Insurance: Mathematics and Economics. Tail bounds for the distribution of the deficit in the renewal risk model
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1 Insurance: Mahemaics and Economics 43 (8 97 Conens liss available a ScienceDirec Insurance: Mahemaics and Economics journal homepage: Tail bounds for he disribuion of he defici in he renewal risk model Georgios Psarrakos Deparmen of Saisics and Acuarial Science, Universi of he Aegean, 83 Samos, Greece a r i c l e i n f o a b s r a c Aricle hisor: Received Ocober 7 Received in revised form Ma 8 Acceped Ma 8 Kewords: Probabili of ruin Defici a ruin Renewal equaion Failure rae DFR IFR Adjusmen coefficien Lundberg condiion Sop-loss premium We obain upper and lower bounds for he ail of he defici a ruin in he renewal risk model, which are (i applicable generall; and (ii based on reliabili classificaions. We also derive wo-side bounds, in he general case where a funcion saisfies a defecive renewal equaion, and we appl hem o he renewal model, using he funcion Λ u inroduced b [Psarrakos, G., Poliis, K., 7. A generalisaion of he Lundberg condiion in he Sparre Andersen model and some applicaions (submied for publicaion]. Finall, we consruc an upper bound for he inegraed funcion Λ u (z dz and an asmpoic resul when he adjusmen coefficien exiss. 8 Elsevier B.V. All righs reserved.. Inroducion We consider he general renewal risk model, ofen referred o as he Sparre Andersen risk model. In his model, he insurer s surplus a ime, which is denoed b U(, is given b N U( = u + c Y k, ( k= where u is he iniial surplus, c is he rae of premium income per uni ime and N is he number of claims in he ime inerval (, ]. The individual claim amouns Y, Y,... are posiive, independen and idenicall disribued (i.i.d. random variables wih common disribuion funcion (d.f. P( = Pr(Y, ail P( = P( = Pr(Y >, densi p( and mean E(Y <. These claim amouns are also independen of N. We assume ha a claim has aken place a ime, wih u being he surplus immediael afer his claim has been paid (see Gerber and Shiu (5. The corresponding inerclaim imes T, T,... are arbirar i.i.d. posiive random variables wih common mean E(T. We assume ha c = ( + θ E(Y /E(T, where θ > is known as he relaive safe loading. The probabili of ulimae ruin is defined b ( ψ(u = Pr inf U( < U( = u, u. ( > address: gpsarr@aegean.gr. The ime of ruin is {, if U( for all > T = inf{ > U( < }, oherwise, and he probabili of ruin is wrien as ψ(u = Pr(T < U( = u. In general, we assume ha E(Y < c E(T, so ha ruin is no cerain o occur. The disribuion of he defici, namel, H(u, = Pr( U(T, T < U( = u, was inroduced b Gerber e al. (987 and represens he probabili ha, saring wih a surplus u, ruin occurs and he defici U(T a he ime of ruin T does no exceed. I is a defecive d.f. wih ail H(u, = ψ(u H(u, = Pr( U(T >, T < U( = u, and saisfies lim H(u, = H(u, = ψ(u <. I is also convenien o define he proper d.f. of he defici, H u ( = H(u, ψ(u = Pr( U(T T <, U( = u (3 wih ail H u ( = H u (. The main purpose of his aricle is o obain improved bounds for he ail of he defici, H(u,. Recenl, Willmo ( and Chadjiconsaninidis and Poliis (7 gave bounds for his /$ see fron maer 8 Elsevier B.V. All righs reserved. doi:.6/j.insmaheco.8.5.4
2 98 G. Psarrakos / Insurance: Mahemaics and Economics 43 (8 97 funcion. In Secion 3, we consruc also lower and upper bounds for H(u,. For his sud, we need some reliabili classificaions, which are briefl reviewed in he nex secion. In Secion 4, we derive wo-sided bounds for a funcion saisfing a defecive renewal equaion, improving and generalising a resul obained b Willmo and Lin (. The ke for our analsis is he funcion Λ u ( = [ψ(u + ψ(uψ(]/(, where = ψ(, inroduced b Psarrakos and Poliis (7. Finall, in Secion 5, we obain an upper bound for he inegraed funcion Λ u (z dz and derive an asmpoic resul in he case where he adjusmen coefficien exiss.. Definiions and preliminaries Le S = N i= X i denoe a compound geomeric random variable, where Pr(N = n = ( n for n =,,,..., and < <. Suppose also ha X, X,... are i.i.d. wih ladder heigh d.f. F and densi f. Then we have ha Pr(S > u = ψ(u, see Bowers e al. (986, Chaper. Furhermore, if E(X = µ <, hen E(S = E(X E(N = µ/(. One expression of he probabili of ruin is he formula of Pollaczeck and Khinchine, see Asmussen (, ψ(u = ( n F n (u, (4 n= where = ψ(. The probabili of ruin also saisfies he defecive renewal equaion, see Willmo and Lin (, ψ(u = ψ(u z df(z + F(u. (5 The soluion of his equaion is ψ(u = where H(u = F(u z dh(z + F(u, (6 + ( n F n (u (7 n= is he probabili of non-ruin, and H(u = H(u = ( n F n (u = ψ(u. n= B Willmo (, we know ha H(u, saisfies he defecive renewal equaion H(u, = whose soluion is H(u, = H(u, df( + F(u +, (8 F(u + dh( + F(u +. (9 + One can easil see ha for =, he relaions (8 and (9 ield (5 and (6, respecivel. A d.f. A(x, x, wih ail A(x = A(x is said o be decreasing (increasing failure rae or DFR (IFR if A(x + /A(x is nondecreasing (nonincreasing in x for an. If A(x is absoluel coninuous wih densi a(x, hen i is DFR (IFR when he failure rae λ A (x = a(x/a(x is nonincreasing (nondecreasing. The exponenial is he onl disribuion ha is boh DFR and IFR, and a mixure of exponenial d.f. is alwas DFR, see Willmo and Lin (, Chaper. A d.f. A(x is called new worse (beer han used or NWU (NBU if A(x + ( A(xA( for ever x,. The DFR (IFR class of disribuions is a subclass of he NWU (NBU class. 3. General bounds In he conex of he renewal risk model, an upper bound for he ail of he defici a ruin, H(u,, in erms of he probabili of ruin was given b Willmo (, Theorem3., who proved ha H(u, [ψ(u + ψ(uψ(]. ( Chadjiconsaninidis and Poliis (7, using he upper bound in (, derive a beer upper bound, ha is H(u, [ψ(u + ψ(u ψ(] [ ψ(][ ψ(u]f(u +. ( ( In he following heorem, we give an upper bound for H(u, ha is igher han he bound (. I is also worh menioning ha he new bound has a simple form similar o (. Theorem 3.. For an u,, i holds ha H(u, [ψ(u + ψ(u ψ(]. ( ψ( This bound is alwas beer han he bound in (. Proof. B Willmo (, relaion (., for an u,, we have ψ(u + ψ(uψ( = H(u, dh(. (3 Since he funcion H(u, is nondecreasing in [, ], (3 ields ψ(u + ψ(uψ( H(u, dh( = H(u, [ ψ(], and afer a lile rearrangemen, he upper bound in ( follows. Nex we verif ha bound ( is alwas igher han bound (. B Proposiion. of Chadjiconsaninidis and Poliis (7 and (, we have [ ψ(u]f(u + [ψ(u + ψ(uψ(]. ψ( Mulipling boh sides of his inequali b [ ψ(]/(, i follows [ ψ(][ ψ(u]f(u + ( ψ( [ψ(u + ψ(uψ(], ( ( ψ( or equivalenl, [ ψ(][ ψ(u]f(u + ( [ ] [ψ(u + ψ(uψ(]. ψ( I is sraighforward o see ha [ψ(u + ψ(uψ(] ψ( [ψ(u + ψ(uψ(] [ ψ(][ ψ(u]f(u +, ( and he proof is complee.
3 G. Psarrakos / Insurance: Mahemaics and Economics 43 ( Observe ha for = in (, we obain he lower bound for he probabili of ruin given b ψ(u F(u F(u. (4 This bound was originall proved b De Vlder and Goovaers (984. B Willmo (, Theorem 3., we know ha if he ladder heigh d.f. F is NWU (NBU, hen a lower (upper bound for he ail of he defici H(u, is H(u, ( F(ψ(u. (5 Nex we improve he bounds given in (5. Proposiion 3.. If he ladder heigh d.f. F is NWU (NBU, hen H(u, ( F(ψ(u F(F(u + F(u +. (6 Proof. Le F be NWU. Then, b (9 we see ha H(u, F( + F(u dh( + F(u +. Appling (6 a he righ side of he las equaion, he lower bound in (6 follows. In he case where F is NBU, he proof is similar. 4. Bounds for defecive renewal equaions In his secion, we derive general bounds for a funcion ha saisfies a defecive renewal equaion. Le B(x be a d.f. wih B( and densi b(x, and le m(x saisf he defecive renewal equaion m(x = m(x db( + υ(x, (7 where υ(x is a coninuous funcion on [, and (,. Equaions of he form (7 arise repeaedl in insurance risk heor and oher areas of applicaion of probabili heor; see Willmo and Lin (, Chaper 9, and references herein. The soluion of he Eq. (7 is given b m(x = υ(x dg( + υ(x, (8 + where G( is he compound geomeric d.f. defined b G( = ( n B n (, n= wih ail G( = ( n B n (. n= Noe ha G( has mass poin G( = a and does no involve he funcion υ(x. B Willmo and Lin (, Corollar 9.., if υ(x ( kb(x for an x and k >, hen m(x ( kg(x/, x. (9 The nex lemma is a special case of a resul in Aposol (974, p. 77, and i is a ke ingredien for our discussion. The proof is similar o he one given b Cai and Garrido (998, Lemma 3. For his reason we omi he proof. Lemma 4.. Le f, f be wo coninuous funcions in he inerval (a, a a wih differen (same monoonici and f a (x dk(x < a, f a (x dk(x < where K is a d.f. Then a a f (x f (x dk(x ( K(a K(a a a a f (x dk(x a f (x dk(x. ( Theorem 4.. Le κ(x be a nondecreasing (nonincreasing funcion. If υ(x ( κ(xb(x, hen m(x ( κ(x G(x κ(xb(x + υ(x. ( Proof. We assume ha κ(x is nondecreasing (nonincreasing. B insering he bound υ(x ( κ(x B(x ino (8, we obain m(x ( + κ(x B(x dg( + υ(x. The funcion κ(x is nonincreasing (nondecreasing in and B(x is nondecreasing in. Appling Lemma 4. in he laer expression and using he relaion (9..6 from Willmo and Lin (, we have x m(x ( κ(x dg( G(x G( + = + G(x G( B(x dg( + υ(x κ(x dg( + + υ(x [ ] G(x ( k(x B(x + υ(x, and he resul follows. [ ] G(x B(x The following resul is an improvemen of he bounds of Willmo and Lin (, Corollar 9.., Corollar 4.. If υ(x ( kb(x, hen m(x ( κ G(x κ B(x + υ(x. Proof. The resul follows immediael b Theorem 4. b seing κ(x = κ. Nex, we derive some new bounds for he ail of defici a ruin, in he renewal model, appling Theorem 4.. In our analsis, he funcion Λ u ( = Λ (u := [ψ(u + ψ(uψ(], which is nonnegaive because he difference ψ(u+ ψ(u ψ( is nonnegaive b Brown (99, plas a ke role. Psarrakos and Poliis (7 obain ha Λ u ( saisfies he defecive renewal equaion Λ u ( = Λ u ( df( + H(u,. (
4 G. Psarrakos / Insurance: Mahemaics and Economics 43 (8 97 Theorem 4.. If he claim size d.f. P is DFR, hen H(u, [ψ(u + ψ(u ψ(] F(u + ψ(u [ψ( F(]. F(uF( If he ladder heigh d.f. F is IFR hen he converse inequali holds. Proof. Le P be DFR. B Willmo (, Corollar 3., i holds ha F(u + H(u, ψ(u, F(u or equivalenl F(u + ψ(u H(u, F( F(u F(. B Szekli (986, he ladder heigh d.f. F is also DFR. Thus, he funcion F(u + ψ(u k( = F( F(u is nondecreasing in, for fixed u. Appling Theorem 4. o Λ u ( in (, ields F(u + ψ(u F(u + ψ(u Λ u ( ψ( F( F(F(u F(F(u + H(u,. Afer some compuaion he lower bound for H(u, follows. If F is IFR, hen we use similar argumens. B Willmo (, Theorem 3., we know ha if he ladder heigh d.f. F is NWU (NBU, hen an upper (lower bound for he ail of he defici H(u, is H(u, ( F(ψ(u [ ] ψ(uψ( ψ(u +. (3 Denoe B (u, he lower (upper bound in (6 and B (u, he upper (lower bound in (3. Theorem 4.3. If F is NWU (NBU, hen for an u, H(u, B (u, ( B (u, H(, u. Proof. B Willmo (, Theorem., see also Psarrakos and Poliis (7, Lemma 3., he funcion Λ u ( saisfies he equaion Λ u ( = which implies Λ u ( = + H(u, dh(, [H(u, F(u + ]dh( F(u + dh(. Using (9, he laer expression gives Λ u ( H(, u = [H(u, F(u + ]dh(. (4 B he definiion of he disribuion of non-ruin in (7 and he general form of he soluion of a defecive renewal equaion (see Asmussen (987, Chaper VI, he funcion Λ u ( H(, u saisfies he defecive renewal equaion Λ u ( H(, u = [Λ u ( H(, u]df( + [H(u, F(u + ]. (5 Moreover, since he d.f. F is NWU (NBU b (6, we have [H(u, F(u + ] ( [ψ(u F(u]F(. Thus, appling Corollar 4. for he defecive renewal equaion in (4, we obain Λ u ( H(, u ( [ψ(u F(u]ψ( [ψ(u F(u]F( + [H(u, F(u + ]. Afer some compuaions he resul follows. Remark. In NWU (NBU case, he lower (upper bound in (5 and appling Corollar 4. in ( ields he upper (lower in (3, respecivel. 5. Furher resuls on he funcion Λ u ( Denoe he funcion N u ( = Λ u (z dz = [π S(u + ψ(u π S (], (6 where π S ( = ψ( d is called he sop-loss premium wih reenion. For more deails on he noion of he sop-loss premium see Cai and Garrido (998. Lemma 5.. The funcion N u ( saisfies he defecive renewal equaion N u ( = + N u ( df( + N u ( F( H(u, z dz. (7 Proof. Inegraing boh sides of ( in he inerval [, implies Λ u (z dz = = = z + z z Λ u (z df( dz H(u, z dz Λ u (z f ( d dz H(u, z dz Λ u (z f ( d dz Λ u (z f ( d dz H(u, z dz.
5 G. Psarrakos / Insurance: Mahemaics and Economics 43 (8 97 B Fubini s heorem we have Λ u (z dz = Λ u (z f (dz d = = Λ u (z f (dz d H(u, z dz ( Λ u (z dz df( + + ( Λ u (z dz df( H(u, z dz ( Λ u (z dz df( ( Λ u (z dz df( H(u, z dz. B he definiion of he funcion N u ( in (6, he resul follows. Le Z u be a random variable wih d.f. H u (see (3. For = in (7, we can compue he mean value of Z u. In paricular, E(Z u = π S(u ψ(u µ, This resul was originall proved b Willmo (, relaion (3.. The following resul gives an upper bound for he funcion H(u, z dz. Theorem 5.. For an u,, i holds ha H(u, z dz {π S(u + ψ(u π S ( [π S (u ψ(u E(S]ψ(} Proof. Clearl, i holds ha N u (F( + H(u, z dz N u (F(. Appling Corollar 4., for he defecive renewal equaion in (7, we have N u ( N u( ψ(u N u ( F( + N u (F( + H(u, z dz. Using (6, afer some simple compuaion he resul follows. We close his secion wih an asmpoic resul. In he Sparre Andersen risk model, he adjusmen coefficien R > is chosen o saisf e Rx df(x =. (8 This equaion is known as he Lundberg condiion, and inegraing b pars implies ha e Rx F(x dx = R. (9 Using he funcion Λ u (, Psarrakos and Poliis (7, obain he following e R H(u, d = R [e Ru ψ(u]. (3 This is a direc generalisaion of (9, since for u =, (3 implies (9. The noion of a direcl Riemann inegrable funcion appears in our las resul. The suggesed references are Feller (97, p. 36 and Asmussen (987, p. 8. Theorem 5.. If for an u, he funcion e R H(u, z dz is direcl Riemann inegrable, hen i holds ha lim er N u ( = e Ru ψ(u R e R df(. Proof. If for an u, he funcion e R H(u, z dz is direcl Riemann inegrable, hen he funcion ] e [N R u (F( + H(u, z dz is also direcl Riemann inegrable. Appling he Ke Renewal Theorem (see for example, Theorem VI.5. of Asmussen (987 o he defecive renewal equaion given in (7 and using Fubini s heorem, one can see ha lim er N u ( = = N u( e R F( d + [ e R N u (F( + H(u, z ] dz d e R df( e R df( = N u( e R F( d + e R df( = N u( e R F( d + R e R df( e R H(u, z dz d z er H(u, z d dz (erz H(u, z dz. (3 Using (6, (9 and (3 in (3, we ge ( lim er N u ( = R N u( + R [e Ru ψ(u] /( H(u, z dz e R df( R = e Ru ψ(u R e R df(, and he proof is complee. We remark ha here are sufficien condiions for a funcion o be direcl Riemann inegrable; see for example Asmussen (987, p. 8 or Willmo and Lin (, p. 57. If he ladder heigh d.f. F is NBUE, i.e. F( d µ F( for an, hen e R H(u, z dz is direcl Riemann inegrable. To verif his, observe ha for all u,, we have H(u, z F(z/( (see, for example, Chadjiconsaninidis and Poliis (7, p. 45. Inegraing wih respec o z in he inerval [, ields H(u, z dz F(z dz µ F(u, which means ha e R H(u, z dz is direcl Riemann inegrable. Noe ha NBU class is a subclass of NBUE class; see for example Willmo and Lin (, Chaper.
6 G. Psarrakos / Insurance: Mahemaics and Economics 43 (8 97 Acknowledgemen The auhor is graeful o an anonmous referee for his/her helpful commens on he exposiion of he paper. References Aposol, T.M., 974. Mahemaical Analsis, Second Ediion. Addison-Wesle, Hong Kong. Asmussen, S., 987. Applied Probabili and Queues. Wile, New York. Asmussen, S.,. Ruin Probabiliies. World Scienific, Singapore. Bowers, N., Gerber, H., Hickman, J., Jones, D., Nesbi, C., 986. Acuarial Mahemaics. Socie of Acuaries, Ihaca. Brown, M., 99. Error bounds for exponenial approximaion of geomeric convoluions. Annals of Probabili 8, Cai, J., Garrido, J., 998. Aging properies and bounds for ruin probabiliies and soploss premiums. Insurance: Mahemaics and Economics 3, Chadjiconsaninidis, S., Poliis, K., 7. Two-sided bounds for he disribuion of he defici a ruin in he renewal risk model. Insurance: Mahemaics and Economics 4, 4 5. Gerber, H.U., Goovaers, M.J., Kaas, R., 987. On he probabili and severi of ruin. ASTIN Bullein 7, Gerber, H.U., Shiu, E.S.W., 5. The ime value of ruin in a Sparre Andersen model. Norh American Acuarial Journal 9 (, Discussions: De Vlder, F., Goovaers, M., 984. Bounds for classical ruin probabiliies. Insurance: Mahemaics and Economics 3, 3. Feller, W., 97. An Inroducion o Probabili Theor and is Applicaions, volume II, Second Ediion. Addison-Wesle, New York. Psarrakos, G., Poliis, K., 7. A generalisaion of he Lundberg condiion in he Sparre Andersen model and some applicaions (submied for publicaion. Szekli, R., 986. On he concavi of he waiing ime disribuion in some GI/G/ queues. Journal of Applied Probabili 3, Willmo, G.E.,. Compound geomeric residual lifeime disribuions and he defici a ruin. Insurance: Mahemaics and Economics 3, Willmo, G.E., Lin, X.S.,. Lundberg Approximaions for Compound Disribuions wih Insurance Applicaions. Springer, New York.
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