Heavy Tails of Discounted Aggregate Claims in the Continuous-time Renewal Model

Transcription

1 Heavy Tails of Discouned Aggregae Claims in he Coninuous-ime Renewal Model Qihe Tang Deparmen of Saisics and Acuarial Science The Universiy of Iowa 24 Schae er Hall, Iowa Ciy, IA 52242, USA February 8, 27 Absrac We sudy he ail behavior of discouned aggregae claims in a coninuous-ime renewal model. For he case of Pareo-ype claims, we esablish a ail asympoic formula, which holds uniformly in ime. Keywords: Asympoics, exended regular variaion, renewal process, uniformiy. Inroducion and he Main Resul Consider a coninuous-ime renewal model, in which claim sizes X k, k = ; 2; : : :, consiue a sequence of independen, idenically disribued (i.i.d.), and nonnegaive random variables wih common disribuion F, while heir arrival imes k, k = ; 2; : : :, consiue a renewal couning process N = # fk = ; 2; : : : : k g ; : (.) We assume ha fx k ; k = ; 2; : : :g and fn ; g are muually independen. To avoid rivialiy, we menion ha X and are no degenerae a. We allow o possibly have a posiive probabiliy a no for pracical usefulness bu for heoreical compleeness. Suppose ha here is a consan ineres force >. Tha is o say, afer ime one dollar becomes e dollars. Then, he aggregae claims form ino a sochasic process of he form A () = X k e ( k) ( k ); ; where for an even E he symbol E denoes is indicaor funcion. Since A () almos surely as, we insead sudy he ail behavior of he discouned process D () = X k e k ( k ); : (.2)

2 We shall derive for he ail probabiliy of D (),, an asympoic formula, which holds uniformly for all for which he renewal funcion = EN = Pr ( k ) is posiive. For his purpose, de ne = f : > g. Wih = inff : > g = inff : Pr ( ) > g, i is clear ha [; ] if Pr ( = = ) > (.3) (; ] if Pr ( = ) =. We shall assume ha he disribuion F on [; ) is exended-regularly-varying ailed, hence heavy ailed. Tha is o say, F (x) = F (x) > holds for all x and here are some consans and, < <, such ha v lim inf x F (vx) F (x) lim sup x F (vx) F (x) v for all v : (.4) We use F 2 ERV( ; ) o signify he regulariy propery in (.4). The class ERV is he union of all classes ERV( ; ) over he range < <. This class has been used o he sudy of precise large deviaions by many people since he work of Klüppelberg and Mikosch (997). I is well known ha ERV is a subclass of he class S of subexponenial disribuions; see Theorem of Goldie (978). The subexponenialiy of a disribuion F is characerized by he relaions F (x) > for all x and F F (x) lim x F (x) Clearly, he class ERV covers he famous class R of disribuions wih regularly-varying ails characerized by he relaions F (x) > for all x and = 2: F (vx) lim x F (x) = v for some > and all v : (.5) I is usually easier o handle disribuions from he class R because of he well-developed Karamaa heory. Alhough he class ERV is marginally larger han he class R, we expec ha asympoic resuls for he ERV case provide more insigh o he sudy in he subexponenial case. For more deails of heavy-ailed disribuions, he reader is referred o Bingham e al. (987) and Embrechs e al. (997). Hereafer, all limi relaionships are for x unless saed oherwise. For wo posiive funcions a() and b(), we wrie a(x) b(x) if lim a(x)=b(x) =. Furhermore, for wo posiive bivariae funcions a(; ) and b(; ), we say ha he asympoic relaion a(x; ) b(x; ) holds uniformly over all in a nonempy se if lim sup a(x; ) x b(x; ) = : 2 2

3 Asympoic formulae ha hold wih such a uniformiy feaure are usually of higher heoreical and pracical ineress. Recall (.2) and (.3). Our main resul is given below: Theorem.. Consider he renewal model inroduced above. If F 2 ERV, hen he relaion holds uniformly for all 2. 2 Some Remarks Pr (D () > x) F xe s d s (.6) Remark 2.. When =, he sum D () reduces o D () = X k e k : (2.) For F 2 ERV( ; ), from inequaliy (3.) below wih x xed, we see ha F (y) = O(y ) for all, < <. Hence, EX < for all, < <. Using his fac we may furher verify ha E (D ()) ^ <. This means ha D () converges almos surely as. Likewise, using he fac EX < and he elemenary renewal heorem, i is easy o verify ha R F e s d s <, irrespecive of wheher or no has a nie mean. Therefore, boh sides of (.6) are well de ned. Remark 2.2. Suppose ha premiums are colleced coninuously a a consan rae c >. Then, he surplus process is S () = xe + c e ( s) ds A () ; ; where x denoes he iniial surplus. De ne he probabiliy of ruin by ime as he probabiliy ha he surplus process ever becomes negaive by ime. Denoe his probabiliy by (x; ). The limi (x; ) = lim (x; ) represens he probabiliy of ulimae ruin. Alhough he pracical relevance of ruin probabiliies is quesionable, hey do provide a good risk measure for insurance business. Clearly, for all 2, Hence, (x; ) = Pr inf S (v) < <v = Pr sup <v D (v) Z v c e s ds > x : (x; ) Pr (D () > x) and (x; ) Pr (D () > x + c=) : Noe ha, by (.4), F (x + c=)e s F xe s holds uniformly for all s 2 [; ). Applying Theorem., we immediaely obain ha he relaion (x; ) 3 F xe s d s (2.2)

4 holds uniformly for all 2. Klüppelberg and Sadmüller (998) rs obained a resul similar o (2.2) wih = for he special case ha fn ; g is a homogeneous Poisson process and F belongs o he class R. For his special case, Tang (25) obained he uniformiy of (2.2). Recenly, Chen and Ng (27) exended he asympoic relaion (2.2) wih = o he case of negaively dependen claims. Remark 2.3. We saed Theorem. in erms of he renewal model, where he innovaions X k, k = ; 2; : : :, denoe claim sizes and hence are nonnegaive. However, in mos siuaions considered in pracice, hese innovaions appearing in (.2) could be real valued. For his more general case, we may insead sudy he ail behavior of he running maximum process We show ha he asympoic formula fd () := sup D (s); : s Pr fd () > x F xe s d s (2.3) holds uniformly for all 2 as long as he righ ail of F is sill exended regularly varying as described in (.4). Acually, for his case, he proof given in Secion 4 unil (4.3) is valid for boh D () and D + () := P X+ k e k ( k ),. Since for all, D () f D () D + (); we see ha (2.3) holds uniformly for all 2 \ [; T ] for an arbirarily xed number T 2. The remaining proof of he uniformiy on of (2.3) can be given by simply copying he par afer (4.3) of he proof of Theorem. wih all D replaced by D f. Remark 2.4. Relaion (.6) unforunaely involves he renewal funcion,, so do relaions (2.2) and (2.3). If he i.i.d. iner-arrival imes have a common nie mean E = =, hen as by he elemenary renewal heorem. This emps us o consider o replace s in (.6) by s. However, his is no feasible in general. Acually, under he condiion F 2 ERV, he di erenial d s in he inegral is on an equal fooing. Thus, we can no ignore an inegral par in he righ neighborhood of. If fn ; g is a homogeneous Poisson process wih inensiy >, hen s = s for all s >. Oher cases where he explici form of he renewal funcion s is available can be found in he lieraure. For example, le have a phase-ype disribuion (of which he Erlang disribuion is a special case) wih densiy given by g(s) = e Ts ; s ; where is a row vecor, T is a marix, and = T wih = (; : : : ; ). The vecor T and he marix T should be chosen such ha (; ) is he iniial disribuion and 4

5 is he inensiy marix of a coninuous-ime Markov jump process wih nie sae space in which one sae is absorbing and he ohers are ransien. In his case, he derivaive of he renewal funcion s, called he renewal densiy, is given by d s ds = e(t+)s : As anoher example, le have a uniform disribuion on (; a). Then, he renewal densiy is given by d s ds = X a es=a e k (k s=a)k : k k: ks=a These formulae are copied from pages 88 and 48 of Asmussen (23). Remark 2.5. We now propose a resul in a general siuaion, in which he renewal funcion s can indeed be simpli ed o s. Consider he discouned process (.2). For any x, denoe by T x = inf f : D () > xg he rs ime when D () up-crosses he level x, where inf? = by convenion. The following is a corollary of Theorem., giving explici approximaions for he ail probabiliy of T x. Corollary 2.. In addiion o he assumpions of Theorem., we assume ha he i.i.d. iner-arrival imes have a non-laice disribuion and a nie mean E = =. Then, Pr ( < T x < ) lim lim sup x R s ) ds = lim lim inf x Pr ( < T x < ) R s ) ds = : (2.4) If F 2 R as de ned in (.5) wih some >, hen relaion (2.4) can be srenghened o lim lim Pr ( < T x < ) x e F (x) = : (2.5) The proof of Corollary 2. is lef o Secion 4. From he proof one sees ha he same resul holds for he case discussed in Remark Lemmas Lemma 3.. Le F 2 ERV( ; ). Then for any and, < <, < <, here are posiive consans c i and d i, i = ; 2, such ha he inequaliy F (y) F (x) c (y=x) (3.) 5

6 holds whenever y x d, and ha he inequaliy F (y) F (x) c 2 (y=x) (3.2) holds whenever y x d 2. Proof. This lemma is a consequence of Proposiion 2.2. of Bingham e al. (987). Acually, wih f = =F we see ha (3.) and (3.2) above are, respecively, (2.2. ) and (2.2.) of Bingham e al. (987). Lemma 3.2. Consider he renewal process fn ; g de ned in (.). I holds for all T 2 and all v > ha lim sup EN v (N>x) = : x 2\[;T ] Proof. Follow he proof of Lemma 5.3 of Tang (24) wih sligh modi caions. Lemma 3.3. Le fx ; X 2 ; : : : ; X n g be n i.i.d. random variables wih common disribuion F 2 S. Then for arbirarily xed numbers a and b, < a b <, he relaion Pr c k X k > x F (x=c k ) holds uniformly for all (c ; : : : ; c n ) 2 [a; b] [a; b]. Proof. See Proposiion 5. of Tang and Tsisiashvili (23). 4 Proofs 4. Proof of Theorem. To be more precise, we assume F 2 ERV( ; ). In he rs half of his subsecion, we prove ha relaion (.6) holds uniformly for all 2 \ [; T ] for an arbirarily xed number T 2. We spli he probabiliy Pr (D () > x) ino wo pars as mx Pr (D () > x) = + Pr X k e k > x; N = n := I + I 2 ; n=m+ n= where m is a emporarily xed ineger. Firs we deal wih I. Recall ha F 2 ERV( ; ). As menioned in he proof of Lemma 4.5 of Tang (25), using a resul of Nagaev (979) we may prove ha for an arbirarily xed number v >, here is some c v > such ha for all n = ; 2; : : : and all x, Pr X k > x c v n v F (x) : 6

7 Therefore, I Pr n=m+ X k > x Pr (N = n) c v F (x) EN v (N>m): By inequaliy (3.2), for some > and all x d 2, Hence by Lemma 3.2, for all x d 2, F (x) c 2 e T F xe T : lim inf m sup 2\[;T ] I R s ) d s c v T e c 2 lim inf sup F m 2\[;T ] xe T EN v (N>m) T ) = : (4.) We urn o I 2. Under he condiion N = n, all k appearing in I 2 are no larger han T. Using Lemma 3.3, i holds uniformly for all 2 \ [; T ] ha mx I 2 = Pr X k e k > x N = n Pr (N = n) = Clearly, n= mx n= Pr X k e k > x N = n Pr (N = n) Pr X k e k > x; N = n := I 2 I 22 : n= n=m+ I 2 = = Pr X k e k > x; N = n n=k Pr X k e k > x; k = F xe s d s : Noe ha I 22 F (x) n=m+ Hence, similar o he proof of (4.), for all x d 2, lim sup m 2\[;T ] n Pr (N = n) : I 22 R s ) d s = : We conclude ha he asympoic relaion (.6) holds uniformly for all 2 \ [; T ]. 7

8 In he second half of his subsecion, we exend he uniformiy of (.6) o he whole inerval. For arbirarily xed numbers and, < <, < <, again by inequaliies (3.) and (3.2), i holds for all x maxfd ; d 2 g and all 2 [; ) ha R F xe s d s R = s ) d s R R F(xe s ) d F (x) s F(xe s ) F (x) c R e s d s R : (4.2) d c 2 e s s d s The righ-hand side of he above ends o as. Therefore, for any " >, here is some T 2 such ha he inequaliy Z F xe s Z T d s " F xe s d s (4.3) T holds for all x maxfd ; d 2 g. Recall (2.). Using Theorem 3. of Tang and Tsisiashvili (24), we obain ha Pr (D () > x) Pr X k e k > x = Z F xe s d s : (4.4) Hence, relaion (.6) holds for =. We are ready o exend he uniformiy of (.6) o he whole inerval. On he one hand, i holds uniformly for all 2 (T ; ] ha Z T Pr (D () > x) Pr (D (T ) > x) F xe s d s Z F xe s d s ( ") T F xe s d s ; where in he second sep we used relaion (.6) wih replaced by T, while in he las sep we used (4.3). On he oher hand, likewise, i holds uniformly for all 2 (T ; ] ha Z Pr (D () > x) Pr (D () > x) F xe s d s Z + F xe s d s ( + ") T F xe s d s ; where in he second sep we used relaion (4.4), while in he las sep we used (4.3). Hence, i holds for all 2 (T ; ] and all large x, say x > x >, ha ( 2") F xe s d s Pr (D () > x) ( + 2") F xe s d s : (4.5) From he rs half of his proof we see ha (4.5) sill holds for all 2 \ [; T ] and all large x, say x > x 2 >. Therefore, (4.5) holds for all 2 and all x > maxfx ; x 2 g. Since " > is arbirary, we have obained he uniformiy of relaion (.6) over all 2. 8

9 4.2 Proof of Corollary 2. Since every rajecory of D () is piecewise consan wih only upward jumps, we have Pr (T x ) = Pr (D () > x) for all 2 \ [; ) and Pr (T x < ) = Pr (D () > x). Hence by Theorem., for all 2 \ [; ), Pr ( < T x < ) = Pr (D () > x) Pr (D () > x) Z F xe s d s F xe s d s = Z F xe s d s ; (4.6) where in he second sep we used he asympoic relaion (.6). There is no problem wih his sep because, wih arbirarily xed, similar o (4.2), lim inf x R F xe s d s R + c 2 s ) d s R e s d s > : e s d s c R For any ` >, by he well-known Blackwell renewal heorem, lim ( s+` s ) = `: s I follows ha, for any " > and all large s, say s > s = s ("; `) >, Therefore, for all x 2 [; ) and > s, Z F xe s d s = ( ")` s+` s ( + ")`: +k` ( + ") F xe s d s +(k )` +(k )` +(k 2)` F F xe [+(k )`] ( + ")` xe s ds = ( + ") Z F xe (s `) ds: Using he de niion in (.4), i holds for all x 2 [; ) and all large s, say s > s 2 >, ha F xe (s `) ( + ")e `F xe s : I follows ha, for all x 2 [; ) and > maxfs ; s 2 g, Z F xe s d s ( + ") 2 e ` Z F xe s ds: A similar lower bound for he inegral R F xe s d s can also be esablished. Hence by he arbirariness of he consans ` and ", he relaion Z F xe s d s Z F xe s ds; ; (4.7) 9

10 holds uniformly for all x 2 [; ). Clearly, he uniformiy of relaion (4.7) indicaes ha R F xe s d s lim lim sup x R From (4.6) and (4.8) we have Pr ( < T x < ) lim lim sup x R s ) ds = lim s ) ds = lim lim x lim inf x R F xe s d s R s ) ds Pr ( < T x < ) R lim sup s ) d s x = : (4.8) R F xe s d s R s ) ds The derivaion above wih lim sup replaced by lim inf is sill valid. This proves (2.4). Likewise, when F 2 R wih some >, R lim lim Pr ( < T x < ) Pr ( < T x < ) F xe s d s x e = lim lim R lim F (x) x s ) d s x e F (x) Z F xe s = lim e lim d s x F (x) Z = lim e s d s e = lim e Z e s ds where in he second sep we used (4.6), in he hird sep we used he dominaed convergence heorem jusi ed by (3.) and (.5), and in he fourh sep we used he argumen of he Blackwell renewal heorem as we did in proving (4.7). This proves (2.5). Acknowledgmens. The auhor wishes o hank he referee for his/her very helpful commens. The suppor of he Old Gold Summer Fellowship from he Universiy of Iowa is acknowledged. References [] Asmussen, S. Applied probabiliy and queues. Second ediion. Springer-Verlag, New York, 23. [2] Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variaion. Cambridge Universiy Press, Cambridge, 987. [3] Chen, Y.; Ng, K. W. The ruin probabiliy of he renewal model wih consan ineres force and negaively dependen heavy-ailed claims. Insurance Mah. Econom. (27), o appear. [4] Embrechs, P.; Klüppelberg, C.; Mikosch, T. Modelling exremal evens for insurance and nance. Springer-Verlag, Berlin, 997. = ; = :

11 [5] Goldie, C. M. Subexponenial disribuions and dominaed-variaion ails. J. Appl. Probabiliy 5 (978), no. 2, [6] Klüppelberg, C.; Mikosch, T. Large deviaions of heavy-ailed random sums wih applicaions in insurance and nance. J. Appl. Probab. 34 (997), no. 2, [7] Klüppelberg, C.; Sadmüller, U. Ruin probabiliies in he presence of heavy-ails and ineres raes. Scand. Acuar. J. 998, no., [8] Nagaev, S. V. Large deviaions of sums of independen random variables. Ann. Probab. 7 (979), no. 5, [9] Tang, Q. Uniform esimaes for he ail probabiliy of maxima over nie horizons wih subexponenial ails. Probab. Engrg. Inform. Sci. 8 (24), no., [] Tang, Q. The nie-ime ruin probabiliy of he compound Poisson model wih consan ineres force. J. Appl. Probab. 42 (25), no. 3, [] Tang, Q.; Tsisiashvili, G. Randomly weighed sums of subexponenial random variables wih applicaion o ruin heory. Exremes 6 (23), no. 3, [2] Tang, Q.; Tsisiashvili, G. Finie- and in nie-ime ruin probabiliies in he presence of sochasic reurns on invesmens. Adv. in Appl. Probab. 36 (24), no. 4,