Upper Bounds for the Ruin Probability in a Risk Model With Interest WhosePremiumsDependonthe Backward Recurrence Time Process

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1 Λ4flΛ4» ν ff ff χ Vol.4, No fl ADVANCES IN MATHEMATICS Aug., 211 Upper Bounds for the Ruin Probability in a Risk Model With Interest WhosePremiumsDependonthe Backward Recurrence Time Process HE Jingmin 1,, WU Rong 2, CUI Jiafeng 3 1. College of Science, Tianjin University of Technology, Tianjin, 3384, P.R. China; 2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 371, P. R. China; 3. College of Science, Tianjin University of Science and Technology, Tianjin, 3222, P. R. China Abstract: In the present paper, we investigate a risk model with interest, in which the claim counting process is a renewal counting process and premiums depend on the backward recurrence time process. We derive two eponential type upper bounds for the ruin probability by martingale methods and recursive techniques. Finally, we study several special cases and get numerical comparisons of upper bounds. Key words: backward recurrence time process; ultimate ruin probability; super-martingale; optional stopping theorem MR2 Subject Classification: 91B3 / CLC number: O211.6; F84 Document code: A Article ID: Introduction We consider a risk model in which the claim counting process is a renewal counting process. Denote by T n,n 1} the sequence of the inter-claim times and S n,n 1} the sequence of the claim times. Then we have n S =, S n = T i. Denote by X n,n 1} the sequence of the amount of claims. Let T n,n 1} independent of X n,n 1}, be positive, independent and identically distributed random variables with common distribution function G =P T, where T is an arbitrary T n,andx n,n 1} be positive, independent and identically distributed random variables with common distribution function F y =1 F y =P X y, with X an arbitrary X n. In the classical compound Poisson model and the Sparren Andersen model, the premium rate is a constant. But in our risk model, the premium rate to be received depends on the time since the last claim. For eample, in auto insurance, many insurance companies are now using credit scores as a factor in rating insurance premiums. If you have managed to build up a no claims bonusncb, then insurance companies will give you a discount on your quote. As long as you are a careful driver, and make no claims, then your NCB will be the single biggest Received date: Revised date: Foundation item: Supported by NSFCNo , No , National Basic Research Program of China973 Program, No. 27CB81495 and the Research Fund for the Doctorial Program of Higher Education. corresponding author: nkjmhe 22@yahoo.com.cn i=1

2 52 μ ρ Ω ffi 4fl way to reduce your insurance. Once claim occurs, the insurance company will reconsider the premium rate. That is to say, the process follows a deterministic path between claim arrival epochs, denoted by a measurable function φt, and it satisfies φt, = + t gsds, t >, where is the initial value of the deterministic path, and gs is a positive continuous function, which ensures that the totality of premiums is increasing in t. We write Nt =supn : S n t} for the claim counting process, then the surplus Ut of an insurance company at time t is given by Ut =u + t Nt gs S Ns ds X i, where u is the initial surplus and the premium rate depends on the current value of the process t S Nt : t } which is called the backward recurrence time process. It is assumed that the positive net profit condition holds, namely, E T i=1 gsds > EX. In this paper, we consider this model with constant interest force δ. Let U δ t bethe value of the surplus at time t, then U δ t =ue δt + t e δt s gs S Ns ds t Ns e δt s d X i. And let V δ t be the discounted value at time zero of U δ t, that is, V δ t =e δt U δ t. We firstly write out the eact epressions for U δ S n,n } and V δ S n,n }. U δ S =u, U δ S 1 =ue δt1 + T1 U δ S 2 =U δ S 1 e δt2 + e δt1 s gsds X 1, T2 U δ S n =U δ S n 1 e δtn + V δ S =u, T1 e δt2 s gsds X 2, Tn e δtn s gsds X n, V δ S 1 =u + e δs gsds X 1 e δt1, Tn V δ S n =U δ S n e = U δ S n 1 e 1 + e 1 e δs gsds X n e δtn Tn = V δ S n 1 +e 1 e δs gsds X n e, δtn 1 i=1

3 4» ±fi, ß, ΦΞ: Upper Bounds for the Ruin Probability 53 Since ruin can occur only at the time of a claim, we define inft :Uδ t < }, T δ =, if U δ t for all t>, to be the time of ruin, and τ δ = infn : Uδ S n < },, if U δ S n for all n N. Let Ψ δ u denote the ultimate ruin probability of U δ t, then Define Ψ δ u =P T δ < U δ = u} = P τ δ < U δ = u} } = P U δ S n < U δ = u n=1 } = P V δ S n < U δ = u. n=1 Ψ δ u; n =P τ δ n U δ = u} n } = P U δ S i < U δ = u i=1 n } = P V δ S i < U δ = u. i=1 Thus, the Monotone Convergence Theorem yields lim Ψ δu; n =Ψ δ u. n In risk theory, a particularly interesting problem is to calculate the ultimate ruin probability. Ruin problems related to the classical risk process with interest have been studied by many authors including Sundt and Teugel [8, 9], Dickson and Waters [5] and Kalashnikov and Konstantinides [6]. Ruin problems in the Sparre Andersen model have also been considered by many authors, such as Dickson and Hipp [3, 4] and Politis [7]. Cai and Dickson [2] studied the Sparre Andersen model with interest and derived two eponential upper bounds for the ultimate ruin probability. The above risk processes are special cases of our risk model. It is fairly difficult to determine the ultimate ruin probability Ψ δ u eplicitly. Therefore, bounds of the ruin probability Ψ δ u are requested. Following Cai and Dickson [2], we also derive two upper bounds for the ultimate probability by martingale techniques and recursive techniques. In addition, we consider several special cases and get numerical comparisons of upper bounds. Throughout this paper, it is assumed that Ee tx eistsfor<t<ξ,andthat lim t ξ Ee tx =. 1 Upper Bound by Martingale Approach In the classical risk model, one uses martingale approach to derive an eponential upper bound for the ruin probability. While, in our risk model, it is difficult to get an eponential martingale. However, we can get a discrete-time super-martingale with respect to V δ S n, that

4 54 μ ρ Ω ffi 4fl is, there eists a positive number R 1, such that e R1V δs n,n } is a discrete-time supermartingale. Using the same argument as Cai and Dickson [2], we can derive an eponential upper bound for the ultimate ruin probability. In this section, we assume that E[ T e δs gsds Xe δt ] >, and that there eists < such that P T e δs gsds Xe δt > if ξ =. The following lemma gives a nice interplay between conditioning and independence, and is etremely useful for proving the following theorem. We can find the similar lemma in Applebaum [1]. Lemma 1.1 Let F be a σ-algebra. If X and Y are two random variables where X is F - measurable and Y is independent of F, then for any non-negative or bounded Borel function f, E[fX, Y F ]=hx, a.s., where h =E[f, Y ] is a Borel function. Lemma 1.2 There eists a unique positive number R 1, such that [ T R 1 e δs gsds Xe δt =1. Proof Let Hr =E [ ep r T e δs gsds Xe δt. From E[ T e δs gsds Xe δt ] > it follows that H = 1, H = E[ T e δs gsds Xe δt ] <, and that Hr is conve and continuous. Further, Hr tends to infinity as r. The case ξ< is obvious. If ξ =, thenwehave T Hr e r P e δs gsds Xe δt as r. From this argument it follows that there eists a unique positive number R 1 such that HR 1 =1. Theorem 1.1 Let R 1 be defined as in Lemma 1.2. Then, for any u, we have Ψ δ u e R1u. Proof By 1, we have Tn+1 V δ S n+1 =V δ S n +e e δs gsds X n+1 e. δtn+1 Let F n = σt 1,,T n,x 1,,X n }. Then, for any n, E [ e R1V ] δs n+1 F n [ Tn+1 } Fn ] = e R1V δs n R 1 e e δs gsds X n+1 e δtn+1 [ Tn+1 = e R1V δs n R 1 } e ] e δs gsds X n+1 e δtn+1 Fn. Since <e 1, using Lemma 1.1 and Jensen s inequality for conditional epectations, we have [ Tn+1 } e ] e R1V δs n R 1 e δs gsds X n+1 e δtn+1 Fn

5 4» ±fi, ß, ΦΞ: Upper Bounds for the Ruin Probability 55 [ Tn+1 } Fn ] e e R1V δs n R 1 e δs gsds X n+1 e δtn+1. Since T n+1 and X n+1 are independent of F n,weget [ Tn+1 e R1V δs n R 1 By Lemma 1.2, we have Hence, [ Tn+1 = e R1V δs n R 1 } Fn ] e e δs gsds X n+1 e δtn+1 e e δs gsds X n+1 e δtn+1. [ Tn+1 e e R1V δs n R 1 e δs gsds X n+1 e δtn+1 = e R1V δs n. E [ e R1V δs n+1 F n ] e R 1V δ S n, which implies that e R1V δs n,n } is a discrete-time super-martingale. Since τ δ n is a bounded stopping time, using the optional stopping theorem for supermartingales, we get On the other hand, we have Hence, we have Ψ δ u; n e R1u. Letting n yields E [ e R1V δs τδ n ] E [ e R1V δs ] = e R1u. E [ e R1V δs τδ n ] E [ e R1V δs τδ n Iτ δ n ] = E [ e R1V δs τδ Iτ δ n ] E [ Iτ δ n ] =Ψ δ u; n. Ψ δ u e R1u. This completes the proof. If gs is a constant number, then our risk model is reduced to the Sparre Andersen model with interest, and the results of Theorem 1.1 and Theorem 2.1 in net section coincide with that of Cai and Dickson [2]. 2 Upper Bound by Recursive Techniques In this section, we derive a different upper bound for the ruin probability by recursive techniques. Similarly, we assume that there eists < such that P T eδt s gsds X > ifξ =.

6 56 μ ρ Ω ffi 4fl Lemma 2.1 There eists a unique positive number R 2, such that [ T R 2 e δt s gsds X =1. Proof Let [ T Hr = r e δt s gsds X. From E[ T eδt s gsds X] >E[ T gsds X] > it follows that H = 1, H = E[ T eδt s gsds X] <, and that Hr is conve and continuous on [,ξ. Further, Hr tends to infinity as r ξ. Thecaseξ< is obvious. If ξ =, thenwehave T Hr e r P e δs gsds Xe δt as r. From this argument it follows that there eists a unique positive number R 2 such that HR 2 =1. Theorem 2.1 Let R 2 be defined as in Lemma 2.1. Then, for any u, Ψ δ u βe [ e R2X] [ T R 2 ue δt + e δt s gsds e R2u, 2 where β 1 =inf t t e R2y df y. e R2t F t In particular, if F is NWUCnew worse than used in conve ordering, then for any u, [ T Ψ δ u R 2 ue δt + e δt s gsds. 3 Proof From the definition of β above, it follows that Thus, by 5 we have Ψ δ u;1 = P X 1 >ue δt1 + = F ue δ + F βe R2 e R2y df y 4 βe R2 E e R2X. 5 T1 βe [ e R2X] ep = βe [ e R2X] E [ ep } e δt1 s gsds dg e δ s gsds R 2 ue δ + R 2 ue δt + T } e δ s gsds dg e δt s gsds.

7 4» ±fi, ß, ΦΞ: Upper Bounds for the Ruin Probability 57 Under an inductive hypothesis, for some integer n 1, we assume that Ψ δ u; n βe [ e R2X] [ T R 2 ue δt + e δt s gsds. By Lemma 2.1 and the fact e δt 1, we have Ψ δ u; n βe [ e R2X] [ T R 2 u + e δt s gsds = βe R2u. 6 Conditioning on T 1 and X 1, we obtain the following recursion formula for Ψ δ u; n +1, T1 Ψ δ u; n +1=E [Ψ δ ue δt1 + = = + F ] e δt1 s gsds X 1 ; n Ψ δ ue δ + e δ s gsds y; n e δ s gsds dg ue δ + ue δ + R eδ s gsds Ψ δ ue δ + Then, by 4 and 6 we have Ψ δ u; n +1 βe R2ueδ + R eδ s gsds ue δ + R eδ s gsds df ydg e δ s gsds y; n df ydg. ue δ + R eδ s gsds e R2y df ydg + βe R2[ueδ + R eδ s gsds y] df ydg = βe [ e R2X] [ T R 2 ue δt + e δt s gsds. Hence, for any n 1, Ψ δ u; n βe [ e R2X] [ T R 2 ue δt + e δt s gsds. Letting n in the equation above, we obtain Ψ δ u βe [ e R2X] [ T R 2 ue δt + e δt s gsds. It follows from Lemma 2.1 and the fact e δt 1that Ψ δ u βe [ e R2X] [ T R 2 ue δt + βe [ e R2X] [ T R 2 u + = βe R2u E [ e R2X] [ T R 2 e δt s gsds e δt s gsds }] e δt s gsds

8 58 μ ρ Ω ffi 4fl = βe R2u e R2u, which gives equation 2. If F is NWUC, then β =[E[e R2X ]] 1 see [1] for details.. By 2 we get [ T Ψ δ u R 2 ue δt + e δt s gsds, which gives equation 3. 3Eamples It is generally difficult to derive the eplicit solution for the ultimate probability Ψ δ u even in some special cases. In this section, we give some numerical eamples to illustrate the application of the two different bounds. We take δ =.5 according to bank rate and assume that gs =c1 + e ηs in the following eamples. Net, we give some numerical solutions about the two risk models to make a comparison between the two upper bounds. Eample 3.1 Let T have an eponential distribution with parameter λ >, and X have an eponential distribution with parameter 1 μ >. From Lemma 1.2 and Lemma 2.1, R 1 satisfies the equation [ δ+η λ ep cr δ + η + δ δ δ + η + 1 ]} 1 d =1, 7 δ 1 rμ δ λ and R 2 satisfies the equation 1 1 rμ [ η ep cr δ + η δ + η + 1 ]} δ 1 λ + d =1. 8 δ δ Take c = 6, η = 1, λ = 1 and μ =1sothatEX =VarX = 1. By 7 and 8, we obtain that R 1 = and R 2 =.13224, then compare upper bounds in Table 1, where Martingale means the upper bound derived by the martingale method, which can be obtained by Theorem 1.1, and Recursion means the upper bound derived by the recursive method, which is the first bound in 2. Table 1 Upper bounds in Eample It can been seen from Table 1 that the upper bounds derived by the recursive method are sharper than those derived by the martingale method. Eample 3.2 Let T have an eponential distribution with parameter λ >, and X have a gamma distribution with shape parameter α> and scale parameter γ>. From Lemma 1.2

9 4» ±fi, ß, ΦΞ: Upper Bounds for the Ruin Probability 59 and Lemma 2.1, it follows that R 1 satisfies the equation [ δ+η λ ep cr δ + η + δ δ δ + η + 1 ]} α γ d =1, 9 δ γ r δ λ and R 2 satisfies the equation γ γ r α [ η ep cr δ + η δ + η + 1 ]} δ 1 λ + d =1. 1 δ δ It is well known that the gamma distribution F with <α 1isNWUC,β 1 = E[e R2X ]= γ γ R 2 α, and the gamma distribution F with α>1 satisfies the following equality β 1 =inf t e ry df y t = γ e rt F t γ r, for all r<γ. Case 1 Set c = 6, η = 1, λ = 1 and α = γ =.75 so that EX =1andVarX = 4 3. It follows easily from 9 and 1 that R 1 = and R 2 = By Theorem 1.1 and Theorem 2.1, the numerical solutions of the upper bounds are derived. See the Table 2. In this case, the epectation of the claim distribution is the same as that in Eample 3.1, but the variance is greater. Table 2 shows that the upper bounds in this case are greater than those in Table 1. Table 2 Upper bounds in Eample 3.2 when α = γ = Case 2 Set c = 6, η = 1, λ = 1 and α = γ =1.25 so that EX =1andVarX =.8. In this case, the epectation of the claim distribution is the same as those in Eample 3.1 and Case 1 in Eample 3.2, but the variance is smaller. We get that R 1 = and R 2 = by 9 and 1. By Theorem 1.1 and Theorem 2.1, the numerical solutions of the upper bounds are derived. See the Table 3. It can been seen from Table 3 that the upper bounds are less than those of Eample 3.1 and Case 1 in Eample 3.2. Table 3 Upper bounds in Eample 3.2 when α = γ =

10 51 μ ρ Ω ffi 4fl Eample 3.3 Let T have an Erlang2 distribution with parameter λ >, and X have an eponential distribution with parameter 1 μ >. From Lemma 1.2 and Lemma 2.1, R 1 is the solution of the equation [ δ+η λ ep cr δ + η + δ δ δ + η + 1 ]} 1 ln d =1, 11 δ 1 rμ δ λ and R 2 satisfies the equation 1 [ η ep cr 1 rμ δ + η δ + η + 1 ]} δ 1 λ + ln d =1. 12 δ δ Take c = 6, η = 1, λ = 2 and μ =1sothatEX =VarX = 1. By 11 and 12, we derived that R 1 =.1826 and R 2 = By Theorem 1.1 and Theorem 2.1, we derive the numerical solutions of the two upper bounds, which are shown in Table 4. Table 4 Upper bounds in Eample Eample 3.4 Let T have an Erlang2 distribution with parameter λ >, and X have a gamma distribution with shape parameter α> and scale parameter γ>. From Lemma 1.2 and Lemma 2.1, R 1 is the solution of the equation [ δ+η λ ep cr δ + η + δ δ δ + η + 1 ]} γ α ln d =1, 13 δ γ r δ λ and R 2 satisfies the equation γ α [ η ep cr γ r δ + η δ + η + 1 ]} δ 1 λ + ln d =1. 14 δ δ Case 1 Take c = 6, η = 1, λ = 2 and γ = α =.75 so that EX = 1and VarX = 4 3. By 13 and 14, we have R 1 = and R 2 = In addition, we also obtain the numerical solutions of the two upper bounds by Theorem 1.1 and Theorem 2.1, which are shown in Table 5. In this case, the epectation of the claim distribution is the same as those in Eample 3.3, but the variance is greater. Table 5 shows that the two upper bounds in this case are greater than those in Table 4. Table 5 Upper bounds in Eample 3.4 when γ = α =

11 4» ±fi, ß, ΦΞ: Upper Bounds for the Ruin Probability 511 Case 2 Set c = 6, η = 1, λ = 2 and α = γ =1.25 so that EX =1andVarX =.8. In this case, the epectation of the claim distribution is the same as those in Eample 3.3 and Case 1 of Eample 3.4, but the variance is smaller. We get that R 1 = and R 2 = by 13 and 14. By Theorem 2.1 and Theorem 3.1, the numerical solutions of the upper bounds are derived. See the Table 6. It can be seen from Table 6 that the upper bounds are less than those in Eample 3.3 and Case 1 of Eample 3.4. Table 6 Upper bounds in Eample 3.4 when γ = α = Since the upper bound derived by recursive method is smaller than e R2u and R 1 is less than R 2 in all the eamples above, the upper bounds derived by the recursive method are sharper than those derived by the martingale method. The same conclusions are shown in Table 1 Table 6. In risk literature, the upper bound derived by martingale method is really good. However, in the more general risk models, we sometimes can find a much better one, which can be shown by the eamples above. References [1] Applebaum, D., Levy Processes and Stochastic Calculus, Cambridge University Press, 24. [2] Cai J., Dickson, D.C.M., Upper bounds for ultimate ruin probabilities in the Sparre Andersen model with interest, Insurance: Mathematics and Economics, 23, 32: [3] Dickson, D.C.M., Hipp, C., Ruin probabilities for Erlang2 risk processes, Insurance: Mathematics and Economics, 1998, 22: [4] Dickson, D.C.M., Hipp, C., On the time to ruin for Erlang2 risk processes, Insurance: Mathematics and Economics, 21, 29: [5] Dickson, D.C.M., Waters, H.R., Ruin probabilities with compounding assets, Insurance: Mathematics and Economics, 1999, 25: [6] Kalashnikov, V., Konstantinides, D., Ruin under interest force and subeponential claim: a simple treatment, Insurance: Mathematics and Economics, 2, 27: [7] Politis, K., Bounds for the probability and severity of ruin in the Sparre Andersen model, Insurance: Mathematics and Economics, 25, 36: [8] Sundt, B., Teugels, J.L., Ruin estimates under interest force, Insurance: Mathematics and Economics, 1995, 16: [9] Sundt, B., Teugels, J.L., The adjustment function in ruin estimates under interest force, Insurance: Mathematics and Economics, 1997, 19: [1] Willmot, G.E., Lin X.S., Lundberg Approimations for Compound Distributions With Insurance Applications, New York: Springer, 2. ψ%9.6*;53+"&47$1!'/2- >@A 1, C B 2, <?= 3 1. οψ Π ρ ρfi, οψ, 3384; 2. ffi ρμρfflρρfi, οψ, 371; 3. οψffl ρ ρ fi, οψ, 3222 :8: E~ΞcKglNT mλ ynn_wyitπvx _wyi DSuq±e Φ [ff s`yi. YffiRQZO}^v MLoHUlNiVΨwΛrb. fl[, Ξc k]vdznhfi GpWJrbNwjFa.,#: [ff s`yi; flωohul; rffi; fl spf