J. Stat. Appl. Pro. Lett. 2, No. 1, 15-22 2015 15 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.12785/jsapl/020102 Martigale Method for Rui Probabilityi a Geeralized Ris Process uder Rates of Iterest with Homogeous Marov Chai Premiums ad Homogeous Marov Chai Iterests Phug Duy Quag Departmet of Mathematics, Foreig Trade Uiversity, 91- Chua Lag, Ha oi, Viet Nam Received: 16 May 2014, Revised: 1 Jul. 2014, Accepted: 7 Jul. 2014 Published olie: 1 Ja. 2015 Abstract: This paper gives upper bouds for rui probabilities of geeralized ris processes uder rates of iterest with homogeous Marov chai premiums ad Hemogeeous Marov chai Iterests. We assume that premium ad rate of iterest tae a coutable umber of o-egative values. Geeralized Ludberg iequalities for rui probabilities of these processes are derived by the Martigale approach. Keywords: Supermartigale, Optioal stoppig theorem, Rui probability, Homogeeous Marov chai. 1 Itroductio I recet years, the classical ris process has bee exteded to more practical ad real situatios. For most of the ivestigatios treated i ris theory, it is very sigificat to deal with the riss that rise from moetary iflatio i the isurace ad fiace maret, ad also to cosider the operatio ucertaities i admiistratio of fiacial capital. Teugels ad Sudt [9,10] cosidered the effects of costat rate o the rui probability uder the compoud Poisso ris model. Yag [12] built both expoetial ad o expoetial upper bouds for rui probabilities i a ris model with costat iterest force ad idepedet premiums ad claims. Xu ad Wag [11] give upper bouds for rui probabilities i a ris model with iterest force ad idepedet premiums ad claims with Marov chai iterest rate. Cai [1] studied the rui probabilities i two ris models, with idepedet premiums ad claims with the iterest rates is formed a sequece of i.i.d radom variables. I Cai [2], the author studied the rui probabilities i two ris models, with idepedet premiums ad claims ad used a fst order autoregressive process to model the rates of i iterest. I Cai ad Dicso [3], the authors give Ludberg iequalities for rui probabilities i two discrete- time ris process with a Marov chai iterest model ad idepedet premiums ad claims. Feglog Guo ad Digcheg Wag [4] used recursive techique to build Ludberg iequalities for rui probabilities i two discrete- time ris process with the premiums, claims ad rates of iterest have autoregressive ovig average ARMA depedet structures simultaeously. P. D. Quag [5] used martigale approach to build upper bouds for rui probabilities i a ris model with iterest force ad idepedet iterest rates ad premiums, Marov chai claims. P. D. Quag [6] used martigale approach to build upper bouds for rui probabilities i a ris model with iterest force ad idepedet iterest rates, Marov chai claims ad Marov chai premiums. P. D. Quag [7] used martigale approach to build upper bouds for rui probabilities i a ris model with iterest force ad idepedet premiums, Marov chai claims ad Marov chai iterests. P. D. Quag [8] also used recursive approach to build upper bouds for rui probabilities i a ris model with iterest force ad Marov chai premiums, Marov chai claims, while the iterest rates follow a fst-order autoregressive processes. I this paper, we study the models cosidered by Cai ad Dicso [3] to the case homogeous marov chai premiums, homogeous marov chai iterests ad idepedet claims. The mai differece betwee the model i our paper ad the Correspodig author e-mail: quagmathftu@yahoo.com Natural Scieces Publishig Cor.
16 P. D. Quag: Martigale Method for Rui Probabilityi a Geeralized Ris Process... oe i Cai ad Dicso [3] is that premiums, iterests i our model are assumed to follow homogeeous Marov chais. Geeralized Ludberg iequalities for rui probabilities of these processes are derived by the Martigale approach. 2 The Model ad the Basic Assumptios I this paper, we study the discrete time ris models with X =X 0 are premiums, Y =Y 0 are claims, I=I 0 are iterests ad X,Y ad I are assumed to be idepedet. To establish probability iequalities for rui probabilities of these models, we cosider two style of premium collectios. O oe had of the premiums are collected at the begiig of each period the the surplus process which ca be rearraged as U 1 U 1 = u. 1 with iitial surplus U1 o = u>0 ca be writte as U 1 = U 1 1 1+I +X Y, 1 1+I + X Y j=+1 1+I j. 2 O the other had, if the premiums are collected at the ed of each period, the the surplus process surplus U 2 o which is equivalet to = u>0 ca be writte as U 2 = u. U 2 1 with iitial U 2 =U 2 1 + X 1+I Y, 3 1+I + [X 1+I Y ] where throughout this paper, we deote b t=a x t = 1 ad b t=a x t = 0 if a>b. j=+1 1+I j. 4 I this paper, we cosider models 1 ad 3, i which X =X 0 is a homogeeous Marov chai, X tae values i a set of o - egative umbers G X =x 1,x 2,...,x m,... with X o = x i ad where 0 p i j 1, + p i j = 1. p i j = P [ X m+1 = x j Xm = x i ],m N,xi,x j G X, We also assume that I =I 0 is homogeeous Marov chai, I tae values i a set of o - egative umbers G I = i 1,i 2,...,i,... with I o = i r ad where 0 q rs 1, + q rs = 1. s=1 q rs = P[I m+1 = i s X m = i r ],m N,i r,i s G I, I additio, Y =Y 0 is sequece of idepedet ad idetically distributed o egative cotiuous radom variables with the same distributio fuctio Fy=PY o y. Based o the previous assumptios, we defie the fiite time ad ultimate rui probabilities i model 1 respectively, by ψ 1 u,x i,i r =P U 1 < 0 ψ 1 u,x i,i r = lim ψ 1 U1 u,x i,i r =P, 5 < 0 U 1 U1. 6 Natural Scieces Publishig Cor.
J. Stat. Appl. Pro. Lett. 2, No. 1, 15-22 2015 / www.aturalspublishig.com/jourals.asp 17 Similarly, we defie the fiite time ad ultimate rui probabilities i model 3 respectively, by ψ 2 u,x i,i r =P U 2 < 0, 7 ψ 2 u,x i,i r = lim ψ 2 U2 u,x i,i r =P < 0 U 2 U2. 8 I this paper, we derive probability iequalities for ψ 1 u,x i,i r ad ψ 2 u,x i,i r by the Martigale approach. 3 Upper Bouds for Rui Probability by the Martigale approach To establish probability iequalities for rui probabilities of model 1, we fst prove the followig Lemma. Lemma 3.1. Let model 1. If ay x i G X,i r G I, Y 1 X 1 1+I 1 1 Xo = x i,i o = i r < 0 ad P Y 1 X 1 1+I 1 1 > 0 X o = x i,i o = i r >0, 9 the there exists a uique positive costat R satisfyig: e R Y 1 X 1 1+I 1 1 Xo = x i,i o = i r = 1. 10 Proof. Defie We have f t= e ty 1 X 1 1+I 1 1 Xo = x i,i o = i r 1, t 0,+ f t= Y 1 X 1 1+I 1 1 e ty 1 X 1 1+I 1 1 Xo = x i,i o = i r, [ f t= Y 1 X 1 1+I 1 1] 2 e ty 1 X 1 1+I 1 1 X o = x i,i o = i r. Hece f t 0. This implies that f t is a covex fuctio with f 0=0 11 ad f 0= Y 1 X 1 1+I 1 1 Xo = x i,i o = i r <0. 12 By P Y 1 X 1 1+I 1 1 > 0 X o = x i,i o = i r >0, we ca fid some costat δ > 0 such that P Y 1 X 1 1+I 1 1 > δ > 0 X o = x i,i o = i r >0. The, we get This implies that f t= e ty 1 X 1 1+I 1 1 Xo = x i,i o = i r 1 e ty 1 X 1 1+I 1 1 Xo = x i,i o = i r e tδ.p.1 Y1 X 1 1+I 1 1 >δ X o =x i,i o =i r Y 1 X 1 1+I 1 1 > δ Xo = x i,i o = i r 1. 1 lim f t=+. 13 t + Natural Scieces Publishig Cor.
18 P. D. Quag: Martigale Method for Rui Probabilityi a Geeralized Ris Process... From 11, 12 ad 13 there exists a uique positive costat R satisfyig 10. This completes the proof. Let: R o = if R > 0 : e R Y 1 X 1 1+I 1 1 Xo = x i,i o = i r =1,x i G X,i r G I. Remar 3.1. e R oy 1 X 1 1+I 1 1 Xo = x i,i o = i r 1. To establish probability iequalities for rui probabilities of model 1, we prove the followig Theorem. Theorem 3.1. Let model 1. Uder the coditio of Lemma 3.1 ad R o > 0, the for ay u>0, x i G X,i r G I, Proof. Cosider the process U 1 V 1 ad S 1 = e R ov 1. Thus, we have is give by 2, we let = U 1 ψ 1 u,x i,i r e R ou. 14 1+I j 1 = u+ S 1 +1 = S1 X j Y j e R +1 ox +1 Y +1 j 1+I t 1. With ay 1, we have S 1 +1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I = S 1 e R ox +1 Y +1 +1 1+I t 1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I = S 1 e R ox +1 Y +1 1+I +1 1 1+I t 1 X 1,X 2,...,X,I 1,I 2,...,I 1+I t 1, 15. From 0 1+I t 1 1 ad Jese s iequality implies S 1 e R ox +1 Y +1 1+I +1 1 1+I t 1 X 1,X 2,...,X,I 1,I 2,...,I S 1 e R ox +1 Y +1 1+I +1 1 t X1,X 2,...,X,I 1,I 2,...,I 1+I 1. I additio, e R ox +1 Y +1 1+I +1 1 X1,X 2,...,X,I 1,I 2,...,I e R ox +1 Y +1 1+I +1 1 X,I e R ox 1 Y 1 1+I 1 1 Xo,I o 1. Thus, we have S 1 +1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I Hece,,=1,2,... is a supermartigale with respect to the σ - filtratio S 1 I 1 = σx 1,...,X,Y 1,...,Y,I 1,...,I S 1. Natural Scieces Publishig Cor.
J. Stat. Appl. Pro. Lett. 2, No. 1, 15-22 2015 / www.aturalspublishig.com/jourals.asp 19 Defie T 1 = mi : V 1 < 0 U 1, with V 1 is give by 15. Hece, T 1 is a stoppig time ad T 1 = mi,t 1 is a fiite stoppig time. Therefore, from the optioal stopplig theorem for supermartigales, we have S 1 T 1 S 1 o =e R ou. This implies that e Rou S 1 S 1 T 1 T 1 S 1 T 1.1 1 T.1 1 T e R o V 1 T 1.1 T 1. 16 From V 1 < 0 the 16 becomes T 1 I additio, ψ 1 u,x i,i r =P Combiig 17 ad 18 imply that = P e Rou U 1 V 1 This complete the proof. Similarly to Lemma 3.1, we have Lemma 3.2. Lemma 3.2. Let model 3. Ay x i G X,i r G Y, if 1 1 T U1 = PT 1. 17 < 0 U1 < 0 = PT 1. 18 ψ 1 u,x i,i r e R ou. 19 Y 1 1+I 1 1 Xo X 1 = x i,i o = i r < 0 ad P Y 1 1+I 1 1 X 1 > 0 X o = x i,i o = i r >0, 20 the, there exists a uique positive costat R satisfyig e R [Y 1 1+I 1 1 X 1] Xo = x i,i o = i r = 1. 21 Let R o = if R > 0 : Remar 3.2. e R oy 1 1+I 1 1 X 1 X o = x i,i o = i r = 1,x i G X,i r G I. X o = x i,i o = i r 1. e R Y 1 1+I 1 1 X 1 Similarly, we establish probability iequalities for rui probabilities of model 3 by provig the followig Theorem. Theorem 3.2. Let model 3. Uder the coditios of Lemma 3.2 ad R o > 0,the for ay u>0,x i G X,i r G r, ψ 2 u,x i,i r e R ou 22 Natural Scieces Publishig Cor.
20 P. D. Quag: Martigale Method for Rui Probabilityi a Geeralized Ris Process... Proof. Cosider the process U 2 give by 4, we let V 2 = U 2 ad S 2 = e R ov 2. Thus, we have 1+I j 1 = u+ S 2 +1 = S2 X j 1+I j Y j e R ox +1 Y +1 1+I +1 1 j 1+I t 1. 1+I t 1, 23 With ay 1, we have S 2 +1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I = S 2 e R ox +1 Y +1 1+I +1 1 1+I t 1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I = S 2 e R ox +1 Y +1 1+I +1 1 1+I t 1 X 1,X 2,...,X,I 1,I 2,...,I. From 0 1+I t 1 1 ad Jese s iequality implies S 2 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I S 2 e R ox +1 Y +1 1+I +1 1 t X1,X 2,...,X,I 1,I 2,...,I 1+I 1. I additio, e R ox +1 Y +1 1+I +1 1 X1,X 2,...,X,I 1,I 2,...,I e R ox +1 Y +1 1+I +1 1 X,I e R ox 1 Y 1 1+I 1 1 Xo,I o 1. Thus, we have S 2 +1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I Hece,,=1,2,... is a supermartigale with respect to the σ - filtratio S 2 I 2 = σx 1,...,X,Y 1,...,Y,I 1,...,I. S 2 Defie T 2 = mi : V 2 < 0 U 2 with V 2 is give by 23. Hece, T 2 is a stoppig ad T 2 = mi,t 2 is a fiite stoppig time. Therefore, from the optioal stoppig theorem for supermartigales, we have S 2 T 2 S 2 o =e R ou. Natural Scieces Publishig Cor.
J. Stat. Appl. Pro. Lett. 2, No. 1, 15-22 2015 / www.aturalspublishig.com/jourals.asp 21 This implies that e R ou S 2 T 2 S 2 T 2 S 2 T 2.1 2 T.1 2 T e R o V 2 T 2.1 T 2. 24 From V 2 < 0 the 24 becomes T 2 I additio, Combiig 25 ad 26 imply that e Rou ψ 2 u,x i,i r =P Thus, 22 follows by lettig i 27. This completes the proof. = P 1 2 T U 2 V 2 = PT 2. 25 < 0 U2 < 0 U2 = PT 2. 26 ψ 2 u,x i,i r e R ou. 27 4 Coclusio Our mai results i this paper, Theorem 3.1 ad Theorem 3.2 give upper bouds for ψ 1 u,x i,i r ad ψ 2 u,x i,i r by the Martigale approach with homogeous Marov chai premiums ad Hemogeeous Marov chai Iterests. To obtai Therem 3.1 ad Theorem 3.2, fst, we obtai importat prelimiary results, Lemma 3.1 ad Lemma 3.2, which give Ludbergs costats. There remai may ope issues - e.g. a extedig results of this article to cosider X =X 0 ad I=I 0 are homogeous Marov chais, Y =Y 0 is a fst - order autoregressive process; b buildig umerical examples for ψ 1 u,x i,r r ad ψ 2 u,x i,r r by the martigale approach; c Let τ m := if 1 U m < 0 m=1,2 be the time of rui. Ca we calculate or estimate quatities such as τ m. Further research i some of these dectio is i progress. Acowledgemet The authors would lie to tha the ditor ad the reviewers for the helpful commet o a earlier versio of the mauscript which have led to a improvermet of this paper. Refereces [1] J. Cai, Discrete time ris models uder rates of iterest. Probability i the gieerig ad Iformatioal Scieces, 16, 2002, 309-324. [2] J. Cai, Rui probabilities with depedet rates of iterest, Joural of Applied Probability, 39, 2002, 312-323. Natural Scieces Publishig Cor.
22 P. D. Quag: Martigale Method for Rui Probabilityi a Geeralized Ris Process... [3] J. Cai ad C.M.D. Dicso, Rui Probabilities with a Marov chai iterest model. Isurace: Mathematics ad coomics, 35, 2004, 513-525. [4] F. Guo ad D Wag, Rui Probabilities with Depedet Rates of Iterest ad Autoregressive Movig Average Structures, Iteratial Joural of Mathematical ad Computatioal Scieces, 62012, 191-196. [5] P. D. Quag, Upper Bouds for Rui Probability i a Geeralized Ris Process uder iterest force with homogeous Marov chai claims, Asia Joural of Mathematics ad Statistics, Vol 7, No.1, 2014, 1-11. [6] P. D. Quag, Upper Bouds for Rui Probability i a Geeralized Ris Process uder Rates of Iterest with homogeous Marov chai claims ad homogeous Marov chai premiums, Applied Mathematical Scieces. Vol 8, No. 29, 2014, 1445-1454. [7] P. D. Quag, Upper Bouds for Rui Probability i a Geeralized Ris Process uder Rates of Iterest with homogeous Marov chai claims ad homogeous Marov chai Iterests, America Joural of Mathematics ad Statistics, Vol 4, No. 1, 2014, 21-29. [8] P.D. Quag, Rui Probability i a Geeralized Ris Process uder Rates of Iterest with Depedet Structures, Joural of Statistics Applicatios & Probability Letters, Vol.3, No.3, 2014, 1-9. [9] B. Sudt ad J. L. Teugels, Rui estimates uder iterest force, Isurace: Mathematics ad coomics, 16, 1995, 7-22. [10] B. Sudt ad J. L. Teugels, The adjustmet fuctio i rui estimates uder iterest force. Isurace: Mathematics ad coomics, 19, 1997, 85-94. [11] L. Xu ad R. Wag, Upper bouds for rui probabilities i a autoregressive ris model with Marov chai iterest rate, Joural of Idustrial ad Maagemet optimizatio, Vol.2 No.2, 2006, 165-175. [12] H. Yag, No-expoetial bouds for rui probability with iterest effect icluded, Scadiavia Actuarial Joural, 2, 1999, 66-79. Natural Scieces Publishig Cor.