The Martingale Method for Probability of Ultimate Ruin Under Quota -(α, β) Reinsurance Model

Transcription

1 J. Stat. Appl. Pro. 5, No. 3, ) 411 Joural of Statistics Applicatios & Probability A Iteratioal Joural The Martigale Method for Probability of Ultimate Rui Uder Quota -α, β) Reisurace Model Bui Khoi Dam 1 ad Nguye Quag Chug 2, 1 Applied Mathematics ad Iformatics School- Haoi Uiversity of Sciece ad Techology, Viet Nam. 2 Departmet of Basic Scieces, Hugye Uiversity of Techology ad Educatio, Viet Nam. Received: 16 Ju. 2016, Revised: 4 Sep. 2016, Accepted: 6 Sep Published olie: 1 Nov Abstract: I this paper, we cosider a quota-α, β) reisurace model i discrete times with assumptio that claim sequece X = X } 1, premium sequece Y =Y } 1 are sequece of idepedet ad idetically distributed radom variables. Furthermore, the sequeces X = X } 1,Y = Y } 1 are assumed to be idepedet. By martigale method we obtai some iequalities for rui probability of the isurace compay, rui probability of the reisurace compay ad joit rui probability. Fially some umerical illustratios are give. Keywords: Quota reisurace, rui probability, joit rui probability, divisio ratio, Ludberg s iequality, supermartigale. 1 Itroductio Rui probability is a mai area i risk theory. Appeared i 1903 i the doctoral dissertatio of Ludberg,F [12].This topic, also of iterest to may authors, appeared i the research of may mathematicias, see [1,3,11,13,14,16] where upper bouds were estimated for rui probability havig expoetial form. Martigale approach is used to estimate the rui probability for such models as: i) the classical risk models, ii) the models of claim sizes ad premiums with sequeces of m-depedet radom variables, iii) the modes of iterest rates which ca be foud i the paper [4, 5, 18, 19, 7, 8]. Besides fidig methods to show the rui probability, the authors also build risk models to reflect the isurace busiess icreasigly more realistic. Oe of the models is Quota reisurace, preseted i the works, see [2, 10, 9]. I this paper, we ivestigate a Quota reisurace model i discrete time that we shall call it Quota -α,β) reisurace or Quota -α,β) share cotract. It is a cotract where the share proportio of the premium is differet from that of the claim, defied as follows: The quota - α, β) reisurace or quota - α, β) share cotract is a cotract betwee the isurace compay ad the reisurer to share premium Y =Y i } i 1 with a fixed proportio α [0,1]. Where Y i is the premium at the ed of the i period,,2,... paid by isured to the isurace compay) ad losses claim sizes) X =X i } i 1 with a fixed proportio β [0,1]. Where X i is the loss at the ed of the i period,,2,... claimed by the isured). Premium Y i : αy i kept by the cedat;1 α)y i trasfered to the reisurer. Claim loss X i : β X i paid by the cedat;1 β)x i paid by the reisurer. α, β [0, 1] are called divisio ratios. Whe the surplus process of the isurace compay is a sequece of radom variablesu } 1 defied by: U = u+α where u deote the isurace compay s iitial capital. Correspodig author chugkhcb@utehy.edu.v Y i β X i, 1.1) Natural Scieces Publishig Cor.

2 412 B. K. Dam, N. Q. Chug: The martigale method for probability of ultimate... [ t P Ui 0 )] is called probability of rui withi fiite time t, deoted by ψ 1) u,t) : [ ψ 1) t u,t)=p Ui 0 )], 1.2) [ P Ui 0 )] is probability of ultimate rui, deoted by ψ 1) u) ad defied as: [ ψ 1) u)=p Ui 0 )]. 1.3) Similarly, we defie the surplus process of the reisurace compay, the probability of rui withi fiite time t ad the probability ultimate rui, respectively, byv } 1,ψ 2) v,t),ψ 2) v): V = v+1 α) Y i 1 β) X i, 1.4) [ ψ 2) t v,t)=p Vi 0 )], 1.5) [ ψ 2) v)=p Vi 0 )], 1.6) where v deote the reisurace compay s iitial capital. Evidetly: lim ψ 1) u,t)=ψ 1) u); lim ψ 2) v,t)=ψ 2) v). t t I the classic model, see [2, 10] where α, β are cosidered equal. I this paper we ivestigate geralized model with α ad β are ay costat i[0, 1] satisfyig coditio 1.9). Recetly, some authors have studied the joit rui probability of isurace ad reisurace compaies, see [15, 9, 20]. We defie the fiite time t ad ultimate joit rui probabilities, respectively, by ψu, v, t), ψu, v): t [ Ui ψu,v,t)=p 0 ) V i 0 )]} 1.7) [ Ui ψu,v)=p 0 ) V i 0 )]} 1.8) ad we have lim ψu,v,t)=ψu,v). t I this paper, we cosider premium calculatio priciple based o the expected value priciple. Hece, we have: EX 1 )<EY 1 ) EX1 )<EY 1 ) βex 1 )<αey 1 ) β EX 1) 1 β)ex 1 )<1 α)ey 1 ) EY 1 ) < α < β EX 1) EY 1 ) + 1 EX 1) 1.9) EY 1 ) This coditio allows us to calculate α, β. I the ext, we shall work with the models 1.1) ad 1.4) uder the followig assumptios: Assumptio 1.1. Iitial capitals u > 0, v > 0. Assumptio 1.2. Let α, β [0, 1] satisfy coditio 1.9). Assumptio 1.3. Let X = ) X i i 1 ad Y = Y i be two sequeces of idepedet ad idetically distributed i.i.d), )i 1 oegative radom variables defied o the same probability space, Ω,F,P ). Furthermore, ) X i i 1 ad Y i )i 1 are assumed to be idepedets. The paper is orgaized as follows: I sectio 2, first we show Lemma 2.1 about existece ad uiqueess of a adjustmet coefficiet, Lemma 2.2 is used to prove Z = exps ) } to be supermatigale, Lemma 2.3 is maximal iequality 1 for oegative supermartigale. Theorem 2.1 will preset probability iequality of the isurace compay. Similarly to Sectio 2, Sectio 3 is dedicated to show probability iequality of the reisurace compay ad to derive some probability iequalities for the joit rui probability. Fially, a umerical example is give to illustrate ψ 1) u),ψ 2) v),ψu,v) i Sectio 4. Natural Scieces Publishig Cor.

3 J. Stat. Appl. Pro. 5, No. 3, ) / Probability iequalities for rui probability of the isurace compay Firstly, we preset three lemmas, which are ecessary for the proof of our mai result i Theorem 2.1). Lemma 2.1. Suppose that αey 1 )>βex 1 ) adp [ β X 1 αy 1 )>0 ] > 0, 2.1) the there exists a positive costat R 0 > 0 which is the uique root of the equatio E exp [ Rβ X 1 α Y 1 ) ]} = ) For all 0<R 1 R 0, we have E exp [ R 1 β X 1 α Y 1 ) ]} ) Proof. Cosider the fuctio gr)=e exp [ Rβ X 1 α Y 1 ) ]} 1 for R [0, ). We have g0)=0, g R)=E β X 1 αy 1 )exp [ Rβ X 1 α Y 1 ) ]}, By assumptio of Lemma 2.1 the g 0) = Eβ X 1 αy 1 ) < 0. Thus, the fuctio gr) is decreasig at 0. Further, ay turig poit of the fuctio is a miimum sice g R)=E β X 1 αy 1 ) 2 exp [ Rβ X 1 αy 1 ) ]} > 0. By Pβ X 1 αy 1 > 0)>0, we ca fid some costat δ 1 > 0 such that Pβ X 1 αy 1 > δ 1 )>0. We use the property that satisfyig if Z is radom variables satisfyig Z 0 the Z Z.1 A with ay A F gr) E exp [ Rβ X 1 αy 1 ) ] }.1 β X1 αy 1 >δ 1 ) 1 E [ expr.δ 1 ).1 β X1 αy 1 >δ 1 )] 1 = expr.δ 1 ).E [ 1 β X1 αy 1 >δ 1 )] 1 = expr.δ 1 ).P β X 1 αy 1 > δ 1 ) 1 2.4) The right- had of 2.4) whe R. Thus lim R gr)=. Cosequetly, there exists a uique positive root of the equatio 2.2), deoted by R 0. Sice, gr) is a covex fuctio with R [0, ) ad g0)=0,gr 0 )=0. If 0<R 1 R 0 the gr 1 ) 0which is equivalet to E exp [ R 1 β X 1 α Y 1 ) ]} 1. The Lemma 2.1 has bee prove. Lemma 2.2. see, [6] page 201) I the probabilistic space Ω,F,P), let U = ) ) U 1,U 2,...,U m,v = V1,V 2,...,V be radom vectors, idepedet of each other, ad f be a Borel s fuctio or m R where E fu,v). If u R m, the fuctio g is defied by E fu,v) if E fu,v) gu)= 2.5) 0 other the g is Borel s fuctio o R m with gu)=e [ fu,v) σu) ], where σu) is σ algebra geerated by vector U. I other words, uder the assumptio we ca computee [ fu,v) σu) ] as if U was a costat vector. Lemma 2.3. see, [17] page 493) Let Y =Y,F ) 0 be a oegative supermartigale. The for all λ > 0 } λp max Y k λ EY 0, 2.6) k } λp supy k λ k EY. 2.7) Natural Scieces Publishig Cor.

4 414 B. K. Dam, N. Q. Chug: The martigale method for probability of ultimate... The followig theorem will give upper bouds for the rui probability of the isurace compay. We put: S = β X i α Y i. 2.8) Theorem 2.1. We cosider model 1.1) such that: i) The assumptios are satisfied; ii) The coditios of Lemma 2.1) are also satisfied. The: a) Z = expr 1 S ) } is a supermartigale; 1 b) The rui probability of the isurace compay withi fiite time ψ 1) u,t) exp R 1 u); c) The ultimate rui probability of the isurace compay ψ 1) u) exp R 1 u). Proof. It follows immediately from result of Lemma 2.1 that there exists a costat R 1 0<R 1 R 0 ), such that: E exp[r 1 β X 1 αy 1 )] } ) We have We ow apply the Lemma 2.2 for the case E [ ] [ Z +1 Z 1,Z 2,...,Z =E exp R1 β X i α Y i ) ] } Z 1,Z 2,...,Z =E Z exp [ R 1 β X +1 αy +1 ) ] Z 1,Z 2,...,Z } = Z E exp [ R 1 β X +1 αy +1 ) ] Z 1,Z 2,...,Z }. 2.10) fu,v)=exp [ R 1 β X +1 αy +1 ) ], U =X 1,...,X,Y 1,...,Y ), V =X +1,Y +1 ), where U,V are mutually idepedet. Correspodig to the value X 1 = x 1,...,X = x,y 1 = y 1,...,Y = y ), u=x 1,...,x,y 1,...,y ), we cosider by assumptio gu)=e exp [ R 1 β X +1 αy +1 ) ]} =E exp [ R 1 β X 1 αy 1 ) ]}, E exp [ R 1 β X 1 αy 1 ) ]} 1, cosequetly I the other had gu)=e exp [ R 1 β X +1 αy +1 ) ] } X 1,...,X,Y 1,...,Y 1. gu)=e exp [ R 1 β X +1 αy +1 ) ] } Z 1,Z 2,...,Z ) Combiig 2.10) ad 2.11) imply that EZ +1 Z 1,Z 2,...,Z ) Z for all ) Natural Scieces Publishig Cor.

5 J. Stat. Appl. Pro. 5, No. 3, ) / SoZ } 1 =expr 1 S )} 1 is o- egative supermartigale. I additio, [ t ] [ ] ψ 1) u,t)=p S u) =P max S [ k) u =P max expr1 S k ) ] } expr 1 u). 2.13) 1 k t 1 k t =1 Applyig Lemma 2.3 for o- egative supermatigalez } 1 =expr 1 S )} 1, we have: ψ 1) u,t) exp R 1 u).ez 1 ) } = exp R 1 u).e exp [ R 1 β X 1 αy 1 ) ])} exp R 1 u). 2.14) By lettig t i 2.14). Thus, This completes the proof. lim t ψ1) u,t) limexp R 1 u)) ψ 1) u) exp R 1 u). 2.15) t Remark 2.1: Our result seems to be ew eve is the case of clasical model whe α = β ; With 0<R 1 R 0 the the right-had side of 2.14) ad 2.15) are the smallest whe R 1 = R 0. Similar to the Sectio 2, we derive iequality for rui probability of the reisurace compay i the Sectio 3 followig. 3 Probability iequalities for rui probability of the reisurace compay We put: S =1 β) X i 1 α) Y i. 3.1) We eed Lemma 3.1 for the proof of Theorem 3.1 below. Lemma 3.1. Uder the coditios 1 α)ey 1 )>1 β)ex 1 ) adp [1 β)x 1 1 α)y 1 ]>0 } > 0, 3.2) the there exists a positive costat R 0 > 0 which is the uique root of the equatio E exp [ R )]} 1 β)x 1 1 α)y 1 = ) For all 0<R 2 R 0, we have E exp [ R 2 1 β)x1 1 α) Y 1 )]} ) The proof of Lemma 3.1 is similar as Lemma 2.1. Now we state theorem below for the case of reisurace compay. Theorem 3.1. We cosider model 1.4) such that: i) The assumptios are satisfied; ii) The coditios of Lemma 3.1) are also satisfied. The: a) Z = expr 2S )} is a supermartigale; 1 b) The rui probability of the reisurace compay withi fiite time ψ 2) v,t) exp R 2 v); c) The ultimate rui probability of the reisurace compay ψ 2) v) exp R 2 v). The proof of Theorem 3.1 is similar to that of Theorem 2.1. Remark 3.1 With assumptios of Theorem 2.1, Theorem 3.1 we have the probability iequality for ultimate joit rui probability. Ideed: ) ψu,v)=p U i 0) V i 0) The iequality 3.5) is corollary of Theorem 2.1 ad Theorem 3.1. ) ) P U i 0) +P V i 0) exp R 1 u)+exp R 2 v). 3.5) Natural Scieces Publishig Cor.

6 416 B. K. Dam, N. Q. Chug: The martigale method for probability of ultimate... 4 Numerical illustratios I this Sectio, we provide two umerical illustratio examples for the probability iequality of ψ 1) u),ψ 2) v) ad ψu,v), with α = ,β = Example 4.1 Let Y = Y } >0 be a sequece of idepedet ad idetically distributed radom variables, with Y 1 havig a distributio: Y P Let X = X } >0 be a sequece of idepedet ad idetically distributed radom variables, with X 1 havig distributio: X P Whe β X 1 αy 1 havig distributio: βx 1 αy P To 2.2) we have the equatio : expr ) expr ) exp r ) exp r ) = ) By Maple the equatio 4.1) has a roof R 0 = Similarly, we have distributio of1 β)x 1 1 α)y 1 1 β)x 1 1 α)y P To the equatio 3.3) gives the equatio: expr ) expr ) exp r ) exp r ) = ) By Maple we fid R 0 = is roof of the equatio 4.2). The followig table shows upper bouds ψ 1) u),ψ 2) v),ψu,v) for a rage of value of u,v: Natural Scieces Publishig Cor.

7 J. Stat. Appl. Pro. 5, No. 3, ) / Table 1: Upper bouds of ψ 1) u),ψ 2) v) ad ψu,v) with X 1 ad Y 1 are discrete radom variables. u,v) ψ 1) u) ψ 2) v) ψu,v) u = 30; v = u = 1550; v = u = 70; v = Example 4.2 Let Y =Y } >0 be a sequece of idepedet ad idetically distributed radom variables, with Y 1 is costat c=1.1. X = X } >0 be a sequece of idepedet ad idetically distributed radom variables, with X 1 has the expoetial distributio: λ exp λ x) if x>0 fx)= 4.3) 0 other x 0 where λ = 1. fx) is desity fuctio of X 1 To coditio of 2.2), we have equatio + 0 exp [ r0.5000x ) x ] dx=1 Use of Maple software, we fid a solutio R 0 = of 4.4). Similarly, to coditio of 3.3), we have equqtio exp r ) 1+r = ) + 0 exp [ r0.5000x ) x ] dx=1 ad fid R 0 = Whe upper bouds ψ 1) u),ψ 2) v),ψu,v) for a rage of value of u,v. exp r ) 1+r = ) Table 2: Upper bouds of ψ 1) u),ψ 2) v) ad ψu,v) with Y 1 = c ad X 1 is cotiuous radom variable. u,v) ψ 1) u) ψ 2) v) ψu,v) u=8;v= u = 10; v = u = 15; v = Coclusios This paper costructed upper bouds for ψ 1) u,t),ψ 1) u),ψ 2) v,t), ad ψ 2) v) i model 1.1) ad 1.4) by the martigale method. Our mai results i this paper ot oly prove Theorem 2.1 ad Theorem 3.1 but also give umerical examples to illustrate for Theorem 2.1 ad Theorem 3.1. Natural Scieces Publishig Cor.

8 418 B. K. Dam, N. Q. Chug: The martigale method for probability of ultimate... Ackowledgemet The authors are very grateful to the aoymous referee ad the editors for their careful readig of the article, correctio of errors, improvemet of the writte laguage, ad valuable suggestios, which made this article much better. Refereces [1] Asmusse, S., Rui probabilities, World Scietific, Sigapore, 2000). [2] Borch, K., The Optimal Reisurace Treaty, ASTIN BULLETIN, 5, , 1969). [3] Buhlma, H., Mathematical Methods i Risk Theory, Berli - Heidelberg - New York Spriger, 1970). [4] Cai, J., Discrete time risk models uder rates of iterest, Probability i the Egieerig ad Iformatioal Scieces. 16, , 2002). [5] Cai, J. ad Dickso, D. C M., Upper bouds for Ultimate Rui Probabilities i the Sparre Aderse Model with Iterest, Isurace: Mathematics ad Ecoomics 32, 61-71, 2003). [6] Chow, Y. S. ad Teicher, H., Probability Theory, Berli - Heidelberg - New York Spriger Verlag, 1978). [7] Dam, B. K. ad Hoag, N. H., Evaluate rui probabilities for risk processes with depedet icremets, Vietam Joural of Mathematical Applicatios, Vol.VI, No.1, , 2008). [8] Dam, B. K. ad Hoag, N. H., Upper bouds of rui probabilities for discrete time risk models with depedet variables, Vietam Joural of Mathematical Applicatios, Vol. VI, No. 2, 49-64, 2008). [9] Dam, B.K. ad Chug, N.Q., The Joit Rui Probability i Risk Model Uder Quota Reisurace ad Excess of Loss Reisurace, submitted to East Asia Joural o Applied Mathematics, 2015). [10] Dickso, D.C.M., Isurace Risk ad Rui, Cambridge Uiversity Press, 2006). [11] Embrechts, P., Kluppelberg, C. ad Mikosch, T., Modellig Extremal Evets for Isurace ad Fiace, Spriger, Berli, 1997). [12] Ludberg, F., I. Approximerad Framstillig av SaolikhetsfuktioeFirst. II. Aterforsakrig av Kollektivrisker, Almquist & Wiksell, Uppsale, 1903). [13] Paulsel, J., O Cramer like asymptotics for risk processes with stochastic retur o ivestmet, The Aals of Applied Probability,Vol.12, N0. 4, , 2002). [14] Paulsel, J., Rui models with ivestemet icome, arxiv: v1math. PR) 25 Ju 2008, 2008). [15] Sacho, S. S., Leo, C. C., Merc, C., Aa, C. ad Maite, M., Effectively Tacklig Reisurace Problems by Usig Evolutioary ad Swarm Itelligece Algorithms, Risks, 2, , 2014). [16] Schmidt, K.D., Lectures o Risk Theory. Techische Uiversität Dresde, 1995). [17] Shiryaev, A. N., Probability, Secod Editio, Spriger-Verlag, 1996). [18] Sudt, B. ad Teugels, J.L., Rui estimates uder iterest force, Isurace: Mathematics ad Ecoomics 16, 7 22, 1995). [19] Sudt, B. ad Teugels, J.L., The adjustmet fuctio i rui estimates uder iterest force. Isurace Mathematics ad Ecoomics 19, 85 94, 1997). [20] Kam, C.Y., Juyi, G., Xueyua, W., O the first time of rui i the bivariate compoud Poisso model, Isurace: Mathematics ad Ecoomics 38, , 2006). Bui Khoi Dam is Professor of Mathematical at Applied Mathematics ad Iformatics School, Haoi Uiversity of Sciece ad Techology. He is referee ad editor of several iteratioal jourals i the frame of pure ad applied mathematics, applied ecoomics. His mai research iterests are: Stochastic processes, Reewal theory, Game theory, Risk theory, ad Applied ecoomics. Natural Scieces Publishig Cor.

9 J. Stat. Appl. Pro. 5, No. 3, ) / Nguye Quag Chug is a lecturer at Hugye Uiversity of Techology ad Educatio, Vietam. His mai research iterests are: Probability ad mathematical Statistics, Risk theory ad Mathematical ecoomics. Natural Scieces Publishig Cor.