E L S E V I E R Insurance: Mathematics and Economics 23 (1998) 157-172

The moments of ruin time in the classical risk model with discrete claim

size distribution

Phi l ippe Picard a,,, C laude Lef~vre b a lnstitut de Sciencefinanci~re et d'Assurances, Universitd Lyon 1, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France b lnstitut de Statistique et de Recherche Opgrationnelle, C.P 210, Universitg Libre de Bruxelles, Boulevard du Triomphe, B-1050, Bruxelles,

Belgium

Received April 1998; received in revised form July 1998; accepted July 1998

Abstract

Firstly exact simple expressions are given for the moments Mr = Eo(Tr l{T<~}) when the initial reserves are equal to zero. Then for positive initial reserves the same moments are expressed very compactly through the Mr's , and the polynomials e,~(O = eXtP(St = n), n = 0, 1 . . . . In both cases the results are exact and valid for any arithmetic claim size distribution. Our main tool is the generalized Appell structure of the polynomials family {en, n ~ N}. Both cases c > )~m and c <_ 3.m are treated. (~)1998 Elsevier Science B.V. All rights reserved.

K©,words: Collective risk theory; Compound Poisson process; Generalized Appell polynomials; Generalized Taylor expansions

1. Introduction

We consider the classical continuous time model of an insurance company with positive claims. The amounts WI, W2 . . . . of the successive claims are i.i.d, and independent of the Poisson process ( N t ) t with parameter ;~ that counts these claims. We denote by

Nt

St = )--~ W/ i=1

the aggregate claim amount at time t,

Rt = u -+- c t - St

the insurer ' s surplus at t ime t, u e N being the initial surplus (for convenience we take u na tura l ) , c the p remium

per unit time. c may be greater or smaller than km where m = E ( W i ) . We denote by {qj , j ~ N*} the c o m m o n

distribution of the W/ 's and by

* Corresponding author. Tel.: +33-4-7243-1175; fax: +33-4-7243-1176; e-mail: picard@univ-lyonl.fr

0167-6687/98/$19.00 (~)1998 Elsevier Science B.V. All rights reserved. PII: S0167-6687(98)00025-0

158 P. Picard, C. Lefdvre /lnsurance: Mathematics and Economics 23 (1998) 157-172

1 ,----, = 2.~qjs j (1.1) -ig(s)

j=l

their generating function. Notice that the W/'s are supposed to be strictly positive but this is not a restriction if the process (Nt)t is conveniently defined. As in Picard and Lef'evre (1997) we introduce the family of polynomials {en, n E N } given by its generating function

O¢3

e xg(s) = ~-~en(X)S n, x ¢ ~ . (1.2)

n=O

Clearly, when t _> 0,

OO

E(s s,) _~ e-)~t(l-g(s)/x) = ~) ~e-~.t en(t)sn

n=O

and thus

P(St = n) = e-Xten(t) . (1.3)

Note that the degree of the polynomial en is n, e0 = 1, en (0) = 8n0. The family {en, n E 1~ } plays a leading part and becomes highly manageable after the introduction of the operator A defined by

Aen+l = en, n > 0, Ae0 = 0, (1.4)

in order to build a generalized Appell structure on this family. This structure will be the key in many proofs since it opens the possibility to write a generalized Taylor expansion

R(x) = Z An R(a) "en (x - a), (1.5) n

for any polynomial R and any constant a. Of course we could express explicitly the en'S through the q~r's (q~r = P(WI + " " + Wr = j) when r > 0 and

q~0 = 3j0) using

()~x)r *r en(x) = 2...---~qn (1.6)

r=0

but the result would be deceptive. The formulae would be more complicated and without any apparent algebraic structure.

In addition to the en'S we introduce the fn 'S defined by

fn(X) -- cx + nen x + , x ~ R , (1.7)

and the operator A = A S - l / c , S-1/c being the shift operator that transforms the function h (x) into h (x - 1/c). It can be proved (see Appendix A or Picard and Lef~vre, 1997) that

A f n + l = fn , n > 0, /~fo = 0, (1.8)

and thus { fn, ~ 1~ } is a family of generalized Appell polynomials too. This opens again the possibility to write (1.5) where/x and f , have been, respectively, substituted for A and en. This possibility will be welcome in some proofs.

The main purpose of the present paper is to give exact expressions for the moments of the ruin time T when u is finite, of course under the condition that T < oo when c > km, since in that case E(T) would have no meaning. This

P. Picard, C. Lefbvre/lnsurance: Mathematics and Economics 23 (1998) 157-172 159

topic seems to have received little attention in the literature with the notable exception of Delbaen (1990). More ancient works by Segerdahl (1939, 1955) or Seal (1969) were concerned with the normal asymptotic approximation of T, and therefore of its moments, only in case of large initial reserves. Comparatively the dual problem of the time required for the surplus to reach a given level has attracted much more attention. In this alternative problem the evaluation of E ( T r I T < ~ ) is far more easy, f.i. using the Laplace transform or martingales, and can be found in several papers or books (see f.i. Gerber, 1979; Delbaen and Haezendonck, 1986; Gerber, 1990; Picard and Lef'evre, 1994).

The scheme of the paper is as follows. In Sections 2 and 3 the initial reserves are supposed to be equal to zero, with c > 3.m or c < Xm, respectively. In both cases exact and very simple expressions can be easily obtained for the moments of T. In Section 4 a probabilistic argument, well adapted to discrete claim size distributions, is used to build P(oo > T > t [ R0 = u) for any t > 0, u > 0, and the subsequent construction of E ( T r l I T < ~ I ) follows in Section 5. Section 6 is devoted to some technical aspects of the computation of these moments. Some useful results on the polynomials en'S and fn 'S are recalled in Appendix A.

Notations, We write E , ( . . . ) for E ( . . . I Ro = u), Pu ('" • ) for P( . . - I R0 = u), 0 = 01 = )~m/c and Or = ),mr/C when the moment mr = E ( W [ ) , r > 1, exists. We also note Mr = Eo(Trl{T<~cl).

2. Evaluat ing Eo(T<I{T<oo}), r > 0 natural, when c > Xm

We start with

(f/ ) /0 Mr = EO(TrlIT<~}) = EO l{x<T" <oo}dx = r t r - l p o ( ~ > T > t)dt,

consequently we need P0(oo > T > t) for any t > 0.

L e m m a 2.1.

Oo

P o ( ~ > T > t ) = e - X t E ( ~ - l ) e n ( t ) l { n > _ c t I, t > 0 . t l : l

(2.1)

Proof. From a classical formula attributed to Prabhu (1961) (see f.i. Seal, 1969; Gerber, 1979 or De Vylder, 1996)

1 [ ct I [ ct P o ( T > t) = -~do Po(St < y )dy = - ~ d o _ ( C t - y ) d y P o ( S t < y)

or in our notation

(5O n

P o ( T > t ) = Z e -~'t ( 1 - -~)en(t)l{n<_ct}. (2.2) n = 0

Since,

Po(T = oo) = 1 )~m

OG

( ) - 1 E(St)ct - Z e - X t en(t) - -~en(t) , (2.3) n = O

the lemma follows from Eqs. (2.2) and (2.3) and

PO(CX~ > T > t) = Po(T > t) - Po(T = ~ ) . []

From this result we get a first formula for Mr.

160 P. Picard, C. Lef~vre /Insurance: Mathematics and Economics 23 (1998) 157-172

Lemma 2.2.

rfo' Mr = (1 - y)yr-2hr(y) dy,

where

h r ( y ) = E e - O ' n / c ) y n r e n ( ~ y ) . n=!

Proof. From (2.1) and Lemma 2.1

fo t Mr = r tr-le-~'t n _ 1 en(t)l{n>_ct}dt

n=l

or

r £ f o "c ' Mr = e_kttr_ 1 n _ 1) en(t)dt n=l

due to the positivity of the terms. The lemma follows after substitution of ny/c for t and permutation of ~ and f . This completes the proof. To explicit hr (y) we need the following lemma. []

Lemma 2.3.

n=lE(Se-(Y/c)g(s))nc--~-nyen ( n y ) = g ( s ) "

Proof. Using Taylor expansion with respect to Zx and the family {fn; n ~ N} defined in (1.7) we get

k k ( ) ( n ) n c..__f_x x + e k ( X ) = Z A n e k ( O ) f n ( X ) = E e k - n - - c c x + n en c

n=0 n=O

and, after multiplication by s k, 0 < s < 1, and summation on k,

OO

eXg(s)=Esne_(n/c)g(s)CXcx+n en (x+cn) . n = 0

Since g and the en'S do not depend on c we may formally substitute c /y for c and write

exg(s)-- l E(se_(Y/c)g(s)) n c ( n ) X n=l CX + ny en X + c y .

The lemma follows when x tends to 0.

The operational expression for hr is the following.

Lemma 2.4.

hr(y) =- ~(r+l) ( - ~ y ) ,

[]

P. Picard, C. Lef~vre /lnsurance." Mathematics and Economics 23 (1998) 157-172

where qJ is the inverse function of ~o defined by

~o(w) = w Yg(eW). C

161

(2.4)

P roof . Putt ing s = e w in (2.4) we have

e~(u') = se-(Y/c)g(s)

and L e m m a 2.3 becomes

oo Ze~O(w)n c ( n ) --en y = g(eW). n=l ny

Using (2.4) again we get

then

E e ~ ( w ) n l e n ( n y ) = w - - q g ( w ) , n=l

OQ

z e vn en y = qt(v)-- v (2.5)

n=l since ~0'(w) is posi t ive near w = 0, w = q~(v) the inverse funct ion of v = ~o(w) can be defined. After r + 1 different iat ions of (2.5) with respect to v, putt ing v = -)~y/c (i.e. w = 0), we get

OG Z (~ ) Iff(r+l)( ~ ) hr(y) = e-(XY/C)nnr en y = - y n=[

and the l emma is proved. []

L e m m a 2.5.

gt" (-k~Yc ) = O2y(1- Oy)-3 ,

gt"'(-A--Yc) =03y(1-Oy) -4+302y2(1-Oy) -5,

q/4)(-A--Yc) =04y(1-Oy)-5+lOO203y2(1-Oy)-e+1503y3(1-Oy)-7 '

~p(5~( -A~yc)~5y~-~y) -~+(~2-~-~5~2~4)~2(~-~y) -7+~5(~2~3~3(~-~y) -8+~4~4(~-~y) -9~

where Oi = Lmi/c, mi = E(WI), 0 = O! and gt is defined in Lemma 2.4.

Proof . Since (1/)Qg(e w) is the generat ing function of the mi's, (2.4) leads to

~o'(0) = 1 Oy, ~0(i)(0) )~mi - - = - - y = - - O i y , i > 1 . C

162 P Picard, C. Lefbvre/lnsurance: Mathematics and Economics 23 (1998) 157-172

Then

q~(~o(w)) = w

implies

q/ (v)q)' (w) = 1,

1 q , ' ( v ) -

~ o ' ( w ) '

e,,(v) = __L_l -¢ ' (w) # ( w ) w ~°'3(w) '

tP"'(v) -- 1 ( - ~ o " ( w ) ] ' -~p" (w) 3 ~o"2(w) q)t(W) ~ ,] w -- g)t4(W~ "1" ~r---~--~(W),

and so on. The lemma follows since v = -)~y/c means w = 0.

Theorem 2.6. When c > )~m,

02 M l - - - - M 2 - -

2c(1 - 0) ' [ c ( 1 - O ) l 2 -- '1"2(1-0----~ '

M3 - - 1 3o2o3 3 o 3 i

[c(1 - 0)] 3 - - .1. 2 (1 - 0----~ -t- 2 (1 ----0) 2 '

M 4 - 1 (~_~5 402+60204 02203 15 04 )

[ c ( 1 - 0 ) ] 4 ,1, 3 ( 1 - 0 ) "1"10(1-0)2"1" 2 ( 1 - - 0 ) 3 "

[]

Proof. Due to Lemma 2.5 we may write

( ~ ) r (1 - yi iff(r+ I) _ y = ZAri i=l

where the Ari'S are constants. Then from Lemmas 2.2 and 2.4,

~fol --y)yr-2tll(r+l) ( ~ ) ~- '~ fo 1 Mr = (1 - y d y = Ari i=l

Since for any k 6 N,

fo I (1 - y)yk 1 (1 - 0y) k+3 dy = (k .1. 2)(k .1. 1)(1 - 0) k+l

yi+r-2 (1 - y) (1 - Oy) i+r+l dy.

(this is easily checked when k = 0 and the general formula follows using differentiation with respect to 0), we get

r r 1

Mr = ~ i ~ l (i "1" r)(i .1. r -- 1)(1 - 0) i+r-l"

P. Picard, C. Lefbvre /Insurance : Mathematics and Economics 23 (1998) 157-172

NOW, for r = 1, Al l = 82, hence M1 = 82/c2(1 - 8). For r = 2,

A21 = 03 A22 = 309,

hence

2 ( 03 M2 = ~ 3 . 2 ( 1 - 8) 2

For r = 3,

A31 = 84,

and so on.

) + 4 . 3 0 C- 8) 3 "

A32 -= 100283, A33 = 158~

163

[]

3. Eva lua t ing EO(Tr), r > 0 na tu ra l , when c < )~m

This unusal case is of some interest if we consider a company with several subsidiaries. Even if the company has a positive non-ruin probability, it may happen that simultaneously one of its subsidiaries runs to ruin with probabili ty 1. In that case the mean time to ruin for this subsidiary may be a significant index concerning the future.

T h e o r e m 3.1. When c < Xm,

1 02 Mi = E0(T) = ~ + 2c02(1 _ 8) '

2 82(3 - 88 + 682) 83(30 - 2) M2 = E0(T 2) = ~-~ + 2c284( l _ 8) 3 + 2c283( 1 _ 0) 2 .

Proof. Here T < oe a.s. and P0(ee > T > t) is given by Eq. (2.2), hence

fo ~ oG FI Eo(Tr)=r t"-'Ee-X'(l-~t)e,,(t)Ii~<c.,,dt

n = 0

n = r ~ - - ~ en(t) dt. r--I --~-t {1

= /ct e )

Using the new variable y = ct/n when n > 0 before permuting ~ and f , we get

then

E0(Tr) = ~7 + ~q7 y r -2 (y _ 1)Ee-Zny/cnren dy,

I1=[

EO(Tr) = --£7 + ~ yr--2(y__ 1)pIr+l) __ Y dy,

due to the definition of kr (y)in Lemma 2.2 and its expression derived in Lemma 2.4. As in Theorem 2.6 we write

r y,

t P ( r + l ) - - Y = E A r l (1 - 07) i+r+l' i = 1

164

hence

r ~llCX~ 1 )y r+i_ 2 r! r ~ (y - Eo(T r) = -~ + -~ Ari (1 - - 0 y ) r + i + l dy.

It is easily proved that

f ~ ( Y - 1 ) y k d y (-1)k { 1 [ 1 Ik = (-]'-~Oy) k+3 -- k + 2 ~ 0 C+I

and finally

1 1 Eo(T) = -~ + -alllO,c

Eo(T 2) = ~ + (A21II -t- A2212),

w h e r e A l l = 02, A21 = 03, A22 = 302 a s in T h e o r e m 2.6.

P. Picard, C. Lefdvre/lnsurance: Mathematics and Economics 23 (1998) 157-172

[]

4. Evaluat ing Pu (c~ > T > t)

Theorem 4.1. Putting go(t, u) = Pu(T > t)and(o(t , u) = Pu(C~ > T > t), then

" ) go(t, u) ~"~)~i/c~ . i = Z.a~ ~u-, - go t + c , O i=0

for any t >_ 0, u natural, the formula being also true when (o is substituted for go.

Proof. Let us 1/c a level n < u has been reached. Thus a renewal argument will lead to

(4.1)

For u > 0 we use Eq. (4.2) to write

u - I u - I u - l - c ~

Z e Z / C e ~ ( - - ~ ) g o ( t , u - - l - c O = Z e Z / C e a ( - - ! ) Z e - ; ~ / C e g ( ~ ) g o ( t - ! , u - - o t - - f l ) ce=O c~=O fl=O

u,( , ) (!) ( , ) = Z ~ , - - u - n Z e o - e, = ~ , - - . , c c

n=0 ct+fl=n

since

( 1 ) c et~ ( ! ) ~-~ eu -- =en(O) =~nO ~+~=n

u ( I t - + 1 ) t 1 go(t, u) = ,._,P(SI/c = n)go - , u - n , u > O, > - . c c

n=0

(4.2)

(4.3)

due to

e -g(s) /c • e g(s)/c = 1

consider the trajectory t ~ St on the interval (0, 1/c). Ruin has not happened if and only if at time

P. Picard, C. Lef~vre / Insurance : Mathematics and Economics 23 (1998) 157-172 165

and the definition of the en's in (1.2). Substituting t + 1 / c for t we get for t > 0, u being a strictly positive natural,

I ! ) ( ( ! ) ( 1 ) 1 + Z e Z / C e ~ t + u 1 . (4.4) q g ( t , u ) = e Z / C e u _ l - ~o t + - 0 - ~o - - - C ' et=O C '

Now it is sufficient to use this formula recursively to get ~o(t, u) explicity expressed as a linear combination of the ~o(t + i / c , 0) , i = 1, 2 . . . . . u. Actually this procedure leads to (4.1). To check that this is true, we simply substitute (4.1) in both members of (4.4) getting

u u - I u - l - c z ~-~ ~i/c D . ( ~ ) ( i ) E ( ! ) ]~~ ( ~ ) ( l + j ) A . ~ ~u-~ -- ~o t + c ' O = eZ/Ceu - e~'J/Ceu_l_c~_j -- ~o t + c , 0 ,

i = 0 ~ = 0 j = O

which is clearly an identity. (When 1 < i < u, the coefficients of ~o(t + i / c , O) are, respectively,

eXi/Ceu-i ( - - ~ )

and

u - i

Z e Z / C e c ~ ( - ~ ) e Z ( i - ' ) / C e u _ c ~ _ i ( - - i - ~ c l )

in the left-hand side and right-hand side members. However, these expressions are equivalent due to formula (1.5) that gives

u--i

e , - i ( x ) = E A ~ e , - i ( x + ! ) e ~ ( - ! ) ,

ee=0

hence

u--i

e u - i ( - ~ ) = ~ = o e U - i - a ( - - ~ + ~ ) e ~ ( - ~ ) "

When i = 0, e u - i ( - i / c ) = 0 and ~0(t, 0) disappears.) Now, when c > )~m, making t tend to infinity in (4.1) gives

Pu ( T = ~x~) V ~ - Z i l c ~ • = Z.~ ,~ ~u- , - P o ( T = ~ ) (4.5) i = 0

and connecting (4.5) with (4.1) we obtain

Pu ( ~ > T > t) ~'~°)~i/co " = Z . ~ ~ ~u-~ - P0 c x ~ > T > t + . i = 0

The proof of the theorem is completed since, when c _< km, T < ~ a.s. and ~ = ~0. []

Remark. From (4.1) and (2.2) we obtain the explicit expression

(!) " Pu ( T > t) = e e u - i - - 1 - en t + . i=o n=0 ct + i

(4.6)

166 P. Picard, C. Lefbvre /lnsurance: Mathematics and Economics 23 (1998) 157-172

This formula is a variant of

u u [ct+u] ( ~ _ _ u ) c t - m + u ( u - ) Pu(T> t ) = e - k t Z e m ( t ) + e - k ' Z Z ej

m=O j = O m = u + l ct j -q- U e m - j t + - - C

,) (4.7)

given initially in Theorem 4.2 of Picard and Lef6vre (1997) and used as starting point in De Vylder (1997a,b). To establish the equivalence between (4.6) and (4.7) we use the generalized Taylor expansion of em with respect

to ZX and j~'s around the value (m - u)/c to get

m - b t /4 - m = ej fm- j t +

c j = 0 c

and

(4.8)

(4.9)

m

e m ( t ) : Z A m - J e m ( ~ - ) f m - j ( t

j=O

u ( ~ _ ~ u _ ) c t + u - m ( u - j ) Z e m ( t ) : E ej ct + u J e m - j t + m = 0 O<j<m<u -- C

Then using (4.8) to transform (4.7) we have

'~,u [,.,+u]~ ( ~ _ ~ ) c t + u - m T ( t + u - j ) > t) = e-kt. . . . , ,-.., ej P,(T j = 0 m=j ct -[- U ----- J e m - j C

which is (4.6) where n has been substituted for m - j and i for u - j.

Re mark . We may notice from (4.5) that, when c > ~,m,

go(u) = Pu(T = c~) = (1 - O) ~..eki/Ceu_i - . i = 0

5. B u i l d i n g Eu(Trllr<~j), r > 0 n a t u r a l , w h e n u > 0

According to Theorem 4.1 the results rely on the formula

t0 ( )f5 ( Eu(TrlIT<~I) = r tr-lpu(oO > T > t)dt = ~-",ki/co . Z..~ "~ "-u-t - r tr-I PO cx~ > T > t + dt i = 0

o r

Eu(Trl{T<oo}) = Bru - rDru

if we introduce the notation

( r / ~"~eki/C e . Bm = ~ u-, -- rtr-t p 0 oo > T> t + dt - - Po(T = ~ ) i=0 [ J - i /c

(5.1)

(5.2)

(5.3)

P Picard, C. Lef~vre/Insurance: Mathematics and Economics 23 (1998) 157-172 167

and

~i/Ce -- t r- l PO T > t + Dr. = ~.. e u-i dt. i = 0 i/c

Lemma 5.1.

t - - I

E ( r ) M r - k L k u ( 1 ) - L r u ( 1 ) P o ( T = ~ ) , Bru = k k = 0

where

"

L k u ( y ) = / _ . . , ~ ~ - I - Y - Y i = 0

and Mr-k is given in Theorem 2.6 (or Theorem 3.1).

(5.4)

(5.5)

Proof. Using the new variable x = t + i / c and the binomial formula to develop ( x - i / c ) r - I ,

Z--., ~ u ( ~ ) [ ~ l ( k ) ( c i\k/ J0~ ( !)2- } ~"~o)~i/c e . r -- 1 ----~ / xr - l -kpo(o0 > T > x )dx - - Po(T = oc) R_ru = u - z - r

i = 0 [ k=O

_ N - ~ ~ i / c , , . r -- - - ,,,2--k - - Po(T = cx~j

i = 0 / k = 0

and the lemma follows due to the definition (5.5). []

Remark. The Lkz,(1)'s and 4~(u) given by (4.9) have exactly the same structure. Thus, when c > km, the Bru, r = 1,2 . . . . can be computed simultaneously with 4flu).

Lemma 5.2.

Dru = -- Lru(y)dy.

Proof. From (5.4) and (2.2) we get

( ° ( n ) ( !1 Dru = E e u _ i - t r - l E e - Z t 1 - en t + l{n<ct+i}dt. i=0 a- i / c ~ ct + i -

Clearly the second summation must be performed only for n = 0, 1 . . . . . i and the integration is restricted to t 6 [ - ( i - n) / c , 0]; thus substituting m for i - n, /0 (!) ( )

e-)~ttr-l eu-i -- f i -m t + m dt (5.6) Dru " ' " m/c c O<m<i<u -

(for the definition of the polynomials f i-m's see (1.7)). Since by generalized Taylor's expansion

u - - m ( ~ C ) ( ) u-m ( m ) ( m ) = Z e u _ m _ k m k fk t_t_m

e , - m ( t ) = Z d x k e u - m - - c fk t + c - c ' k = 0 k = 0

168 P Picard, C. Lef~vre/lnsurance: Mathematics and Economics 23 (1998) 157-172

after substitution of i for m + k we get

U

(c°) eu-m (t) = Z e u _ i -- f i -m t q- I

i=m

Inserting (5.7) into (5.6) we have

Dru = e-Xttr- l eu_m (t)dt m=0 m/c

or, after substitution of y for - c t / m , and i for m,

fO~= 0 { i ' ~ r-1 ( i ) ( ~ ) Dru = e (~'i/c)y ~- -c y ) eu-i - -c y -- dy

and the lemma follows.

Joining (5.2), Lemmas 5.1 and 5.2 we get the main theorem.

Theorem 5.3.

fo' Eu(Trl{T<°e}) = Z Mr-kLku(1) - Po(T = oO)Lru(1) + r Lru(y)dy, k=0

where Lru(Y) has been defined in (5.5) and Mr-k is given in Theorem 2.6 (or Theorem 3.1).

Corollary 5.4.

L' Eu(Tllr<~l) = M1Lou(1) - Po(T = ex~)Llu(1) + Llu(y)dy,

fo 1 E,(T211r<~I) = M2Lou(1) + 2M1Liu(1) - Po(T = ¢~)L2u(1) + 2 L2u(y)dy.

(5.7)

[]

Remark. As already noticed in Delbaen (1990) we see that rth conditional moment of T exists if and only if Mr exists, i.e. if and only if mr+l exists.

6. Final expressions for Eu(TrllT<~}), r > 0 natural, when u > 0

Following Picard and Lef~vre (1997), De Vylder (1997a) showed that ap(u) = 1 - qg(u) can be easily computed when c > )~m. The same procedure gives the Lku (1)'s at the same time as well, and the only difficulty left is the evaluation of fd Lku (y)dy. The recursive formula in Lemma 6.1 will solve this problem.

Lemma 6.1.

So ' ( So ) Lku(y)dy = ~ y ~ 8kl q- (k -- 1) L k - l , u - j ( y ) d y - Lk - l , u - j (1 ) q~r for k > 1. O<r<j<u

P. Picard, C. LefOvre /lnsurance : Mathematics and Economics 23 (1998) 157-172 169

Proof. For k > 1, using generating functions, we write from (5.5)

2 Lku(y)su = ~-~ - Y-l(seZy/c)ieu-i - sU-i = Z - Y-l(se((Z-g(s))/c)~)i" u=O O<i<u<e~ i=0

Hence

)~)ZsU fo Lku(y)dy Z (g ( s ) L)s i I c ) J 0 e((Z-g(s))/c)iyyk-ldy (g(s) . . . . u=O i=0

and after integration by parts

=~=olSi(e((~-gls))/c)i(--~)k-l--rSkl)--(k--1)fol(Se(()v-g(s))/c)y)i(--~)k-lyk-2dy}.

Using again (6.1) with k - 1 substituted for k we get

vc 1 1 Lk-l.u(1) -- 3kl -- (k - 1) Lk-l.u(y)dy s" s u Lk,,(y)dy - g ( s ) ~

(when k = 1, the last integral disappears). We insert the expansion

g(s) -

1 ~ ( ~ ) r vc oc r~O g ) -- l--'~'~'~"~tT*r sJ 1 - - = ~..~Z...,~J = - - 2 Z q*rs'

= r=O j = r O<_r<j<_~o

in (6.2) and identification in s u then completes the lemma.

Corollary 6.2.

fo 1 L "1"" *r Li.(y)dy=-~ Z ( 1 - o . . - j t ))qj, O<r<_j<u

fo i [ r + l 1 } *r LZu(y)dy = y ~ / ~ - - - ( 1 - LO.u-j(l)) - ~LI.,-j(1) qj . Osr<_j<u

(6.1)

(6.2)

[]

Proof. The first formula is obvious. For the second, from Lemma 6.1 we get

s0 ' Is0 / L2.(y)dy = -2 ~ Ll.._j(y)dy -- L l , u - j ( 1 ) q,jr. O<r<j_<u

Now, due to the first formula,

Z folLl'u-J(y)dyq~ r= ~ ~ l(1-Lo'u-j-j'(1))q~r'q*J r O<r<j<u O<r<_j<u O<r'<j'<u-j

and after summation on j + j ' = n

1 =-~ Z (t-Lo'u-n(1))q*n(r+")

O<r, r1<u y4-rl <n-<bl

(6.3)

170 P. Picard, C. Lef~vre /Insurance." Mathematics and Economics 23 (1998) 157-172

and summation on r and r '

= ~ Z ( 1 - L o , u _ n ( 1 ) ) Z ( i + 1)q *i. n = 0 i = 0

Coming back to (6.3)

f01 1 1 .r L2u(y )dy=~-~ E ( 1 - L o ' u - n ( 1 ) ) ( i + l ) q * i - - x Z L l ,u - j (1 )q j O<i<n<u O<_r<_j<u

hence the lemma. []

Joining Corollaries 5.4 and 6.2 with Theorems 2.6 and 3.1 we obtain explicit formulae for the first moments. For instance, the means are given as follows.

Theorem 6.3. When c > Lm,

02Lou(1) Eu(TI{T<~}) - 2c(1 - 0) (1 - O)Llu(1) + ~ ~ (1 - Lo,u-j(1))qj

O<_r<j<u

and, when c < Xm,

( ~ ) l *r 02 LOu(1) + Z (1 - LO,u-j(1))qj , Eu (T) = + 2c02--(-( - O) XO<r<_j<_u

where

Lku(1) = Z e ~'i/c -- eu-i -- •

i = 0

Appendix A

Lemma A.1. A is shift invariant (i.e. A S a = Sa A where S a is such that Sah(x) -~- h(x + a)).

Proof. From

Ae xg(s) = A en(X)S n = s en-I (x)s n-I = se xg(s),

\ n = 0 / n = l

we get

ASaexg(s) = Ae(X+a)g(s) = eag(S)AeXg(s) = eag(S)seXg(s) = Sa(seXg (s)) = SaAexg(s)

and A Sa en = Sa Aen, n > O, follows.

Lemma A.2.

X fn+~ = f,,, n >__ O,

[]

where

X = AS -l/~,

P. Picard, C. Lef~vre/Insurance: Mathematics and Economics 23 (1998) 157-172

cx t .1 f n (X) -- cx +--~e" X + c "

171

Proof. From O 0

Z e n ( x )n s n - I = (eXg(S) ) , s = xg ' (s)e xg(s) ,

n = l

we get

H

nen (x) = x ~ j X q j e , 7 _ j ( x ) , j=l

and

n - I

Now

n > l ,

A (nen (x)/x) = Z jXqj en- 1 - j (X) = (n -- 1 ) en- 1 (x)/x. j = t

Afn(x ) = AS -I/c en x + = A S ( ' - lVc c X - n e , , ( x ) cx

= S (n-l)/c (en-l (x) (n cx- 1) - - e n - l ( X ) ) = S(n-l)/c ( c x - n + e,,-l(x)) = f n - l ( x ) .

Lemma A.3. For any constant a and any polynomial R of degree n,

II I1

R(x) = Z A i R ( a ) e i ( x -- a) = Z A i R ( a ) f i ( x - a). i = 0 i = 0

[]

Proof. {ei, i E N } and therefore {S -ae i , i E N } are bases for the vector space of the polynomials. Using convenient constant t~i's we write for the polynomial SaR,

H

S a R = ~-~otiei,

i = 0

then, for r _< n, t? ?l

A r R = Z o t i A r s - a e i = Z o t i S - a e i _ r ,

i=r i=r

tT

Ar R(a) = Zot ie i_r (a -- a) = Olr,

i=r

since ei-r(O) = 6jr. The first result follows. The proof of the second result is the same,/X and fi being, respectively, substituted for A and el. []

172 P Picard, C. Lef~vre/Insurance: Mathematics and Economics 23 (1998) 157-172

References

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14, 163-179. Picard, Ph., Lef'evre, C. 1997. The probability of ruin in finite time with discrete claim size distribution. Scandinavian Actuarial Journal 1, 58-69. Prabhu, N.U., 1961. On the ruin problem of collective risk theory. Ann. Math. Statist. 32, 757-764. Seal, H.L., 1969. Stochastic Theory of a Risky Business. Wiley, New York. Segerdahl, C.O., 1939. On homogeneous random processes and collective risk theory. Thesis, Stockholm. Segerdahl, C.O., 1955. When does ruin occur in the collective theory of risk. Skand. Aktu. Tidskr. 38, 22-36.