6 Let X be a positive random variable which is regularly varying with

some 1 . - Case 1. If for some 1 j d , jXX e , then , X for some

measure and we have

1 .jF a

a (11)

- Case 2. If, for some 1 k d , kXX 1 , then , X for some

measure and we have

1 1 1 | | .kd k

d F dk

1 1 (12)

- Case 3. If XX 1 , then , X for some measure and we have

*: :

*1 1* *

| | ,

k di d i d

i i k

a a

Fk d k

a 1 (13)

where for 1 i d , :i da is the i th larger component of a and

: :

* 1: 1 1inf 1, 1 : .

k di d i d

k d i i k

a a

k k d ak d k

Proof. Let A Fa . We have

100 R. BIARD

1 1

: 0 0 .d d

i i i i

i i

A a x a x

x

Case 1. Let jXX e for some 1 j d . Since X , there exists a function

0L such that P X u u L u . Moreover, from Karamata's Theorem (see e.g.

Embrechts et al. (1997), Theorem A3.6 p 567), we have for large u ,

1P X u u L uu

. Let dB be a Borel set bounded away from 0 . We

have

1 1 1| |lim limj j ju u

P X u P X uB P X u P u X B

e e e

1

1lim jB u Xuu L u r dF r

e

1lim j XBu

u L u r dF ru

e

11

lim jBuu L u r u ur L ur dr

e

1 .jBr r dr e

Hence, we have , X with defined for all Borel set dB bounded

away from 0 as

1 .jBB r r dr e

Since we have

1

i jj

i di j

r A a a r

e

1 .jr a

the first result follows.


101

Case 2. In this case, using the same way, we obtain that , X with

defined for all Borel set dB bounded away from 0 as

1/| | ,k kBB r r dr 1 1

and since 1d a 1 ,

1 1/| | /| |k k kr A d k d k d r 1 1 1

1 1 | | ,kd k

r dk

1

and the second result follows. Case 3. In this case we get that , X with defined for all Borel set

dB bounded away from 0 as

1/| | .B

B r r dr 1 1

It remains to find : /| |r r A1 1 .

1 1

/| | /| | 0 | | 0 .d d

i i

i i

r A a r a 1 1 1 1

Denote by :k da the k th larger component of a . Let 1 1k d and assume

that : 1:/ | |k d k da r a 1 . Then

: :

1 1

/| | /| | /| |d k

i d i d

i k i

r A a d k r a kr

1 1 1 1

: :

1 1/| | .

k di d i d

i i k

a a

rk d k

1

Let

: :1: 1 11, 1 :

k di d i dk d i i k

a aK k d a

k d k

. K is lower

bounded by 1 and 1d K (since : 1 /d da d ), so there exists a K -minimal element

102 R. BIARD

denoted by *k . Thus,

*: :

*1 1* *

/| | /| |

k di d i d

i i k

a a

r A rk d k

1 1 1

and the third result follows.

3. COMPUTATION OF RUIN PROBABILITIES IN THE PRESENCE OF

DEPENDENCE

3.1 A simple model of dependence

In this Subsection, we investigate a simple model of dependence between the lines

of business. For each claim occurrence, we allow the claim amount of a branch either be

independent of the others or equal to a common random variable. This model is inspired by

Biard et al.., 2008 who have introduced a model of dependence between claim amounts in

univariate setting. Explicitly, the distribution of 1 ,..., dX XX is such that, for

1 j d ,

0 1 ,j j j jX I W I W

where, - 0

jj dW is an i.i.d. non-negative random vector with common

distribution W , for some 1 ,

- and 1j

j dI is a vector of independent Bernoulli random variables

with same parameter 0,1p , and independent from 0j

j dW .

Let F be the c.d.f. of W . Note that dependence is only measured through the parameter p .

Here, we assume that 1d a 1 .

Lemma 3.1 Let 1 , 1 X for some 0 and some Radon measure 1 .

Let 2 , 2 X for same 0 and some Radon measure 2 . Moreover we assume

that, for some function L , there exists 1 2, 0c c such that, for large u

1 1| | ,P u c u L uX


103

and 2 2| | .P u c u L uX

If 1X and 2X are independent, then 1 21 2, 1 2

1 2 1 2

c c

c c c c

X X .

Proof. Let dA be a Borel set bounded away from 0 For 1,2i ,

,i i X , so from Definition 2.3

.lim| |i

iu i

P uAA

P u

X

X (14)

Since | |i iP u c u L uX ,

,lim i i iu

u L u P uA c A

X

with 01/L L . So from Hult and Lindskog (2006b) Proposition A.1,

1 2 1 1 2 2 .limu

u L u P uA c A c A

X X

By independence and regular variation, 1 2 1 2 1 2| | | | | | .P u P u P u c c u L u X X X X

Hence,

1 2 1 2

1 21 1 1 2 1 2

,lim| |u

P uA c cA A

P u c c c c

X X

X X

and the result follows. Proposition 3.2 Under the assumptions of this subsection, we have, for 0T and

large u ,

, , 1 1 1dd u T p d d

104 R. BIARD

1

1 1 1 1 .d

k d k

k

dd kk p p d k d d T F uk

Proof. Since W , there exists slowly varying function L such that, for 0u ,

P W u u L u .

By construction X is composed of a sum of random variables of the form

0

1, , \ 1 01, , \

ii i

i i d i iI Ii i d

W W

X e e

for some subset of 1, ,d . For all subset of 1, ,d ,

| |P u c u L u X for some constant c .

Thus, from Lemma 3.1 ,MR X with , for all Borel set dB bounded away

from 0 ,

.lim| |u

P uBB

P u

XX

For 1, , 1k d , let

0

1

.d

k ik i

i k

W W

X 1 e

Let 01

d iii

WX e and 0d WX 1 .

Let 1A d F 1 . Note that the set 1A d F

1 is symmetric in each

direction. Denote by M the random variable which counts the number of random variables

equal to 0W in X . We have, for large u ,

| |A P u P uA X X

0

|d

k

P M k P uA M k X

0

1d

k d k k

k

dk p p P uA

X ( A is symmetric in each direction)


105

0

1 | | .| |

kdk d k k

kk

dP uA

k p p P uP u

X

XX

Since for 1, , 1k d , 0| | | |k kP W u P W u1 1 and for 1, ,i d ,

| |iiP W u P W ue , we have, from Lemma 3.1, for 1, , 1k d ,

,k

k X with, for all Borel set dB bounded away from 0

0, 1| |,

| |

k kk

k

B d k BB

d k

1

1

where, - from Lemma 2.6 Case 2,

0, ,lim| |

kk

u k

P W uBB

P W u

1

1

and

10, 1 | | ,k k

d kA d

k

1

- and, from Lemma 2.6 Case 1 with 1 1a d ,

11

1

,lim| |u

P W uBB

P W u

e

e

and

11 1 .A d

Moreover, we have, for all Borel set dB bounded away from 0 0 1 ,B B

106 R. BIARD

so 10 1 ,A d

- and from Lemma 2.6 Case 2 with k d ,

0, ,lim| |d d

u

P W uBB B

P W u

1

1

so |1| .d A d

Moreover, by independence and regular variation, we have for large u , for

1 1k d ,

| | ,kkP X u d k F u

1

and

| | ,dP X u F u

1

and

0 .P X u dF u

Thus, we get

0

| | | | .d

k d k kk

k

dkA P X u p p A P u

1 X

Moreover, since from Proposition 2.5, we have for 0T and large u

1, , | | ,d u T T d F P u 1 X

we get the result.


107

Corollary 3.3 When no transfer is allowed ( 0 ), we get, for large u and 0T ,

,01

, 1 1 1 .d

d k d kd

k

dku T d p p p d k d T F u

This result corresponds to or (6).

Corollary 3.4 When transfer is allowed without restriction ( 1 ), we get, for

large u and 0T ,

,10

, 1 .d

k d kd

k

dku T p p k d k T F u

This result corresponds to sum (4).

3.2 A Poisson shock model

In the Subsection, we study a classical Poisson shock model; when a claim occurs, it

may affect either one specific line of business or all the lines. Explicitly, let X be a non-negative random variable which is regularly varying with some 1 . For 1 j d , we

assume that the specific claims of the business line j arrive at the jump times of a Poisson

process 0jt tN with intensity j . Let j

kX be the k th specific claim amount of

line j . Assume that, for 1 j d and 1k , 1jkkX is an i.i.d. sequence with common

distribution X . Thus, the specific aggregate claim amount process of the line j is, for

0t ,

1

.

jNtjj

t kk

S X

We assume that the claims which affect all the lines of business arrive at the jump

times of a Poisson process 00t tN with intensity 0 . Let 0 0

k kXX 1 be the vector of

the k th claim amounts of this kind. We assume again that 01k kX is an i.i.d. sequence

with common distribution X . Here, for simplification, we have assumed that all the lines

of business pay the same amount for a common claim. Thus, the common aggregate claim

amount process is, for 0t ,

108 R. BIARD

0

0 0

1

.Nt

t kkS X

We also assume that all 0jt tN and 1

jkkX , 0, ,j d are independent.

From the compound Poisson process properties, we are able to get a risk process of the type

of (3), since we can write

0

0

1 1 1 1

,

jN N Ndt t tj

t k j kkk j k k

X S X e X

where 0

d jtj

N t N is a Poisson process with intensity

0 1 d , and 00 1

d jk k k j j kkj

X X X e with, 1k k an

i.i.d. sequence of random variables independent of all others random variables and with

/jkP j for 1k and 0 j d .

Proposition 3.5 Under the above assumptions, we have, for 0T and large u ,

*: :

*1 10, * *

1

, 1 ,

k dj d j d

dj j k j jd

j

a a

u T a T F uk d k

where for 1 j d , :j da is j th larger component of a and

: :

1 1* 1:inf 1, 1 : .

k dj d j d

j j kk d

a a

k k d ak d k

Proof. From Proposition 2.5, we have for 0T and large u

, , | | .d u T T F P u a X

Let A Fa .


109

From Hult and Lindskog (2006a), Section 4, we have

0

| | | | 1 1 ,P u P X u

X 1

and

00

1

0

| |

| | 1

dj

jj

A A

A

1

1

where

0 ,lim| |u

P X uAA

P X u

11

and, for 1 j d ,

.lim| |

jj

u j

P X uAA

P X u

e

e

We get the result using 2.6, Case 1 and Case 3. Corollary 3.6 When no transfer is allowed ( 0 ), we get, for large u and

0T ,

0,0

1 1

, .mind

j j jd

j d j

u T a a T F u

This result corresponds to or (6).

Corollary 3.7 When transfer is allowed without restriction ( 1 ), we get, for

large u and 0T ,

0,1 , 1 1 .d u T d T F u

This result corresponds to sum (4).

110 R. BIARD

4. OPTIMAL ALLOCATION PROBLEMS

Throughout this Section, we assume 0 . So we write , 0d d . This case

corresponds to the probability that at least one of the line business becomes negative before

T without money transfer. In this section, we suppose that the company owns a global initial reserve u to

allocate to the d lines of business in order to minimize its finite-time ruin probability.

Explicitly, we have the following optimal problem :

1

0,1

undertheconstraint 1

, ,min

d

dd

a a

u T

a

(15)

Here, we are going to minimize the asymptotics of ,d u T we denote by

,d u T . Thus the problem (1) becomes :

1

0,1

undertheconstraint 1

, ,min

d

dd

a a

u T

a

(16)

We investigate the following cases. Case 1. the company is composed of d lines of business and they are mutually

independent; explicitly, the model is the Subsection 3.2 one with 0 0 .

Case 2. the company is composed of two lines of business and their dependence

structure is described by the Poisson shock model of Subsection 3.2. Case 3. the company is composed of three lines of business, one is independent

from the others and the two others are dependent via the Poisson shock model of

Subsection 3.2.

4.1 Case 1

In this Subsection, we start with the model of Subsection 3.2 wherein 0 is

assumed to be equal to zero. That is to say that X is composed with d mutually

independent random variables, so the business lines are mutually independent too. Proposition 4.1 Under the above assumptions, we have for 0T and large u ,


111

1

, .d

j jd

j

u T a T F u

Proof. Take 0 0 in Corollary 3.6.

The following proposition gives the optimal allocation of our optimization problem

(2). Proposition 4.2 Under the assumptions of the Subsection 4.1, the solution of (2) is,

for all 1 i d ,

11*11

1

.i

i

dj

j

a

Proof. Let1

: 0,1dd i ii

g g a

a a . g is a continuous,

differentiable and strictly convex function on 0,1 d . Using the method of Lagrange

multipliers, we find one *a which minimizes g on

Since g is strictly convex, on the non empty open convex set 1 1da a

this minimum is unique. In Figure 2, for 2d , we represent, 1a and 2a as a function of 1 /

( 1 / varies from 0 to 1). Both cases 2 and 5 are plotted. As expected, we

allow a larger part of the initial reserve to the riskier line of business. We can also note that

when is increasing, then initial reserves become more similar.

1 1, , 0,1 , 1 .d d da a a a

112 R. BIARD

Figure 2: Optimal solution of the Case 1 for 2d

4.2 Case 2

In this Subsection, we investigate the two dimensional case wherein the dependence

structure is described by the model of Subsection 3.2. Let 0,1a such that 1a a and

2 1a a . Proposition 4.3 Under the above assumptions, we have for 0T and large u ,

0 1 22 , min ;1 1 .u T a a a a T F u

Proof. Take 2d in Corollary 3.6.

Proposition 4.4 Under the assumptions of the Subsection 4.2, the solution of (16) is

0 1 2

11 1 0 1 2

111 0 2 11

10 1 1 0 2 1

1 10 1 21 1

1if | | ,

2

* if ,

if .

a


113

Proof. Let, for 0 1/2a

0 1 21 1 ,g a a a

and for 1/2 1a

0 2 12 1 .g a a a

1g (resp. 2g ) is differentiable and strictly convex on 0,1/2 (resp. 1/2,1 ). Moreover

1 21/2 1/2g g for all 1 0,1/2a and 2 1/2,1a ; 1 2g a g a . Let

1

2

0 1/2

.1/2 1

g a x

g ag a a

Thus, g is continuous on 0,1 and g is strictly increasing on

0,1/2 1/2,1 . So, g is strictly convex on 0,1 . As a consequence, on the non

empty open convex set 0,1 , there exists a unique *a which minimizes g . Since

0 1g g , we have

10,1/21

2* 1/2,12

1 2

argmin ifg 1/2 0 ,

argmin ifg 1/2 0 ,

1/2 ifg 1/2 0 and g 1/2 0 ,

g

a g

that is to say

* *1 1 1 1

* * *2 2 2 2

1 2

/ 0 ifg 1/2 0 ,

/ 0 ifg 1/2 0 ,

1/2 ifg 1/2 0 and g 1/2 0 .

a g a

a a g a

Since 2,

argmin argming , we get the result.

114 R. BIARD

In Figures 3, 4 and 5, for 2d , we represent, 1a and 2a as a function of 1 / (we fix 0 and 1 / varies from 0 to 01 . in the three figures, both

cases 2 and 5 are plotted. Figures 3, 4, and 5 represent respectively cases 0 0.1 , 0 0.3 and 0 0.5 .

Remark 4.5 There are three different forms of the optimal allocation in

Proposition 4.4. - When 0 is large, or when the two lines of business are very similar, we

allocate half of the reserve to each line. Actually, both high positive

dependence and close parameters conduce to a similar behavior of the two

processes. We can observe this behavior on the figures. In Figures 2, 3, we

observe a plateau when 1 is closed to 2 and this plateau becomes

wider when 0 , so the dependence, increases. When the dependence is

too high, as in Figure 4, we only observe a plateau.

- When 1 is large, compared to 0 and 2 , the optimal solution is

the same as in the case where the two lines of business are independent and

where 2 is switched with 2 0 , and, as expected, we allocate

more to the first line of business. In Figures 2, 3, it corresponds to the part

after the plateau. We have also the symmetric case, when 2 is large

compared to 0 and 1 , which is corresponds to the part before the

plateau in Figures 3, 4. We can also note that an increase of conduces to a decrease of the difference

between the two reserves.

Figure 3: Optimal solution of the Case 2 for 2d and 0 0.1


115



4.3 Case 3

In this Subsection, we assume that the insurance company has three lines of

business, two dependent through the common shock model of Subsection 3.2, and one

independent from the two others. Explicitly, we have (with a simple adaptation of the

Subsection 3.2 model) : 1

,N t

t kk

S X

where N t is a Poisson process with intensity 0 1 2 3 , and

302 0 1

ik k k k i i ki

X X X 1 e with,

116 R. BIARD

- 0 , 0 3i i ,

- for 0 3i , 1ik kX is an i.i.d sequence with common distribution

X ,

- and with all 1ik kX , 0 3i independent and independent from N t ,

- 1k k an i.i.d. sequence of random variables independent of all others

random variables and with /ikP i for 1k and 0 3i .

Denote by 3 the ruin probability associated with the above model.

Proposition 4.6 Under the above assumptions, we have, for 0T and large u ,

0 1 2 1 1 2 2 3 33 , min ; .u T a a a a a T F u

Proof. Let A Fa .

We have, for large u

30 0

02 2

1

| | | | | | | | 1 1 .i

ii

i

P u P X u P X u P X u

X 1 e 1

Moreover , X with

30

2 1,21

02

| |

,| | 1

ii

i

A A

A

1

1

where

21,2

2,lim

| |u

P X uAA

P X u

1

1

and, for 1 j d ,

.lim| |

jj

u j

P X uAA

P X u

e

e


117

We get the result using 2.6, Case 2 and Case 3. Proposition 4.7 Under the assumptions of Subsection 4.3, the solution of (2) is as

follows. - If 0 1 2| | , then

10 1 2 1

1 * 2 *1 1

0 1 2 31 1

13 13 *

1 10 1 2 31 1

21

,22

.

2

a a

a

- If 0 1 2 , then

11 11 *

11 11 0 2 311 1

10 2 1

2 *11 1

1 0 2 311 1

13 13 *

11 11 0 2 311 1

,

,

.

a

a

a

- If 0 2 1 , then

118 R. BIARD

10 1 1

1 *1 1 1

0 1 2 31 1 1

12 12 *

1 1 10 1 2 31 1 1

13 13 *

1 1 10 1 2 31 1 1

,

,

.

a

a

a

Proof. Fix 3 0,1a . Let, for 130,1a a ,

1 0 1 3 1 1 1 2 3 2 3 31 min ,1 1 .g a a a a a a a a

Using the same way as in the proof of Proposition 4.4, we get 1 * 33 1

0,1

argmina g a g .

Then 3 *3

0,1

argmina g and we get the result.


119

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capital and ruin probabilities. Technical Report 1441, School of ORIE, Cornell University. HULT, H. and LINDSKOG, F. (2006b). On regular variation for infinitely divisible

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RESNICK, S. (2007). Heavy-tail phenomena: probabilistic and statistical

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