An Extension of Panjer s Recursion

Transcription

1 1 A Extesio of Paer s Recursio Kaus Th. Hess, Aett Liewad ad Kaus D. Schidt Lehrstuh für Versicherugsatheatik Techische Uiversität Dresde Abstract Sudt ad Jewe have show that a odegeerate cai uber distributio = {q } N0 satisfies the recursio q +1 = a + b q + 1 for a N 0 if ad oy if is a bioia, Poisso or egativebioia distributio. This recursio is of iterest sice it yieds a recursio for the aggregate cais distributio i the coective ode of risk theory whe the cai size distributio is iteger vaued as we. A siiar characterizatio of cai uber distributios satisfyig the above recursio for a N 0 with 1 has bee obtaied by Wiot. I the preset paper we exted these resuts ad the subsequet recursio for the aggregate cais distributio to the case where the recursio hods for a N 0 with k for arbitrary k N 0. Our resut are of iterest i catastrophe excess of oss reisurace. 1 Itroductio A cai uber distributio is a probabiity easure : BR [0, 1] satisfyig [N 0 ] = 1. Obviousy, there is a oe to oe correspodece betwee the coectio of a cai uber distributios ad the coectio of a sequeces {q } N0 R + satisfyig =0 q = 1 ; therefore, we sha write = {q } N0. For k N 0, defie N k := { N 0 k}. A odegeerate cai uber distributio = {q } N0 is said to be the Paer distributio with paraeters a, b R ad k N 0 if it satisfies the iear differece equatio q +1 = a + b q + 1 for a N k ad the iitia coditios q = 0 for a N 0 \N k ; i this case we write = Paera, b; k. The Paer distributio Paera, b; k is aso said to be a Paer distributio of order k, ad the coectio of a Paer distributios of order k is caed the Paer cass of order k.

2 2 The Paer cass of order 0 is iportat sice Paer 1981 has show that, i the coective ode of risk theory, the aggregate cais distributio ca be coputed by recursio whe the cai uber distributio is a Paer distributio of order 0 ad the cai size distributio is cocetrated o N 0 ad hece is a cai uber distributio itsef. I the preset paper, we exted Paer s recursio to the Paer cass of order k with arbitrary k N 0 Sectio 4. Sudt ad Jewe 1981 have show that the Paer cass of order 0 is idetica with the coectio of a odegeerate bioia, Poisso, or egativebioia distributios, ad Wiot 1988 has idetified a distributios of the Paer cass of order 1. I the preset paper, we idetify a distributios of the Paer cass of order k with arbitrary k N 0 Sectio 3. The proofs of both of these resuts rey o a differetia equatio which turs out to characterize the oet geeratig fuctio of a Paer distributio Sectio 2. With regard to appicatios, et us ote that Paer distributios of order k with k 1 are suitabe for portfoios of risks which are subect to catastrophe excess of oss reisurace where, as a rue, the priority is exceeded oy whe at east k cais occur. 2 A differetia equatio I the preset sectio we characterize the Paer distributio Paera, b; k by a differetia equatio for its probabiity geeratig fuctio. This resut wi be used to idetify a distributios of the Paer cass of order k ad to exted Paer s recursio for the aggregate cais distributio i the coective ode of risk theory. The probabiity geeratig fuctio : [0, 1] [0, 1] of a cai uber distributio = {q } N0 is defied as t := q t For a N 0, we have q = 0/!. 2.1 Theore. For a, b R ad k N 0 ad a odegeerate cai uber distributio = {q } N0, the foowig are equivaet: a = Paera, b; k. b For every N, satisfies the differetia equatio 1 a t h t = =0 a + b h 1 t + q k k! t k c with t [0, 1 ad the iitia coditios h 0 = 0 for a N 0 \N k. satisfies the differetia equatio 1 a t h k+1 t = k+1 a + b h k t with t [0, 1 ad the iitia coditios h 0 = 0 for a N 0 \N k.

3 3 Proof. t k q k t k =! ad hece Assue first that a hods. The we have t = = =k+1 =k+1 q a + b a + b + 1 =k = at q =k = at t +! q 1 t + 1 q t + a + b a + b t +1 q 1 =k 1 t 1! 1 at a t = + b 1 k t + q k! t k Therefore, a ipies b. Obviousy, b ipies c. Assue ow that c hods. By iductio, we obtai 1 at +1 t = +1 a + b for a N k. Lettig t := 0, the previous idetity yieds q +1 = a + b q + 1 t t 1 for a N k ad the iitia coditios yied q = 0 for a N 0 \N k. Therefore, c ipies a. Theore 2.1 is kow i the case k = 0 ; see Schidt 2001; Satz For a cai uber distributio = {q } N0, the bioia oet of order N 0 of is defied to be β [] := =0 q = = q The bioia oet β [] is fiite if ad oy if i t 1 t exists, ad i this case we have β [] = 1! The foowig resut is iediate fro Theore 2.1: 2.2 Coroary. Assue that = Paera, b; k with a < 1. The β [] for a N 0. is fiite

4 4 3 The distributios of the Paer cass of order k A cai uber distributio = {q } N0 is the bioia distributio B, ϑ with paraeters N ad ϑ 0, 1 if q = ϑ 1 ϑ hods for a N 0. the Poisso distributio Pα with paraeter α 0, if α α q = e! hods for a N 0. the egativebioia distributio NBβ, ϑ with paraeters β 0, ad ϑ 0, 1 if q = β ϑ β ϑ hods for a N 0. the ogarithic distributio Logϑ with paraeter ϑ 0, 1 if q = 1 ϑ og1 ϑ hods for a N. the exteded egativebioia distributio ENB, β, ϑ with paraeters N, β, +1 ad ϑ 0, 1] if q = β + 1 ϑ 1 β ϑ β ϑ hods for a N. the exteded ogarithic distributio ELog, ϑ with paraeters N 2 ad ϑ 0, 1] if 1 ϑ q = 1 ϑ = hods for a N. These distributios wi be refered to as basic cai uber distributios.

5 5 3.1 Rearks. I the ratio defiig the probabiities of the exteded egativebioia distributio, the uerator is either stricty positive for a N or stricty egative for a N ad we aso have β + 1 =0 ϑ = =0 β ϑ = 1 ϑ β Therefore, the exteded egativebioia distributio is we defied. For N 2, we have = = 1 = Therefore, the exteded ogarithic distributio is we defied. Wiot 1988 used the ter exteded trucated egativebioia distributio for ENB1, β, ϑ ; see aso Kuga, Paer ad Wiot 1998 ad Wiot ad Li Athough there is a obvious ustificatio for this terioogy, we prefer to oit the adective trucated here sice our resuts suggest to reverse the order of trucatio ad extesio ad to cosider exteded egativebioia distributios ad trucated exteded egativebioia distributios. Apparety, the distributios ENB, β, ϑ ad ELog, ϑ with N 2 have ot bee cosidered before; see Johso, Kotz ad Kep The proof of the foowig ea is straightforward: 3.2 Lea. Every basic cai uber distributio is a Paer distributio. Tabe 1 beow cotais for every basic cai uber distributio cosidered as Paera, b; k the paraeters a, b, k ad the probabiity geeratig fuctio : TABLE 1 Basic cai uber distributios a b k t B, ϑ ϑ 1 ϑ +1 ϑ 1 ϑ 0 1 ϑ + ϑt Pα 0 α 0 e α1 t NBβ, ϑ ϑ β 1 ϑ 0 Logϑ ϑ ϑ 1 β 1 ϑt 1 ϑ og1 ϑt og1 ϑ ENB, β, ϑ ϑ β 1 ϑ 1 ϑt β 1 1 ϑ β 1 ELog, ϑ ϑ ϑ = = 1ϑt 1ϑ β+ 1 β+ 1 ϑt ϑ

6 6 3.3 Rearks. Tabe 1 shows that there exists Paer distributios Paera, b; with a = 1 ; this has first bee observed by Wiot 1988 who discovered ENB1, β, 1 = Paer1, β 1; 1. For ENB, β, 1 = Paer1, β 1; the bioia oet of order is fiite if ad oy if N 0 \N, ad for ELog, 1 = Paer1, ; the bioia oet of order is fiite if ad oy if N 0 \N 1. This shows that Coroary 2.2 caot be exteded to the case a = 1. For k N 0 ad a cai uber distributio = {q } N0 satisfyig q k > 0 ad =k+1 q > 0, defie { } k q := 1 k 1 q N 0 The k is a odegeerate cai uber distributio satisfyig k t = t k 1 q t 1 k 1 q The distributio k is said to be the k trucatio of. 3.4 Reark. Let N be a rado variabe with distributio = {q } N0 satisfyig q k > 0 ad =k+1 q > 0. The k is the coditioa distributio of N uder the evet {N k}. Tabe 2 beow cotais the defiitios of the k trucatios of the basic cai uber distributios: I particuar, we have TABLE 2 k trucatios of basic cai uber distributios k k B, ϑ N 0 \N B, ϑ; k Pα N 0 Pα; k NBβ, ϑ N 0 NBβ, ϑ; k Logϑ N Logϑ; k ENB, β, ϑ N ENB, β, ϑ; k ELog, ϑ N ELog, ϑ; k B, ϑ = B, ϑ; 0 Pα = Pα; 0 NBβ, ϑ = NBβ, ϑ; 0 Logϑ = Logϑ; 1 ENB, β, ϑ = ENB, β, ϑ; ELog, ϑ = ELog, ϑ;

7 7 The foowig resut is iediate fro Lea 3.2: 3.5 Lea. If is the k trucatio of a basic cai uber distributio, the beogs to the Paer cass of order k. We sha see that the coverse of Lea 3.5 hods as we. 3.6 Lea. Assue that = Paera, b; k. The k+1 a + b > 0 Moreover, a + b 0 ipies a < 1, ad a + b < 0 ipies a 1. Proof. The first iequaity is iediate sice is odegeerate. Let us ow assue that a > 0 ad a + b 0. The we have, for a N k, ad hece q +1 = a + a + b + 1 q q +1 k a k q k a q Sice the series =k k+1/+1 a k diverges for a 1, we obtai a < 1. Let us ext assue that a > 0 ad a + b < 0. The we have, for a N k, ad hece q +1 = k a + k+1 a + b + 1 q k + 1 a q q a k q k+1 k + 1 Sice the series +1 1 =k k+1 a k diverges for a > 1, we obtai a 1. We ca ow characterize the distributios of the Paer cass of order k : 3.7 Theore. For k N 0 ad a odegeerate cai uber distributio, the foowig are equivaet: a beogs to the Paer cass of order k. b is the k trucatio of a basic cai uber distributio. Proof. Assue first that a hods ad cosider = {q } N0 = Paera, b; k. By Theore 2.1, we have d og k dt t = k+1 a + b 1 at for a t [0, 1 ad 0 = 0 for a N 0 \ N k. equatio, we distiguish three cases depedig o a : To sove the differetia

8 8 The case a = 0 : I this case, Lea 3.6 yieds b 0, ad we obtai k t = c ebt for soe c R. The geera soutio of this differetia equatio has the for t = c t + c k e bt ad the iitia coditios together with 1 = 1 yied t = e b1 t bt bt e! 1 b b e! Therefore, we have Pb; k. The case a < 0 : I this case we obtai for soe c R. Sice k t = c 1 at k+1+b/a q +1 = k a + k+1 a + b + 1 hods for a N k, Lea 3.6 yieds the existece of soe N k satisfyig q +1 = 0 < q. We obtai = a+b/ a, ad hece k t = c 1 at k The geera soutio of this differetia equatio has the for q t = c t + c k 1 at ad the iitia coditios together with 1 = 1 yied t = 1 1 a + a 1 a t a 1 a a 1 a a 1 a t Therefore, we have Ba+b/ a, a/1 a; k. The case a > 0 : I this case we obtai k t = c 1 at k+1+b/a for soe c R. To sove this differetia equatio, we distiguish five cases depedig o b :

9 9 The case b > a : I this case we have a + b 0 ad Lea 3.6 yieds a 0, 1. Lettig β := a+b/a, we obtai β 0, ad k t = c 1 at k+β The geera soutio of this differetia equatio has the for t = c t + c k 1 at β ad the iitia coditios together with 1 = 1 yied t = β 1 at 1 a β + 1 β a β a Therefore, we have NBa+b/a, a; k. The case b = a : I this case we obtai k t = c 1 at k 1 a β at for soe c R. The geera soutio of this differetia equatio has the for t = c t + c k og1 at ad the iitia coditios together with 1 = 1 yied t = k 1 og1 at og1 a 1 a at Therefore, we have Loga; k. The case +1 a < b < a with N\N k+1 : I this case we have a + b < 0 ad Lea 3.6 yieds a 0, 1]. Lettig β := a+b/a, we obtai β, +1. Proceedig as i the case b > a, but takig ito accout the possibiity of a = 1, we obtai t = β at β at β a β a Therefore, we have ENB, a+b/a, a; k.

10 10 The case b = a with N 2 \N k+1 : I this case we have a + b < 0 ad Lea 3.6 yieds a 0, 1]. We obtai k t = c 1 at k 1 for soe c R. The geera soutio of this differetia equatio has the for t = c t + c k 1 at 1 og1 at ad the iitia coditios together with 1 = 1 yied 1 at t = =k 1 a Therefore, we have ELog, a; k. The case b < k+1 a : This case is ipossibe because of Lea 3.6. Therefore, a ipies b. The coverse ipicatio foows fro Lea 3.5. =k 4 Paer s recursio For a cai uber distributio F = {f } N0 For N 0, defie f := ad for a, N 0, et 1 if = 0 ad = 0 0 if = 0 ad N i=0 f 1 i f i if 0 F := {f } N0 The F is a cai uber distributio satisfyig F t = F t The distributio F is said to be the covoutio of order of F. For cai uber distributios = {q } N0 ad F = {f } N0, defie { } Cop, F := q f N 0 The Cop, F is a cai uber distributio satisfyig Cop,F t = F t The distributio Cop, F is said to be the copoud distributio of F uder.

11 Reark. Let N be a rado variabe such that the distributio of N is a cai uber distributio, et {X } N0 be a sequece of rado variabes which is i. i. d. ad idepedet of N, ad defie S := N =1 X I the coective ode of risk theory, N is iterpreted as the uber of cais, X is iterpreted as the cai size of cai ad S is iterpreted as the aggregate cai size of the portfoio. If the distributio F of each X is a cai uber distributio, the F is the distributio of =1 X ad Cop, F is the distributio of S. The foowig resut exteds Theore 2.1: 4.2 Theore. For a, b R ad k N 0 ad a odegeerate cai uber distributio = {q } N0, the foowig are equivaet: a = Paera, b; k. b For every cai uber distributio F = {f } N0 satisfyig f 0 = 0 ad for every N, Cop,F satisfies the differetia equatio 1 a F t h t = i=1 a + b i h i t i F i t + q k t F k with t [0, 1 ad the iitia coditios h 0 = 0 for a N 0 \N k. Proof. Assue first that a hods ad et G := Cop, F. The we have G t = F t ad hece Gt = F t F t Because of Theore 2.1, this yieds 1 a F t Gt = 1 a F t F t F t = a+b F t + q k k F t k 1 F t = a + b G t F t + q k F t k This is the differetia equatio of b i the case = 1, ad the geera case ow foows by iductio. Furtherore, Theore 2.1 yieds 0 = 0 for a N 0 \ N k. Sice F 0 = f 0 = 0, differetiatio of both sides of the idetity G t = F t yieds G 0 = 0 for a N 0 \N k. Therefore, a ipies b.

12 12 Assue ow that b hods. Cosider the cai uber distributio F = {f } N0 satisfyig f 1 = 1 ad hece f 0 = 0. Sice F t = t, we have Cop, F = ad the differetia equatio becoes 1 at h t = k a + b h 1 t + q k! t k By Theore 2.1, this yieds = Paera, b; k. Therefore, b ipies a. As a cosequece of Theore 4.2, we obtai the foowig extesio of Paer s recursio for the probabiities of a copoud distributio: 4.3 Coroary. Assue that = Paera, b; k. If F = {f } N0 is a cai uber distributio satisfyig f 0 = 0 ad if G = {g } N0 = Cop, F, the g = 0 hods for a N 0 \N k ad the idetity g = a + b g f + q k f k hods for a N k. =1 Proof. By Theore 4.2, we have g = 0 for a N 0 \N k as we as 0 = a + b i i G 0 i F i 0 + q k 0 F k i=1 for a N k. The ast idetity yieds g = a + b i g i f i + q k f k for a N k. i=1 I a siiar way we obtai a recursio for the bioia oets of a copoud distributio: 4.4 Coroary. Assue that = Paera, b; k with a < 1. If F = {f } N0 is a cai uber distributio satisfyig f 0 = 0 ad if G = {g } N0 = Cop, F, the the idetity β [] G = 1 1 a =1 hods for a N k such that β [] F a + b is fiite. β [ ] G β [] F + q k β [] F k The resuts of this Sectio are kow i the case k = 0 ; see Schidt 2001; Abschitt 7.3. Coroary 4.4 is a variat of a resut of DePri 1986 who obtaied a recursio for the ordiary oets i the case k = 0 ; see aso Schidt 1996; Theore Reark. The ipicatio a = b of Theore 4.2 as we as Coroary 4.3 ad Coroary 4.4 ca be exteded to arbitrary F = {f } N0.

13 13 Refereces DePri, N Moets of a cass of copoud distributios. Scad. Actuar. J Johso, N.L., Kotz, S., ad Kep, A.W Uivariate Discrete Distributios. Wiey, New York ad Chichester. Kuga, S.A., Paer, H.H., ad Wiot, G. E Loss Modes. Wiey, New York ad Chichester. Paer, H.H Recursive evauatio of a faiy of copoud distributios. ASTIN Bu. 12, Schidt, K.D Lectures o Risk Theory. Teuber, Stuttgart. Schidt, K.D Versicherugsatheatik. Spriger, Beri Heideberg New York. Sudt, B., ad Jewe, W.S Further resuts of recursive evauatio of copoud distributios. ASTIN Bu. 12, Wiot, G.E Sudt ad Jewe s faiy of discrete distributios. ASTIN Bu. 18, Wiot, G.E., ad Li, X.S Ludberg Approxiatios for Copoud Distributios with Isurace Appicatios. Spriger, Beri Heideberg New York. Kaus Th. Hess, Aett Liewad ad Kaus D. Schidt Lehrstuh für Versicherugsatheetik Techische Uiversität Dresde D Dresde E ai: schidt@ath.tu-dresde.de