Numerical Algorithm for the Stochastic Present Value of Aggregate Claims in the Renewal Risk Model

Transcription

1 Gn. Mah. Nos, Vol. 9, No. 2, Dcmbr, 23, pp. 4- ISSN ; Copyrigh ICSRS Publicaion, 23 Availabl fr onlin a hp:// Numrical Algorihm for h Sochasic Prsn Valu of Aggrga Claims in h Rnwal Risk Modl Dyi Ma Collg of Scinc, China Thr Gorgs Univrsiy Yichang , China mdysav@63.com (Rcivd: / Accpd: 24--3) Absrac For h sochasic prsn valu of aggrga claims in h rnwal risk modl, a numrical algorihm is consrucd basd on h Mon Carlo and random procss principl. Th basic ida and dsign procss of his algorihm is daild. Th numrical simulaion rsuls show ha i is consisnc wih h hory analysis rsul undr diffrn paramr diffrn disribuion. Th numrical algorihm rsuls can dircly s h ruin probabiliy of his rnwal risk modl, o dvlop suppor in h acual dcision. Kywords: Rnwal risk modl, Numrical algorihm, Mon Carlo, Exponnial lévy procss. Inroducion Risk fr and risky invsmn modl is a classical modl o an insurr. Suppos ( ) h pric procss of h invsmn modl is a gomric lévy procss { R, } In ohr words,. { R (, } is a lévy procss which sars wih, has

2 Numrical Algorihm for h Sochasic 5 indpndn and saionary incrmns, and is sochasically coninuous. In mahmaical financ, his assumpion on pric procss is widly usd [, 2, 3, 4]. For h gnral hory of lévy procss, w rfr h radr o s Sao[5], and Applbaum [6]. To h rnwal risk modl [7] wih succssiv claims, X, X 2, Λ, ar indpndn idnically disribud (i.i.d.) random variabls. Th random variabl disribuion F on [, ]. A h sam im, hir arrival ims, τ τ 2 Λ, consiu a rnwal couning procss N( = #{ n =,2, Λ τ n },. Thn h amoun of aggrga claims up o im as follows: N ( S( = = X ( k), (.) k For convninc, suppos ha all sourcs of randomnss, { X, X 2, Λ }, { N(, } and { R(, } ar muually indpndn. Thn h sochasic prsn valu of fuur aggrga claims up o im can b xprssd as s) D( = ds = X ( k) k= τ ( k )) ( τ ( k ) ), (.2) In his papr, w shall focus on h numrical simulaion of D (, and compar o h hory valu. W shall also considr numrical simulaion o ruin probabiliy. Th rs of papr consiss of fours scions. Scion 2 shows h numrical simulaion modl of his papr; Scion 3 prsns numrical simulaion algorihm, is h ky conns of his papr; Scion 4 givs h rsuls of numrical simulaion; Th las scion concluds h papr. 2 Numrical Simulaion Schm of h Rnwal Risk Modl Suppos an insuranc businss commncing a im wih iniial walh x >, hn h modl of h cash flow of prmiums lss claims is considrd as a compound rnwal procss wih h form C ( = c S(, (.3) Whr c is a fixd ra of prmium paymn and { S (, } is a compound rnwal procss dfind in (.). Bcaus h pric procss of h ( ) invsmn modl is h gomric lévy procss { R, }, h walh procss of h insurr is dfind as [8] R( U ( = ( x + This papr discusss h modl: U ( = x + dc( dc( ) (.4) (.5) Th modl (.5) is oo compound o do numrical simulaion. In ordr o consruc a numrical simulaion schm, pu (.), (.2), (.3) o (.5), U ( = x + = x + c dc( d = x ds( + d( S( )

3 6 Dyi Ma al. = x + c = x + c d Nk = ( ) ) d k= X τ ( k )) ( k) τ ( k ) τ ( k )) X ( k) (.6) Whr x is h iniial walh, c is a fixd ra of prmium paymn, { R(, } is a lévy procss, τ (k) is h kh claim ims, N( = #{ n =,2, Λ τ ( n) }, X (k) is h kh claim sizs. Th schm (.6) is h numrical simulaion schm for h sochasic prsn valu of aggrga claims in h rnwal risk modl. This papr mainly rsarchs h rsuls in h numrical simulaion schm wih diffrn paramr diffrn disribuion. 3 Numrical Algorihm of h Rnwal Risk Modl In ordr o numrical simulaion on random procss of h modl (.6), i nds o assum concr disribuion. Wihou loss of gnraliy, his papr assums R( = µ + σw (, whr µ, σ ar consans, W ( is Brownian moion. X (k) ~ lognormal disribuion or wibull disribuion, N( ~ Poisson disribuion. EX (k) = λ, EN ( = θ. According o h numrical simulaion schm, h algorihm flowchar is dfind as follows. Bgin Inpu Esima w( Esima N(, Tao(N() Esima R( Esima X(k) Esima ingral Esima sum Oupu U( End Figur : Th algorihm flowchar of h rnwal risk modl

4 Numrical Algorihm for h Sochasic 7 () In Esimaing w( Modul Evry im always bgin from and nd a im, h im inrval is.. Evry momn gnraor on random follows normal disribuion, and hn accumula h valus o im which is h valu of Brownian moion w(. Finally w( is sandardizd. (2) In Esimaing R( Modul R( = µ + σw (, in which µ dnos avrag incom of invsmn, and σ dnos varianc of invsmn incom. So h usual assumpion is µ >. Th paramrs µ, σ will dircly affc h probabiliy of ruin. (3) In Esimaing Ingral Modul R( is simad, which mans h valus of ingrand in [,] ar all known. So h ingral valu can b obaind by using rapzoidal ingraion or fas ingraion mhod. (4) In Esimaing N( and Tao N( Modul N( follows h Poisson disribuion, bu h momn of vns can b rcordd if i dircly gnras a random which follows h Poisson disribuion. In his papr, a random can b g which follows h similar Poisson disribuion hrough xponnial disribuion in [,]. Th spcific procss is ha vry im a random which follows h xponnial disribuion is obaind. if lss han, again producing a random, hn summaion, if lss han, rpa las sp, ohrwis nd of h procss. So N( is h numbr of random numbrs minus on, Tao(N() is h cumulaiv sum of random numbr. (5) In Esimaing X(k) Modul X(k) dnos h siz of vry claim amoun which may follows diffrn disribuions. This papr mainly considrs wo yps of disribuion: lognormal disribuion and wibull disribuion. Diffrn disribuion will lads o diffrn probabiliy of ruin. (6) In Esimaing Summaion Modul Afr obaining N(, R(Tao(k)) and X(k), h rsuls can b achivd by cumulaiv sum. 4 Numrical Simulaion Rsuls In ordr o obsrv h rsuls of diffrn paramr diffrn disribuion, firsly assum X (k) ~ lognormal disribuion. a) x =, c = 5, λ =. 5, θ =. 5, µ =. 5, σ =, W ( moion, =, sp lngh.. b) x =, c = 5, λ =, θ =, µ =. 5, σ =, W ( moion, =, sp lngh..

5 8 Dyi Ma al. c) x =, c = 5, λ =. 5, θ =. 5, µ =. 5, σ =, W ( moion, =, sp lngh.. d) x =, c = 5, λ =, θ =, µ =. 5, σ =, W ( moion, =, sp lngh x x x Figur 2: X(k) is h lognormal disribuion From h fig.2, i can b sn ha whn c λθ >>, On avrag, don' los mony, or c λθ <<,i may b ruin. Bu fig. only prsns h on random procss. In ordr o furhr discuss h rlaion bwn c and λθ, undr h sam condiion, rpa ims. Tabl : Rpa ims a) b) c) d) Ruin probabiliy.6893 From abl, if h iniial walh is oo lil and X(k) is h lognormal disribuion, ruin probabiliy rlaivly high wihou h rlaion bwn c and λθ. Scondly assum X (k) ~ wibull disribuion. a) x =, c = 5, λ =. 5, θ =. 5, µ =. 5, σ =, W ( moion, =, sp lngh..

6 Numrical Algorihm for h Sochasic 9 b) x =, c = 5, λ =, θ =, µ =. 5, σ =, W ( moion, =, sp lngh.. c) x =, c = 5, λ =. 5, θ =. 5, µ =. 5, σ =, W ( moion, =, sp lngh.. d) x =, c = 5, λ =, θ =, µ =. 5, σ =, W ( moion, =, sp lngh x x 5 5 x Figur 3: X(k) is wibull disribuion From h fig. 3, i can b sn ha h rsuls similar o h fig.. Similar, undr h sam condiion, rpa ims. Tabl 2: Rpa ims a) b) c) d) probabiliy No ha h rsuls ar no happnd. In h liraur, if X (k) is havy aild and c λθ <, h rsul ruin probabiliy is on whn. Th numrical simulaion and hory analysis ar consisnc. From anohr prspciv, c can b simply hink o rciv mony, λ is h man of claims siz, θ is h man of claims ims. Whn c λθ <, in ohr words, h aggrga claims ar grar han incom,ruin is no avoid.

7 Dyi Ma al. 5 Conclusion Th sochasic prsn valu of aggrga claims in h rnwal risk modl is widly usd in mahmaical financ. In h liraur, mos rsarchrs wr inrsd in hory analysis, bu ignor numrical analysis. In his papr, w firsly consruc a numrical simulaion schm for h sochasic prsn valu of aggrga claims in h rnwal risk modl. Basd on h numrical simulaion schm, w dsign a numrical simulaion algorihm for i. Diffrn paramr diffrn disribuion lads o diffrn numrical simulaion rsuls. Ths rsuls ar all consisnc wih h hory analysis rsuls. Rfrncs [] J. Cai, Ruin probabiliis and pnaly funcions wih sochasic ras of inrs, Sochasic Procsss and hir Applicaions, 2(24), [2] J. Paulsn, Ruin modls wih invsmn incom, Probabiliy Survy, 5(28), [3] J. Paulsn and H.K. Gjssing, Ruin hory wih sochasic rurn on invsmns, Advancs in Applid Probabiliy, 29(997), [4] K.C. Yun, G. Wang and R. Wu, On h rnwal risk procss wih sochasic inrs, Sochasic Procsss and hir Applicaions, 6(26), [5] K. Sao, Lévy Procss and Infinily Divisibl Disribuions, Cambridg Univrsiy Prss, Cambridg, (999). [6] D. Applbaum, Lévy Procss and Sochasic Calculus, Cambridg Univrsiy Prss, Cambridg, (24). [7] J. Li, Asympoics in a im-dpndn rnwal risk modl wih sochasic rurn, Journal of Mahmaical Analysis and Applicaions, 387(22), [8] Q. Tang, G. Wang and K.C. Yun, Uniform ail asympoics for h sochasic prsn valu of aggrga claims in h rnwal risk modl, Insuranc: Mahmaics and Economics, 46(2),