On the Renewal Risk Model with Constant Interest Force

Transcription

1 Jornal of Rik Analyi and Crii Rpon, Vol. 3, No. (Ag 3, 8894 On h Rnwal Rik Modl wih Conan Inr For K.K. Thampi Dparmn of Saii, SNMC, M.G.Univriy, Krala68356, INDIA. hampinm@yahoo.o.in M.J. Jaob Dparmn of Mahmai, NITC, Cali6736, INDIA. mjj@ni.a.in Abra In hi papr, w onidr a rnwal rik modl wih onan inr for for an inran porfolio. W di qaion for h rvival probabiliy and i LaplaSilj ranform hav bn obaind. W provid rriv algorihm for h ppr and lowr bond for h rin/rvival probabiliy ndr inr for. Finally, w driv an xponnial ingral qaion for h rvival probabiliy. Som pial a ar alo did. Kyword: Conan inr for, Gnralizd Exponnial diribion, LaplaSilj ranform, Probabiliy of Rin, Rriv allaion.. Inrodion Th laial ompond Poion rpl pro, i i ofn amd ha h rpl riv no inr ovr im. B h larg porion of h rpl of h inran ompani om from invmn inom. Th impa of invmn rik on h rin probabiliy and ohr i of boh horial inr and praial imporan. In rik hory, hr i onidrabl inr in h dy of invmn inom. In hi papr w rl givn by Snd and Tgl (995, 997 and h ida givn in Jingmin.al ( and Cai and Dikon ( o dy h rin problm for h Gnralizd Exponnial diribion. W aim o find h bond for h lima rin probabiliy by rriv hniq. Yang (999 onidrd a dir im rik modl wih a onan inr for and a nonxponnial ppr bond for rin probabilii wr obaind. Paln and Gjing (997 onidrd a diffion prrbd laial modl, a Lndbrg yp inqaliy wa obaind by aming a ohai invmn inom. Rnly hr ha bn onidrabl inr in xnding rl from laial rik hory, o mor flxibl gnral modl. A gnral modl involv h ampion ha h inrlaim im ar indpndn and idnially diribd, b no narily xponnial. Th rling rpl pro i rfrrd o a a rnwal rik pro or o alld Sparr Andrn modl propod by Sparr Andrn (957. Analyi of Sparr Andrn pro i mor diffil han h laial modl, b rmarkabl progr ha bn mad. On of h la of diribion for h inrlaim im i Erlang diribion. Vario ap of rin in Erlangian rik modl ar did in Dikon and Hipp (998,, Chng and Tang (3, Sn and Yang (4 and Li and Garrido (4a. Thampi al.(7 hav onidrd anohr rnwal rik pro in whih laim or a Gnralizd xponnial diribion.th mo rlad rfrn on h rin probabiliy for rnwal rik modl orrponding o or a i Dong and Wang (6 whih dlivrd ingrodiffrnial qaion for h rvival and rin probabilii wih ngaiv rik m. Thy obaind Pblihd by Alani Pr Copyrigh: h ahor 88

2 xa xprion and ppr and lowr bond for h rin probabiliy. Thr xi a va lirar on h laial rik modl wih onan inr for. Compard o hi hr ar a rmarkably fw papr on rnwal rik modl ndr inr for. On obvio raon i ha h hory i mh mor ompliad in h rnwal rik modl. W mnion ha hr ar papr whih ar dvod o h rin probabiliy of rnwal rik modl wih inr for. W do no plan o i hr a ompl li of rfrn. Rnly, Konaninid al. ( onidrd h rin probabiliy ψ ( wih h onan inr for >. Thy ablihd an aympoi xprion for h lima rin probabiliy. Hr w apply h mhodology in Snd and Tgl (995, Cai and Dikon ( and Dong and Wang (6 o driv bond for rin/rvival probabilii for a parilar la of rnwal rik modl ndr inr for. Th prpo of hi papr i o driv om xplii xprion for rin/rvival probabilii for a parilar la of rnwal rik pro ndr inr for. W am ha h laim inrarrival im hav Gnralizd xponnial diribion. W how ha hniq ha an b applid o prod xplii rl for rin probabilii in laial rik modl ndr inr for an alo b applid whn inrlaim im i Gnralizd xponnial. W hav obaind an ingral qaion aifid by h rin/rvival probabiliy ndr inr for. Alo w driv a opl alrnaiv xprion, an xponnial ingral qaion for h rvival probabiliy and h ohr a ond ordr diffrnial qaion aifid by h LaplaSilj ranform of h nonrin probabiliy. Th olin of h papr i mmarizd a follow. In ion, w o h mahmaial prliminari ha hlp o drib h modl in bqn ion. In ion 3, w apply h hniq dvlopd in Snd and Tgl (995 o driv a diffrnial qaion for h rvival probabiliy in h pial a whn α =. In ion 4, w obain bond for h rin probabiliy by rriv hniq and find ha h drivaion i om wha ompliad in gnral, b l o whn w am α =. An xponnial ingral qaion for h rvival probabiliy i drivd in ion 5. Th gnralizd xponnial diribion ha bn inrodd by Gpa and Knd (999. A random variabl X ha h gnralizd xponnial diribion wih paramr α and if i ha diribion fnion x α F(x; α, =(, x>, α >, >. wih orrponding dniy fnion x α x f(x;α, =α(, x >, α >, >. Th gnralizd xponnial diribion i dnod by GE(α,. Th GE diribion ha many ni propri and i an b d a an alrnaiv o Gamma and Wibll diribion in many iaion(gpa and Knd(. Prliminari W onidr a rik pro in whih laim or a a rnwal pro. L {T i} i= b a qn of indpndn and idnially diribd random variabl, whr T i dno h im nil h fir laim h and, for i>, dno h h im bwn h (i h and i laim. W am ha Ti ha a GE (n, diribion wih dniy fnion n g( = n(, >, > whr n i a poiiv ingr, and diribion fnion n G( = (, >, >. For mo of hi papr, w illra ida by rriing or anion o h a in whih n=. Of or, whn n=, w hav h laial rik modl. L U ( dno h val of h rrv a im. U ( i govrnd by du ( = d U ( d dx(, ha i ( (v U ( = dx(v. ( whr =U ( and i a prmim ha inran ompany riv pr ni im. In addiion o h prmim inom, h inran ompany alo riv inr of i rrv wih onan for and X( dno h amlad amon of laim orring in h im inrval (, ], ha i X( = N( j= X W am ha laim nmbr pro j {X(} i omprid of rnwal who inrlaim {N(} im {T,T...} hav GE(, diribion. Th individal laim amon X,X... indpndn of {N(} ar poiiv, indpndn and idnially diribd random variabl wih ommon mlaiv Pblihd by Alani Pr Copyrigh: h ahor 89

3 diribion fnion F( x = P{X x} and man m=e(x. Finally, if = ( v = dv = if > For onvnin w will drop h indx, whn h for of inr i zro. Thn ( i rdd o h al rik pro U( = X(. ( Now w dfin h im of rin by τ =inf{:u (<}. L ψ ( dno h probabiliy ha h ompany i rind a omim aring wih iniial rrv. So ψ ( = P{ U U ( < } = P{τ < U (=}. W dno φ ( = ψ (, h rvival probabiliy, ha i, h probabiliy ha rin nvr or. Conidr h rik rrv pro U ( = X(. L Y=TX,i=,,... and m (= E{ } b i i i y Y i h momn gnraing fnion of. Conidr h qaion T i X m i y ( = E{ }E{ } =. (3 Clarly m y ( =. Thi qaion may hav a ond roo. If h a olion o (3, = xi, hn i i niq and poiiv. Thi qaion i h dfining qaion for h adjmn offiin. 3. Ingral Eqaion for Srvival Probabiliy In hi ion, w driv an ingral qaion for φ (. Sppo ha h fir laim or a im T= and h amon of laim i X = x, h rpl j afr h paymn of fir laim i x. By onidring h im and amon of fir laim, h ondiional probabiliy ha h ompany will rviv i ( T ( φ ( x. Th w hav ha i φ ( = E{φ (U(T } = E{φ ( X }, ( Y i ( ( ( φ ( = g ( φ ( xdf(xd. Th hang of variabl =, w obain φ ( = / / / / ( ( ( ( ( φ ( xdf(xd. Diffrniaing xprion ( wih rp giv ( φ ( = (4 / / / / ( ( ( ( ( φ (xdf(xd. (5 Diffrniaing xprion (5 again wih rp o and rarrang h rm φ ( = ( ( ( ( ( / / / / ( ( φ (xdf(xd ( φ (xdf(x. (6 Inring φ ( and φ ( ino (6 and making a opl of implifiaion w arriv a h ingrodiffrnial qaion ( φ ( = ((φ ( φ( φ (xdf(x. (7 Ingraing boh id of (7 wih rp and afr implifiaion ( φ ( = ( φ ( φ ((φ ( ( φ (vdv φ (x(f(xdx, whih w rwri a ( φ ( ( φ ( ( φ (vdv (8 Pblihd by Alani Pr Copyrigh: h ahor 9

4 =( φ ( φ ( ( φ ( φ (x(f(xdx. (9 A limi, h lf hand id of h xprion (9 nd o a onan B(=( ψ (d, and h righ hand id bom ( (φ ( φ ( m. I i aily n ha B( =, w obain B(((φ ( m ( φ ( =. Inring ( ino (8, w g ( φ ( = ( φ ( m B(( ( φ (vdv φ (x(f(xdx. ( Ingraing hi xprion again wih rp o giv φ ( = (3 φ (v dv (64 ( ( v B(( m vφ (v dv ( φ (vdv φ ( Whn =, ( rd o φ( = φ (vx (F(x dxdv ( φ(vdv φ( ( m m φ(x F (xdx, (3 wih h ingrad ail diribion of F i givn by x F (x = [F(y]dy. m m and φ ( = r olion of h qaion (3. 3. Lapla Tranform whr r i h poiiv Th apparan of onvolion in (3 gg ha i i br o Lapla ranform o g an xplii xprion for nonrin probabiliy whn h inr for i zro. So w inrod and v ˆφ( = φ(vdv, y ˆf ( = df(y. Taking h Lapla ranform of h ingral qaion (3, w obain φ( m ˆφ( =. (4 f ˆ( ( Sh xprion an b aily invrd ing mahmaial ofwar pakag. Rriv Callaion of Rin Probabilii In hi ion, w driv bond for h rin probabiliy in rnwal rik modl by rriv hniq. Exa olion for h rin probabiliy ψ ( ar diffil o find. W find h bond for φ ( by dirizing h ingral qaion (. For any h> and k =,,... w hav wih (h (h φ (hk φ (hk φ (hk, (h φ (hk = h (kh php 3(k k k (h (h p φ (jhh pkh φ (jhh j= j= φ k h (h 3 4 j= p (j φ (jh φ ( p kh k (h p5 φ ((kjh f j(h j= (hk = (kh p kh p f (h (h 5 k k (h (h p φ (jhh pkh φ (jhh j= j= k h (h 3 ( 4 j= p (j φ (jh φ ( p kh Pblihd by Alani Pr Copyrigh: h ahor 9

5 f (h= k hk h(k k (h p5 φ ((kjh f j (h, j= (F(ydy. whr p =3, p =, p = 64, p 4 3 = B(( m and p = m. Th algorihm implmn lowr and ppr bond ivly. Th wo idd bond for h rvival probabiliy i of horial imporan, a i an hardly b d for dir nmrial ompaion. For h inr fr modl, howvr, h algorihm giv fl wo idd bond for φ(. A lighly diffrn mhod i d o obain h bond for h rin probabiliy in a Claial rik modl wih inr in whih G( =,, >. For ompond Poion modl wih inr, hr xi a niq poiiv olion, γ, h ha γx = E{ γx } dx. (5 x Th olion o qaion (5 i alld h adjmn offiin for ompond Poion rik modl modifid by inr. Th righ hand id of qaion (5 i implifid o γ y 5 γ γ dy = γ Γ,, y n y whr Γ(n, x = y dy, n>, x i h x inompl Gamma fnion. Thorm 4. Am ha h adjmn offiin γ >, xi, hn γ γ a ψ ( a, (6 for, whr a =inf x [,x γx x γy x F(ydy F(ydy and γx x x [,x γ y x a = p F(ydy. F(ydy Proof: Th proof an b givn by going along h am lin of h proof of (Thorm 5.4., Rolki.al wih om obvio modifiaion. Thi i known a wo idd Lndbrg bond for h rin fnion ψ (. Frhrmor, for all γ γ ψ ( < a if a > ψ ( and a <ψ ( if a<ψ (. No ha, F(y dy γm a = = ψ (, (7 γ y m x (γ F(ydy and analogoly for a ψ (. L X hav an xponnial diribion wih β x F(x =. In hi a, an xplii formla for ψ ( i alo availabl, namly Γ, β( ψ ( =. (8 β Γ, W Frobni ri mhod o driv hi formla, whih i onidrd o b h impl mhod. Nmrial xampl ar givn o illra h appliaion of xplii formla and i ppr bond in ompond Poion rik modl wih inr. For nmrial illraion, w =., = and β =3. For inr faor, w onidr wo diffrn val for =.5 and.. Th orrponding val of h adjmn offiin γ ar.879 and.3466 rpivly. W ompar h ppr bond wih xa val of rin probabilii in laial rik modl ndr inr for in Tabl and. Tabl: Rin Probabilii whn =.5 Exa Rin Uppr Bond Rin ( = Pblihd by Alani Pr Copyrigh: h ahor 9

6 Tabl: Rin Probabilii whn =. Exa Rin Uppr Bond Rin ( = Alrnaiv Exprion for φ ( In hi ion w driv a opl of alrnaiv xprion for h nonrin probabiliy. Afr implifiaion and rarrangmn on (, w obain h qaion mb(( φ ( φ ( = ( ( ( ( L φ (vdv ( dv v C(=, whih i implifid o φ (x(f(xdx. (9 C( =. Mliplying by C( on boh id of (9, and ingra from o, w g φ ( φ ( = ( m B( v v ( φ (ydy dv 3 v v φ (vy(f(y dy dv. li m, ( bom ( m φ( = φ( ( ( 3 v v m { φ(vxdf (x}dv. ( Thrfor h mlipliaion of C( wih φ ( lad o an xponnially diminihing rvival probabiliy fnion. Thorm: Th LaplaSilj ranform of qaion φ ( aifi h following ond ordr diffrnial qaion: ˆ ˆ φ ( ( 3 φ ( [ 3( (( f(]φ ˆ ˆ ( = ( φ ( φ (, ( whr ˆφ ( = φ (d d, φ ˆ ( = φ ˆ ( d, d d ˆ φ ( and φ ( = φ ( = φ ˆ ( =. d d Proof: Conidr h ond ordr diffrnial qaion obaind in (7. ( φ ( = (( φ ( φ( φ (xdf(x. Taking Lapla ranform on boh id of h qaion, w g L{φ (xf(x}, L{φ (}L {φ (} L{φ (} φ ( φ ( =L {φ (} 3 L {φ (} L {φ (} L{φ (} L{φ (} (3 whr L(φ ( = φ ˆ (. Thn w hav h following idnii L(φ ( = φ ( L (φ (, L(φ ˆ ( = L (φ ( φ (, L(φ ˆ ( = L (φ ( (L (φ ( φ (, L(φ ˆ ( = L (φ ( 4φ ( φ ˆ (. ( an b aily provd by inring h idnii in (3. Th idnii ar provd by ingraion by par. Rmark: Whn =, w g h am xprion a in (3. Tha i ( φ( φ ( ˆφ( =. (f( ˆ Pblihd by Alani Pr Copyrigh: h ahor 93

7 W an limina h rm φ ( by onidring h maximm of h aggrga lo pro aoiad wih h rpl pro whn = and i follow ha φ ( φ( = m. In onlion, h rl in hi papr giv bond for h rin/rvival probabiliy in a Sparr Andrn rik modl wih onan inr for. Th hniq ha i applid o prod xplii rl for h rin probabiliy in h laial rik modl ndr inr for an alo b applid whn inrlaim im hav Gnralizd xponnial. In ion 4, a rriv algorihm i drivd o obain h h bond for φ (. Th algorihm yild boh lowr and ppr bond ivly. Th bond for h rin probabiliy in h ompond Poion modl ar givn wih nmrial illraion. A opl of alrnaiv xprion for φ ( ar alo drivd. Aknowldgmn W wold lik o hank h anonymo rfr for h onriv ggion and ommn on h prvio vrion of hi papr. Rfrn. Andrn, S.E. On h olliv rik hory of rik in h a of onagion bwn h laim. Tran. of h XV Innal. ongr of Ari,, (957 : 9.. Cai, J., Dikon, D.C.M. Uppr bond for lima rin probabilii in h Sparr Andrn modl wih inr. Inran: Mahmai and Eonomi, 3(, (3: Chng, Y., Tang, Q Momn of rpl bfor rin and dfii a rin in h Erlang( rik pro. Norh Amrian Aarial Jornal, 7, (3. 4. Dikon, D.C.M., Hipp, C. Rin probabilii for Erlang( rik pro. Inran: Mahmai and Eonomi,, (998: Dikon, D.C.M., Hipp, C. On h im o rin for Erlang( rik pro. Inran: Mahmai and Eonomi, 9, (: Dong, Y., Wang, G. Rin probabiliy for rnwal rik modl wih ngaiv rik m. Jornal of Indrial and Managmn Opimizaion, (, (6: Gpa, R.D., Knd, D., Gnralizd Exponnial diribion, Arl & Nw Zaland J. Sai., 4(, (999: Gpa, R.D., Knd, D., Gnralizd Exponnial diribion: Diffrn mhod of imaion. J. Sai. Comp. Siml., 69, ( : Jingmin, H.E.,Rong, W.U., Jiafng, CUI. Uppr bond for h rin probabiliy in a rik modl wih inr for who prmim dpnd on h bakward rrrn im pro. Advan in Mahmai, 4(4, ( : Konaninid, D.G., Ng, K.W., Tang, Q. Th probabilii of abol rin in h rnwal rik modl wih onan for of inr, Jornal of Applid Probabiliy, 47(, (: Li, S. and Garrido, J. On rin for h Erlang(n rik pro. Inran: Mahmai and Eonomi, 3, (4a: Palon. J and Gjing. H.K. Opimal hoi of dividnd barrir for a rik pro wih ohai parn on invmn, Inran: Mahmai and Eonomi,, (997: Rolki, T., Shmidli, H., Shmid, V., Tgl, J. Sohai Pro for Inran and Finan. John Wily & Son ( Sn, L. and Yang, H. On h join diribion of rpl immdialy bfor rin and dfii a rin for Erlang( rik pro. Inran: Mahmai and Eonomi, 34, (4: Snd, S., and Tgl, J. Rin ima ndr inr for. Inran: Mahmai and Eonomi, 6, (995: Snd, S., and Tgl, J. Th adjmn fnion in rin ima ndr inr for.inran: Mahmai and Eonomi, 9, (997: Thampi,K.K., Jaob, M.J., Raj, N. Rin Probabilii ndr Gnralizd Exponnial diribion. AiaPaifi Jornal of Rik and Inran, (, (7 : Yang, H. Nonxponnial bond for rin probabilii ndr inr for. Sandinavian Aarial Jornal,, (999: Pblihd by Alani Pr Copyrigh: h ahor 94