Second-Order Asymptotic Expansion for the Ruin Probability of the Sparre Andersen Risk Process with Reinsurance and Stronger Semiexponential Claims

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1 Ieraoal Joral of Sascs ad Acaral Scece 7; (: 4-45 p:// do:.648/j.jsas.7. Secod-Order Asympoc Expaso for e R Probably of e Sparre Aderse Rs Process w Resrace ad Sroger Semexpoeal Clams Rovsa Alyev Deparme of Operao Researc ad Probably Teory, Appled aemacs ad Cyberecs, Ba Sae Uversy, Ise of Corol Sysems of ANAS, Ba, Azerbaja Emal address: alyevrovsa@yaoo.com To ce s arcle: Rovsa Alyev. Secod-Order Asympoc Expaso for e R Probably of e Sparre Aderse Rs Process w Resrace ad Sroger Semexpoeal Clams. Ieraoal Joral of Sascs ad Acaral Scece. Vol., No., 7, pp do:.648/j.jsas.7. Receved: arc, 7; Acceped: arc 8, 7; Pblsed: Aprl 4, 7 Absrac: I s sdy e Sparre Aderse rs process w resrace s cosdered. Te secod-order asympoc expaso for e r probably s obaed, we e clam szes ave e srogly semexpoeal dsrbo. oreover, mercal examples cases proporoal resrace ad exsess sop loss resrace are provded. Keywords: Sparre Aderse Rs Process, Resrace, R Probably, Secod-Order Asympoc Expaso, Semexpoeal Dsrbo. Irodco Cosder e srpls process N R = + c, ( were R, s e srpls of e srer a me, = R > e al srpls of srace compay, c > e cosa rae per me a wc e premms are receved,,,... are depedely ad decally dsrbed (..d. posve radom varables represeg dvdal clam N = max : ξ ξ amos. Cog process { } deoes e mber of clams p o me, were e clam er-arrval mes or mes bewee clams ξ, are assmed..d. posve radom varables. Frer, we assme a e seqeces { ξ }, ad { }, are depede. Sppose a, also c > m, so a r s o cera o occr, were m = E ad = Eξ. To provde s codo s sggesed ac = ( + m, were > s safey loadg coeffce. We er-clam mes ξ, are ave a expoeal dsrbo w mea / λ, wc = s eqvale o a, N as a Posso dsrbo w parameer λ, s case ( s called classcal rs process or Cramer-Ldberg model acaral lerare. I case, we er-clam mes ξ, ave a arbrary dsrbo o [,, oer words, N a ordary reewal process, ( s called a Sparre Aderse rs process. Tere are exss lerare sdes, were some mpora problems coeced w Sparre Aderse rs process were solved (see, for example, Asmsse S. (, Albrecer H., Claram.., ármol. (6, Aleševcee A., Leps R., Šalys J. (9, Alyev R. T., Jafarova V. (9, Gerber H. U., S E. W. (5, Hald., Scmdl H. (4, L S., Garrdo J. (4, L S., Dcso D. C.. (6, Lo S., Tasar., Tso A. (8, Scmdl H. (. Te lmae r probably ψ (, wc s e ma global caracersc of e reewal rs model, s gve by ψ( = P{f R < R = }. Noe a e classcal sdes major role e sdy of e probably of r s e so-called adjsme coeffce or e Ldberg coeffce. Adjsme coeffce s defed as e posve solo of e caracersc eqao w respec o r:

2 Ieraoal Joral of Sascs ad Acaral Scece 7; (: were r = + с r ( r = rx ( r E e e df ( x Noe a a specal place e sdy of e probably of r aes e case of large clams. Noe a, for e modellg of large clams s sed w eavy-aled r dsrbos. I e case of eavy als ( r E( e =. Hece, e caracersc eqao ( becomes meagless. So s case a ew approac s reqred. I s dreco we meo Embrecs P., Veraverbee N. (98, Balras A. (999, Aleševcee e al. (9 ec. I sdy Embrecs P., Veraverbee N. (98 was obaed asympoc eqvalece as for r probably ψ (, e we eqlbrm fco F ( = F d, > m belog o e class of sbexpoeoal dsrbos (see, defo seco : were ψ( ~ F e ( с = m I sdy Balras A. (999 e rae of covergece for asympoc relao (3 was obaed, we F belogs o a sbclass of sbexpoeal dsrbos ad ξ, ave a expoeal dsrbo. Based o resls of paper Borovov A. (, sdy Aleševcee A., Leps R., Šalys J. (9 e secod-order beavor relao (3 s vesgaed, e case clam sze dsrbo belog o e srogly semexpoesoal class. I s ow a srace compaes also sre er rss o aoer compay. Ts ype of srace s called resrace. Bascally ere are some ypes of resrace coracs: proporoal resrace, excess of loss resrace ad excess sop loss resrace. Eac ype of resrace are descrbed by a fco ( x, we descrbes e amo pad by e srace compay e eve of a clam valex ad ( x x (see, for example, Dcso D. C., Waers H. R. (996, Dcso D. C., Waers H. R. (997, Dcso D. (5, p Proporoal resrace. If a rasferor compay self sasfes a cera fraco < β of eac clam, ad e remag sare β resrace compay, e s d s called a proporoal resrace. Parameer β s called e reeo lm. I s case, e loss of e rasmsso compay s β for clam. For e proporoal resrace = β, < β ad. (3 ( ( / β F x = F x.. Excess sop loss resrace. I s case, resrace compay pays clams exceedg a cera level, ad order o sre emselves agas large losses, o defy some of e pper level L. I s case ( = m max{ ;}, L. I s o dffcl o { } deerme fco F ( x s were Coseqely,. Dsrbo fco of { ( } F x = P x = P + P + P 3 { (, }, {, } P = P x F x x < = P x = F, x> { (, } {, L} P = P x < + L = P x < +, x < = F ( + L F (, x > { (, } {, } P = P x > + L 3 = P L x > + L, x < = F ( + L F ( L, x >, F x x < F( { } x = P x = F x + L, x> Tal fco of F x s, F x x < F ( x = P{ ( > x} = F x + L, x> Now sppose a e srer effecs resrace ad a e amo pad by e srer we e clam, occrs s (, were (. We wll assme rogo a resrace premms are calclaed w a loadg facor, were. Te assmg a resrace premm are pad coosly, e srer s srpls a me, s deoed by R, were N = + (, R c = (4

3 4 Rovsa Alyev: Secod-Order Asympoc Expaso for e R Probably of e Sparre Aderse Rs Process w Resrace ad Sroger Semexpoeal Clams m c = c c = + + E (5 ( ( Te prpose of s paper s o vesgae of e lmae r probably. a Resl ψ ( = P{f R < R = } Le s rodce some classes of dsrbo fcos (see, Borovov A. (. Defo. Dsrbo fco F o[, belogs o e class L of dsrbos w log als, f for ay fxed y as F( + y~ F( Defo. Dsrbo F o [, s called sbexpoeal, ad deoed F S, f F( x = F( x > for allx > ad Were F ( ( x F F x ~ F x, as x F ( ( x x F( x F d *( = F ( x = Dsrbo fco F ceered o (, ad belogs o e class S or L, f e fco F+ ( = F( I{ } belogs o e correspodg class, were I { } s dcaor of e se { }. Cosder e followg class of fcos (see, Borovov A. (: Defo 3. Dsrbo fco F belogs o e class Se of semexpoeal dsrbos, f F x e Q x =, were Q( x = x α L( x, α ad L( x a slowly varyg fco a fy ad L( x, as x, f α =. Frermore, α Q( x Q( x~ Q x Q( x + as x, > ε for x x every ε > ; Q( x Q( x + Q( x = o( as x,. x Defo 4. Dsrbo fco F belogs o e class Se of srogly semexpoeal dsrbos, f F s semexpoeal w parameer < α <. Examples for srogly semexpoeal dsrbos sc as e Webll dsrbo ad Beadera II-ype als wc are as follows, respecvely: α λx F( x = e, x, λ >, < α < ; λ λ F x x x x α α α α = exp,, λ >, < α <. I paper, Borovov A. ( was proved e followg eorem: Teorem (Borovov. Le X, X,... depede ad decally dsrbed radom varables w oarmec dsrbo fco F Se ad EX <, EX <. Te for as X P sp X > = F d + af ( + o( F ( (6 X X X = EX Were EX b ( a =, b = E sp X, dfx EX EX = = = ( X F - vold covolo of fco of F ( x w self. Le s gve followg lemmas from Foss S., Korsov D., Zacary S. (9 ad Aleševcee A., Leps R., Šalys J. (9. Lemma. Le F L ceered o[, ad Gceered o (,]. Te we, F G( ~ F(. Lemma. Le Z oegave radom varable w FZ L ad Y oegave radom varable, o depedg of Z, sc a EZ <. Te we FZ Y d = FZ d EYFZ ( + o( FZ (. Te ma resl of s paper ca be formlaed e form of e followg eorem. Teorem. Le e seqeces { ξ }, ad { }, be wo depede seqeces of radom varables, sc a varables eac seqece are depede ad decally dsrbed. oreover, F Se, m = E < ad = Eξ <. Te as Were ψ ( = F d + a F ( + o( F ( (7 e a e b c = + e e e ( ξ e = E c, =, X

4 Ieraoal Joral of Sascs ad Acaral Scece 7; (: ( = ξ = c ξ = = b E sp ( c df c defed from (5. Proof. I s easy o see a for e proporoal ad excess sop loss resrace ypes e codo F Se provdes F( Se. Coseqely, ca be sed e sceme of e proof of Teorem from Aleševcee A., Leps R., F Šalys J. (9. Sce, ξ s Se S L ad posve radom varable, accordg o Lemma, as ca be obaed: O e oer ad F ( ~ F ( ( ( c ξ (8 F ( c ξ Se, accordg o e defo of e class of srogly semexpoeal dsrbos. I s o dffcl o see a E ( c ξ c > E. Ideed, from (5 ca be obaed: <, or c = ( + m ( + E ( = ( m + ( + E( O e oer ad by defo (. Terefore, m. Coseqely, ag o acco E be obaed: ca c = ( m + ( + E > ( + E( > E( Sce, c ξ ave o-arfmec dsrbo ad ( c ξ E <, so sg Teorem ca be obaed: ψ( = P{f R < R = } = P sp ( ( c ξ > = = e b = E F d F ( o( F ( ( c ξ ( c ξ ( c ξ = ( c ξ e e e b = e F d F ( ( ( o( F ( c ξ e e (9 were ( ξ e = E c, =,, b = E sp ( ( c ξ. = Applyg Lemma o e egral obaed: F d ca be c ξ = ( ( (. c ξ ξ + F d F d c E F o F ( Tag o acco ( (9 saeme of Teorem ca be obaed. Ts compleed e proof of Teorem. Remar. I e parclar case (x=x,.e. we ere s wo resrace, Teorem mples Teorem of paper Aleševcee A., Leps R., Šalys J. (9. Corollary. Le codos of Teorem be sasfed ad ξ as a expoeal dsrbo w a al x F ( x = e, x. Te as ξ ψ ( = ψɶ ( + o( F ( ( E ( ψɶ ( F d F ( = e + e Really, s case, e cosa b e asympoc expaso (9 ca be explcly comped sg e formla Pollacze-Kc (see, Aleševcee A., Leps R., E Šalys J. (9. Terefore, coeffce s a = ad e from (7 ca be obaed (. 3. Nmercal Examples I s seco we ca cosder e followg mercal examples. Example. Le ξ as a expoeal dsrbo w a x al F ( x = e, x ad ( x = x,.е. wo resrace. ξ I s case = Eξ = ad sg ( ad Corollary of sdy Aleševcee A., Leps R., Šalys J. (9 as ca be obaed: were ψ( = ψɶ ( + o( F ( ( were

5 44 Rovsa Alyev: Secod-Order Asympoc Expaso for e R Probably of e Sparre Aderse Rs Process w Resrace ad Sroger Semexpoeal Clams E ψɶ ( = F d + F ( e e ( e E c ξ = E cξ = с m = m Le also e radom varable as Webll dsrbo x3 w al F, x = e x ad =, 7. Te resls of calclaos vales of fco ψɶ ( a dffere vales of e al capal are gve e followg able: Таble. ( = ; m = 6 ; =,7 ; c =, ~ ψ ( Example (Proporoal resrace. Uder codos of Example cosder proporoal resrace. Le =, 7 ad e relave srace premm resrace compay be =,8. I s ow a reeo lm ms sasfy e eqaly (see, Dcso D. (5, p. 99: β > I or example β >,358. Le β =. Coseqely, we ca draw e followg ables for e fco ψɶ ( from ( for dffere vales of e al capal : Таble. ( = ; m = 6 ; =,7 ; =,8; β =, 7; c = 6, ~ ψ ( Таble 3. ( = ; m = 6 ; =,7 ; =,8; β =, 5; c = 4, ~ ψ ( Таble 4. ( = ; m = 6 ; =,7 ; =,8; β =, ; c =, ~ ψ ( Example 3 (Exsess sop loss resrace. Uder codos of Example cosder exsess sop loss resrace. Le =, 7; ad =,8. I s o dffcl o see, a E( = F ( x dx = F ( x dx + F ( x + L dx = F ( x dx + F ( x dx + L E ( xf ( x dx xf ( x dx xf ( x L dx = = + + = = xf ( x dx + xf ( x dx L F ( x dx, + L + L Coseqely, we ca draw e followg ables for e fco from ( for dffere vales of e al capal ad parameers ad L: Таble 5. ( = ; m = 6 ; =,7 ; =,8; = 5; L = 3; c = 9, ~ ψ (,45,8,6,97,688,59,45,33 Таble 6. ( = ; m = 6 ; =,7 ; =,8; = 5; L = 4; c = 8, ~ ψ (,464,688,9,874,66,5,4,3 Таble 7. ( = ; m = 6 ; =,7 ; =,8; = 5; L = 5; c = 8, ~ ψ (,368,58,6,83,63,49,387,3 Noe: If e rasmsso compay wll pay all clams p o a cera lm, ad e clams exceedg e lm s sg o resrace compaes. If s rle apples o eac clam, e s ype of resrace s called e excess of loss resrace. Parameer s called e lm of reeo. For e excess loss resrace ( x = m{ x, }. I s o dffcl o see a s case F F ( x, x < ( x = F ( x =., x Terefore, F( Se. Coseqely, ma resl of prese sdy Teorem does o clde excess of loss resrace. 4. Coclso I prese paper e Sparre Aderse rs process w resrace s cosdered. Secod-order asympoc expaso for e r probably s obaed, we clam szes ave e srogly semexpoeal dsrbo. Te obaed resl sows a, s asympoc expaso re also for e proporoal resrace ad excess sop loss resrace ypes, b s o sasfed for e excess loss resrace. Nmercal examples provded paper sow a, as are srace effec o e probably of r. Noe a, sce, mos cases ere are o aalyc expressos avalable for e defc dsrbo ad s momes a e me of r fre sdes ca be oba asympoc expasos for ese caracerscs sg meods of prese sdy.

6 Ieraoal Joral of Sascs ad Acaral Scece 7; (: Acowledgeme Te aor express s as o Professor T. A. Kayev, TOBB Uversy of Ecoomcs ad Tecology (Trey, for s sppors ad valable advces. Refereces [] Albrecer H., Claram.., ármol. (6. O e dsrbo of dvded paymes a Sparre Aderse model w geeralzed Erlag ( erclam mes. Israce: aemacs ad Ecoomcs 37, pp [] Aleševcee A., Leps R., Šalys J. (9. Secod-order asympocs of r probables for semexpoeal clams. Laa aemacal Joral, 49, 4, pp [3] Alyev R. T., Jafarova V. (9. O e momes of e Sparre Aderse srpls process ad s average vale. 3 Ieraoal Cogress o Israce: aemacs ad Ecoomcs, Isabl, Trey, p. 39. [4] Asmsse S. (. R Probables. World Scefc, 38 p. [5] Balras A. (999. Secod-order asympocs for e r probably e case of very large clams. Sb. a. J., 4, pp [6] Borovov A. A. (. O sbexpoeal dsrbos ad asympocs of e dsrbo of e maxmm of seqeal sms. Sb. a. J., 43, pp [7] Dcso D. C., Waers H. R. (996. Resrace ad r. Israce: aemacs ad Ecoomcs, 9,, pp [8] Dcso D. C., Waers H. R. (997. Relave resrace reeo levels. ASTIN Blle, 7,, pp ASTIN BULLETIN, Vol. 7, No.. 997, pp [9] Dcso D. Proporoal Resrace. Ecyclopeda of Acaral Scece 6. [] Dcso D. Israce Rs ad R, Cambrdge Uversy Press, 5, 9 p. [] Embrecs P., Veraverbee N. (98. Esmaes for e probably of r w specal empass o e possbly of large clams. Israce: aemacs ad Ecoomcs,, pp [] Foss S., Korsov D., Zacary S. (9. Covolos of log-aled ad sbexpoeal dsrbos, Prepr avalable a p:// [3] Gerber H. U., S E. W. (5. Te Tme Vale of R a Sparre Aderse odel. Nor Amerca Acaral Joral, 9 (, pp [4] Hald., Scmdl H. (4. O e maxmsao of e adjsme coeffce der proporoal resrace. ASTIN Blle, 34, pp [5] L S., Garrdo J. (4. O r for e Erlag ( rs process, Israce: aemacs ad Ecoomcs 34 (3, pp [6] L S., Dcso D. C.. (6. Te maxmm srpls before r a Erlag ( rs process ad relaed problems, Israce: aemacs & Ecoomcs 38 (3, pp [7] Lo S., Tasar., Tso A. (8. O resrace ad vesme for large srace porfolos. Israce: aemacs ad Ecoomcs, 4,, pp [8] Scmdl H. (. O mmzg e r probably by vesme ad resrace. A. Appl. Probab., 3, pp