A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

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1 A Uniform Asympoic Esimae for Discouned Aggregae Claims wih Subeponenial Tails Xuemiao Hao and Qihe Tang Deparmen of Saisics and Acuarial Science The Universiy of Iowa 241 Schaeffer Hall, Iowa Ciy, IA 52242, USA Ocober 26, 27 Absrac In his paper we sudy he ail probabiliy of discouned aggregae claims in a coninuous-ime renewal model. For he case ha he common claim-size disribuion is subeponenial, we obain an asympoic formula, which holds uniformly for all ime horizons wihin a finie inerval. Then, wih some addiional mild assumpions on he disribuions of he claim sizes and iner-arrival imes, we furher prove ha his formula holds uniformly for all ime horizons. In his way, we significanly eend a recen resul of Tang (27, J. Appl. Probab. 44, Keywords: Asympoics; Poisson process; renewal process; subeponenial disribuion; uniformiy. 1 Inroducion Consider he renewal risk model, in which claim sizes X k, k = 1, 2,..., consiue a sequence of independen, idenically disribued (i.i.d., and nonnegaive random variables wih common disribuion F, while heir arrival imes τ k, k = 1, 2,..., independen of X k, k = 1, 2,..., consiue a renewal couning process N = #{k = 1, 2,... : τ k },. (1.1 Tha is o say, he iner-arrival imes θ 1 = τ 1, θ k = τ k τ k 1, k = 2, 3,..., consiue anoher sequence of i.i.d., nonnegaive, and no-degenerae-a-zero random variables. If {N, } is a homogeneous Poisson process, hen his model reduces o he commonly used compound Poisson model. Aggregae claims form a random sum X( = N X k,. Suppose ha Corresponding auhor. uemiao-hao@uiowa.edu; phone: ; fa:

2 here is a consan ineres force r >. The discouned aggregae claims are epressed as he sochasic process D r ( = e rs dx( = where he symbol 1 E denoes he indicaor funcion of an even E. X k e rτ k 1 (τk,, (1.2 From (1.2 we see ha {D r (, } corresponds o a special case of he sochasic inegral Z = e Rs dp s,, where {R, } and {P, } are wo independen sochasic processes fulfilling cerain requiremens so ha Z is well defined. When boh of hem are Lévy processes, Gjessing and Paulsen (1997 gave a wealh of eamples showing he eac disribuion or asympoic ail probabiliy of Z. Relaed discussions on he disribuion of Z can also be found in Dufresne (199, Paulsen (1993, 1997, and Nilsen and Paulsen (1996, among ohers. However, we noice ha all hese references did no pay paricular aenion o he imporan case ha {P, } has heavy-ailed jumps. In his paper, we are ineresed in he asympoic ail behavior of D r ( for all for which he renewal funcion λ = EN = Pr (τ k is posiive. Define Λ = { : λ > } = { : Pr (τ 1 > } for laer use. We shall assume ha he claim-size disribuion F on [, is subeponenial, denoed by F S. Tha is o say, F ( = 1 F ( > holds for all and he relaion F lim n ( F ( holds for all (or, equivalenly, for some n = 2, 3,..., where F n denoes he n-fold convoluion of F. I is well known ha every subeponenial disribuion F is long ailed, denoed by F L, in he sense ha he relaion F ( y lim F ( holds for all (or, equivalenly, for some y. Moreover, he class S covers he class ERV = n = 1 of disribuions wih eended-regularly-varying ails. By definiion, a disribuion F on [, is said o belong o he class ERV if F ( > holds for all and here are some consans α and β, < α β <, such ha he relaions v β lim inf F (v F ( lim sup 2 F (v F ( v α (1.3

3 hold for all v 1. Noe ha relaions (1.3 wih α = β define he famous class R of regularlyvarying-ailed disribuions wih regulariy inde α. Anoher useful disribuion class is A, which was inroduced by Konsaninides e al. (22. By definiion, a disribuion F on [, is said o belong o he class A if F S and, for some v > 1, lim sup F (v F ( < 1. (1.4 Since relaion (1.4 is saisfied by almos all useful disribuions wih unbounded suppors, we remark ha he class A almos coincides he class S. In conclusion, R ERV A S L. For more deails of heavy-ailed disribuions in he cone of insurance and finance, he reader is referred o Embrechs e al. (1997. Hereafer, all limi relaionships hold as ends o unless saed oherwise. For wo posiive funcions a ( and b (, we wrie a ( b ( if lim a ( /b ( = 1. Furhermore, for wo posiive bivariae funcions a (, and b (,, we say ha he asympoic relaion a (, b (, holds uniformly over all in a nonempy se if lim sup a (, b (, 1 =. Tang (27 invesigaed he ail probabiliy of he sochasic process (1.2 and proved ha, if F ERV, hen he relaion Pr (D r ( > F (e rs dλ s (1.5 holds uniformly for all Λ. This formula ransparenly capures all sochasic informaion of he claim sizes and heir arrival imes. However, we poin ou ha he assumpion F ERV unforunaely ecludes some imporan disribuions such as lognormal and Weibull disribuions. In he cone of ruin heory, Tang (25 and Wang (27 obained some similar asympoic resuls as (1.5 for he finie-ime ruin probabiliy bu for a fied ime horizon Λ. Our goal in his paper is o eend he work of Tang (27 from he class ERV o he class S so ha lognormal and heavy-ailed Weibull disribuions are included. The class ERV enjoys some favorable properies like inequaliies (3.1 and (3.2 in Tang (27, which play a crucial role in esablishing he main resul of Tang (27, bu he class S does no possess such properies. Therefore, o achieve he desired eension we have o employ differen approaches. The res of his paper consiss of four secions: Secion 2 presens our main resuls and Secions 3, 4, and 5 prove hem, in urn, afer preparing some necessary lemmas. 3

4 2 Main Resuls The firs main resul of his paper is given below: Theorem 2.1 Consider he discouned aggregae claims described in relaion (1.2. If F S, hen relaion (1.5 holds uniformly for all Λ T = Λ [, T ] for arbirarily fied T Λ. In he ne wo main resuls below, we eend he se over which relaion (1.5 holds uniformly o he maimal se Λ. Theorem 2.2 Consider he discouned aggregae claims described in relaion (1.2. If F A and Pr (θ 1 > δ = 1 for some δ >, hen relaion (1.5 holds uniformly for all Λ. The echnical assumpion on he disribuion of θ 1 in Theorem 2.2, hough no nicelooking, causes no rouble for real applicaions since δ can be arbirarily close o. For a disribuion F on [, wih a finie posiive epecaion µ, denoe by F e ( = 1 µ F (sds,, is equilibrium disribuion funcion. Our hird main resul is given below: Theorem 2.3 Consider he discouned aggregae claims described in relaion (1.2, in which {N, } is a homogeneous Poisson process wih inensiy λ >. If F S and F e A, hen he relaion Pr (D r ( > λ holds uniformly for all (, ]. F (e rs ds (2.1 We remark ha he assumpions F S and F e A in Theorem 2.3 are saisfied by almos all useful heavy-ailed disribuions such as Pareo (wih finie epecaion, lognormal, and Weibull disribuions. Le us illusrae he usefulness of he uniformiy of (2.1. Denoe by τ( = inf{ : D r ( > }, >, he ime when D r ( up-crosses he level. Clearly, τ( is a defecive random variable wih oal mass Pr (τ( < = Pr (D r ( < 1. Le all condiions of Theorem 2.3 hold. We firs consider he asympoic behavior of he Laplace ransform of τ (. For every u >, use inegraion by pars and he ideniy Pr (τ( = Pr (D r ( > for all o ge Ee uτ( = u Pr (D r ( > e u d. 4

5 Subsiuing he uniform asympoic relaion (2.1 ino he above hen changing he order of inegrals, we have Ee uτ( λ e us F (e rs ds. This gives an eplici asympoic epression for he Laplace ransform of τ (. We hen consider he limiing disribuion of τ ( condiional on (τ ( <. For every fied >, by Theorem 2.3, Pr (τ ( τ ( < = Pr (D r ( > Pr (D r ( > F (ers ds F (e rs ds. (2.2 If F R α for some α >, hen using Theorem A3.2 of Embrechs e al. (1997 we see ha he convergence F (e rs F ( e αrs (2.3 holds uniformly for all s [,. Therefore, dividing boh inegrands on he righ-hand side of (2.2 by F ( hen plugging (2.3, we obain Pr (τ ( τ ( < 1 e αr, meaning ha he limiing disribuion under discussion is eponenial. 3 Proof of Theorem 2.1 Lemma 3.1 Le X k, 1 k n be n independen random variables, each X k disribued by F k. If here are n posiive consans l k, 1 k n, and a disribuion F S such ha F k ( l k F ( holds for k = 1,..., n, hen for arbirarily fied numbers a and b, < a b <, he relaion ( n n Pr c k X k > F k (/c k holds uniformly for all (c 1,..., c n [a, b] [a, b]. Proof. The proof can be given by going along he same lines of he proof of Proposiion 5.1 of Tang and Tsisiashvili (23 wih some obvious modificaions. Lemma 3.2 Consider he renewal couning process {N, } defined in (1.1. There eis some h > such ha Ee hn < holds for all. 5

6 Proof. I is shown in Sein (1946 ha, for arbirarily fied >, here eiss some h > such ha Ee hn <. For every, we can find a posiive ineger k such ha (k 1 < k. Inducively applying Lemma 2.2 of Cai and Kalashnikov (2, we can obain i.i.d. random variables N (1,..., N (k wih common disribuion as ha of N such ha N N k p k i=1 N (i + k 1, where for wo random variables X and Y, he relaion X p Y means ha Pr(X > Pr(Y > for all. Therefore, Ee hn <, as claimed. Proof of Theorem 2.1: Arbirarily choose some posiive ineger N. Clearly, for Λ T, ( N ( n Pr (D r ( > = + Pr X k e rτ k >, N = n n=1 n=n+1 = I 1 (,, N + I 2 (,, N. Firs consider I 2 (,, N. We have ( n I 2 (,, N Pr X k e rτ 1 >, τ n < τ n+1 = = n=n+1 n=n+1 n=n+1 n=n Pr ( n Pr X k e rs >, τ n τ 1 s < τ n+1 τ 1 Pr (τ 1 ds ( n Pr X k > e rs Pr (N s = n 1 Pr (τ 1 ds ( n+1 X k > e rs Pr (N s = n dλ s. Applying Lemma 1.3.5(c of Embrechs e al. (1997 o he above, for every ε > and some C ε >, I 2 (,, N C ε (1 + ε F (e rs E(1 + ε N s 1 (N s Ndλ s C ε (1 + ε E(1 + ε N T 1 (NT N F (e rs dλ s. By Lemma 3.2, we can choose some ε sufficienly small such ha E(1 + ε N T ha E(1 + ε N T 1 (NT N as N. Therefore, for all >, <. I follows lim sup I 2 (,, N N Λ F =. (3.1 T (ers dλ s 6

7 Ne consider I 1 (,, N. Using Lemma 3.1, i holds uniformly for all Λ T ha ( n n I 1 (,, N Pr ( X k e rτ k >, N = n n=1 n=n+1 = I 11 (, I 12 (,, N. Clearly, for all Λ T, I 11 (, = Pr ( X k e rτ k >, N k = F (e rs dλ s. (3.2 For I 12 (,, N, similarly o he derivaion for I 2 (,, N, we have I 12 (,, N I follows ha, for all >, n=n+1 n+1 n=n n Pr ( X k e rτ 1 >, N = n F (e rs dλ s F (e rs Pr (N s = n dλ s n=n (n + 1 Pr (N T n. lim sup I 12 (,, N N Λ F =. (3.3 T (ers dλ s From (3.1, (3.2, and (3.3 we conclude ha he asympoic relaion (1.5 holds uniformly for all Λ T. 4 Proof of Theorem 2.2 Lemma 4.1 If a disribuion F on [, saisfies (1.4 for some v > 1, hen here are posiive consans p, C, and such ha he inequaliy holds uniformly for y. F (y F ( Cy p (4.1 Proof. This is a resaemen of Proposiion of Bingham e al. (1989. See also (2.3 in Chen e al. (25. 7

8 Lemma 4.2 If a disribuion F on [, saisfies (1.4 for some v > 1, hen lim lim sup F (e rs dλ s F (ers dλ s =, (4.2 where he posiive consan r and he renewal funcion λ s, s, are he same as inroduced in Secion 1. Proof. For every Λ, apply inequaliy (4.1 o obain ha, for, F (e rs dλ s F = (ers dλ s This implies (4.2. F (e rs /F (e r dλ s F C 2 (ers /F (e r dλ s e pr(s dλ s = C 2 epr( s dλ s e prs dλ s. e prs dλ s Lemma 4.3 Under he condiions of Theorem 2.2, we have Pr (D r ( > k=n+1 F (e rs dλ s. (4.3 Proof. Arbirarily choose some posiive ineger N such ha Nδ Λ. Since Pr (θ 1 > δ = 1, we have ( N ( Pr (D r ( > Pr X k e rτ k + X k e r(k Nδ e rτ N >. (4.4 Wrie Σ δ = k=n+1 X ke r(k Nδ, whose disribuion does no depend on N. Corollary 3.1 of Chen e al. (25, ( Pr (Σ δ > = Pr X k e rkδ F ( e rkδ > F (. F ( Applying Hence, by inequaliy (4.1, here is some consan C > such ha Pr (Σ δ > C F ( for all [,. Ne we come back o (4.4. Inroduce a new random variable Σ δ independen of {X k, k = 1, 2,...} and {τ k, k = 1, 2,...} wih a ail saisfying Pr ( Σδ > = min { C F (, 1 },. Therefore, Σ δ p Σδ, and ( N Pr (D r ( > Pr X k e rτ k + Σ δ e rτ N >. (4.5 8

9 To apply Lemma 3.1, we choose some M 1 > and derive ( N ( N Pr X k e rτ k + Σ δ e rτ N > = Pr X k e rτ k + Σ N δ e rτ N >, (θ i M 1 i=1 ( N + Pr X k e rτ k + Σ N δ e rτ N >, (θ i < M 1 = J 1 (, N, M 1 + J 2 (, N, M 1. (4.6 i=1 I is easy o prove by inducion on N ha ( N J 1 (, N, M 1 Pr X k e rτ k + Σ δ e rτ N ( N > Pr (θ i M 1. (4.7 i=1 Subsiuing (4.7 ino (4.6 and rearranging he resuling inequaliy, we have ( N Pr X k e rτ k + Σ δ e rτ N J 2 (, N, M 1 > ( N. 1 Pr i=1 (θ i M 1 Furher subsiuing his ino (4.5, applying Lemma 3.1 o J 2 (, N, M 1, and leing M 1, we obain ha Pr (D r ( > N Pr ( X k e rτ k > + Pr ( Σδ e rτ N > Pr ( X k e rτ k > + Nδ Pr ( Σδ > e rs Pr (τ N ds F (e rs dλ s + C F (e rs Pr (τ N ds. (4.8 Apply inequaliy (4.1 again o obain ha, for some M 2 Λ (, Nδ] and all large, F Nδ (ers Pr (τ N ds CF ( e rm 2 Ee pr(τ N M 2 F (e rs dλ M2 C Ee pr(τ N M 2, (4.9 s F (e rs dλ s λ M2 as N. From (4.8 and (4.9, he asympoic relaion (4.3 follows immediaely. Proof of Theorem 2.2: According o Lemma 4.2, for every ε > here eiss some T > such ha he inequaliy T Nδ T F (e rs dλ s ε F (e rs dλ s (4.1 9

10 holds for all large. By Theorem 2.1 and inequaliy (4.1, i holds uniformly for all (T, ] ha Pr (D r ( > Pr (D r (T > T F (e rs dλ s ( F (e rs dλ s (1 ε T F (e rs dλ s. Likewise, by Lemma 4.3 and inequaliy (4.1, i holds uniformly for all (T, ] ha Hence, for all (T, ] and all large, (1 2ε Pr (D r ( > Pr (D r ( > F (e rs dλ s ( + F (e rs dλ s (1 + ε T F (e rs dλ s. F (e rs dλ s Pr (D r ( > (1 + 2ε F (e rs dλ s. (4.11 By Theorem 2.1 again, he inequaliies in (4.11 also hold for all Λ T (hence for all Λ and all large. As ε > is arbirary, we complee he proof. 5 Proof of Theorem 2.3 Konsaninides e al. (22 invesigaed he asympoic behavior of he ruin probabiliy of he compound Poisson model. In heir model, he surplus process is epressed as S r ( = e r + c e r( s ds X k e r( τk 1 (τk,, where is he iniial surplus, c > is he consan rae a which he premiums are coninuously colleced, and {X k, k = 1, 2,...}, {τ k, k = 1, 2,...}, and r are he same as appearing in relaion (1.2. The couning process {N, } generaed by {τ k, k = 1, 2,...} is a homogeneous Poisson process wih inensiy λ >. The ruin probabiliy is defined as ( ψ r ( = Pr inf S r ( <. << 1

11 Theorem 2.1 of Konsaninides e al. (22 shows ha, if F e A, hen ψ r ( λ r Based on relaion (5.1 we produce he following resul: F (y dy. (5.1 y Lemma 5.1 Consider he discouned aggregae claims described in relaion (1.2, in which {N, } is a homogeneous Poisson process wih inensiy λ >. If F e A, hen Pr (D r ( > λ r Proof. In erms of he model of Konsaninides e al. (22, ( ( ψ r ( = Pr D r ( c e rs ds I follows ha By (5.1 and inegraion by pars, ψ r ( µλ ( Fe ( r sup << F (y dy. (5.2 y >. ψ r ( Pr (D r ( > ψ r ( c/r. (5.3 Changing ino c/r in he above yields ha ψ r ( c/r µλ ( Fe ( c/r F e (y dy r c/r y 2 Since F e A L, F e (y dy = µλ y 2 r (K 11 ( K 12 (. c/r K 11 ( K 21 (, K 12 ( K 22 (. In order o infer ψ r ( ψ r ( c/r, i suffices o show ha lim sup K 12 ( K 11 ( = µλ r (K 21 ( K 22 (. < 1. (5.4 Since F e A, here eis some ε, < ε < 1, such ha F e (v /F e ( 1 ε holds for all large. Hence, for all large, K 12 ( K 11 ( = n=1 v n F e (y v n 1 F e ( y dy 2 n=1 v n (1 ε n 1 n=1 v n 1 v n F e (v n 1 v n 1 F e ( y dy = v 1 2 v 1 + ε. y dy 2 This proves (5.4. Therefore by (5.1 and (5.3, relaion (5.2 follows immediaely. 11

12 Lemma 5.2 For a disribuion F on [, wih a finie posiive epecaion, if relaion (1.4 wih F replaced by F e holds for some v > 1, hen Proof. Clearly, lim lim sup F (e rs ds F (e rs ds =. (5.5 F (e rs ds F (e rs ds = e r 1 y df e (y 1 df y e (y = F e(e r e r F e( dy F e(y dy. y 2 F e(y e r y 2 By (5.4, here is some consan C > such ha, uniformly for all >, Therefore, (5.5 holds. F (e rs ds F (e rs ds C F e(e r e r F e( C e r. Proof of Theorem 2.3: The proof can be given by copying he proof of Theorem 2.2 wih he only modificaion ha we use Lemmas 5.1 and 5.2 insead of Lemmas 4.2 and 4.3. References [1] Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular Variaion. Cambridge Universiy Press, Cambridge, [2] Cai, J.; Kalashnikov, V. NWU propery of a class of random sums. J. Appl. Probab. 37 (2, no. 1, [3] Chen, Y.; Ng, K. W.; Tang, Q. Weighed sums of subeponenial random variables and heir maima. Adv. in Appl. Probab. 37 (25, no. 2, [4] Dufresne, D. The disribuion of a perpeuiy, wih applicaions o risk heory and pension funding. Scand. Acuar. J. 199, no. 1-2, [5] Embrechs, P.; Klüppelberg, C.; Mikosch, T. Modelling Eremal Evens for Insurance and Finance. Springer-Verlag, Berlin, [6] Gjessing, H. K.; Paulsen, J. Presen value disribuions wih applicaions o ruin heory and sochasic equaions. Sochasic Process. Appl. 71 (1997, no. 1, [7] Konsaninides, D.; Tang, Q.; Tsisiashvili, G. Esimaes for he ruin probabiliy in he classical risk model wih consan ineres force in he presence of heavy ails. Insurance Mah. Econom. 31 (22, no. 3,

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