On Finite-Time Ruin Probabilities in a Risk Model Under Quota Share Reinsurance

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1 Applied Mathematical Sciences, Vol. 11, 217, no. 53, HIKARI Ltd, On Finite-Time Ruin Probabilities in a Risk Model Under Quota Share Reinsurance Bui Khoi Dam 1 and Nguyen Quang Chung 1,2 1 Applied Mathematics and Informatics School Ha Noi University of Science and Technology, Ha Noi, Viet Nam 2 Department of Basic Sciences Hung Yen University of Technology and Education, Hung Yen, Viet Nam Copyright c 217 Bui Khoi Dam and Nguyen Quang Chung. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we investigate a quota share reinsurance risk model which is an extension of the one introduced in [14]. Assuming that the premium incomes and claim amounts take values in finite sets of non-negative numbers, we derive the explicit formula for the joint ruin probability of the insurer and the reinsurer. Moreover, we give upper bounds of ruin probabilities by martingale and inductive methods. In particular, we show that there exists a quota share level that minimizes the joint ruin probability and the upper bound of ruin probability of each insurance companies. Finally, a numerical illustration is given. Mathematics Subject Classification: Primary 91B3; Secondary 6H3, 6G4 Keywords: Quota share, ruin probability, joint ruin probability, Martingale process, recursive equation 1 Introduction We consider a discrete-time model for an insurer s surplus. The insurer s surplus in period n, n 1 denoted as U n is defined by: n n U n u + Y i X i (1.1

2 261 Bui Khoi Dam and Nguyen Quang Chung where u (u > is the insurer s initial surplus; Y n denotes the premium income during the nth period (i.e, from time (n 1 to time n, Y {Y n } n> is a sequence of independent and identically distributed (i.i.d. non-negative random variables; X n denotes the claim amount during the nth period, X {X n } n> is a sequence of i.i.d. non-negative random variables and is independent of Y. The process {U n } n> defined by (1.1 is called a surplus process (see [14] and [18]. It is an extension of the surplus process considered in [7] with the new assumption that the premium income is a sequence of i.i.d non-negative random variables. Nowadays, modern insurance businesses can allow insurers to invest in risk-free or risky assets. Therefore, in [1], [2], [11], [18] and [16], the authors extended the above surplus process by including the interest rate. In the classical risk models, claims are assumed to be paid by one insurer. However, insurers can transfer risks from one primary insurer (the ceding company or cedent to another one (the reinsurance company through reinsurance contracts. For that reason, [5], [6], [9] and [19] extended the classical surplus process in consideration of reinsurance. Quota share reinsurance is a common type of proportional reinsurance. It is a contract between the cedent and the reinsurer to share premium incomes and claim amounts with the same proportion. There are many problems related to quota share reinsurance contracts such as [15], [12], [3], [19], [4] and [5]. The previous problems concerned only the interest of one party (either the cedent or the reinsurer. In our paper, we consider the interests of both the insurer and reinsurer. By considering the effect of quota share reinsurance on surplus processes, we extend surplus process (1.1. The major difference between our model and the previous ones is that premium incomes are assumed to be a sequence of non-negative random variables. To make it fair to both the cedent and the reinsurer, we introduce their joint ruin probability. Starting from the idea in [8], we derive an explicit formula for this probability under some constraints. In particular, we determine the quota share (retention level that minimizes the joint ruin probability and the upper bound of ruin probability of each party. As [4], we establish adjustment coefficients of both the cedent and reinsurer as functions of the quota share level. By applying the martingale and the inductive methods, we obtain the upper bounds of the ruin probabilities of both insurance companies. These bounds are tighter than the one in the surplus process (1.1. This paper is organized as follows. A short description of the model and some notions are presented in Section 2. Section 3 is devoted to the construction of the explicit formula and optimal problem for the joint ruin probability of the cedent and reinsurer. The upper bounds of ruin probabilities obtained

3 On finite-time ruin probabilities in a risk model 2611 by martingale and inductive methods are given in Sections 4 and 5. Finally, a numerical illustration is given. 2 The risk model We consider the surplus process given by (1.1 and assume that the insurer enters into a quota share reinsurance contract. We denote α (α [, 1] as the quota share level. The cedent s surplus and reinsurer s surplus at period n are denoted U n and V n, respectively. Then, U n and V n can be expressed as n n U n u + α Y i α X i, V n v + ( 1 α n Y i ( 1 α n X i, (2.1 where u, v are initial surpluses of the cedent and the reinsurer. The new surplus process (2.1 is an extension of the one in (1.1. We say that the cedent s (the reinsurer s ruin occurs at period n (n 1, 2,... if the cedent s (the reinsurer s surplus at period n falls to zero or below. i.e. U n (V n. (2.2 Hence, the finite-time ruin probability with the insurer s initial surplus u, which we denote as ψ n (1 (u, α, is defined by ( n ψ n (1 (u, α P (U k. (2.3 Similarly, ψ n (2 (v, α is the finite-time ruin probability with the reinsurer s initial surplus v and is defined by ( n ψ n (2 (v, α P (V k. (2.4 We define the finite-time joint ruin probability of the cedent and the reinsurer as the probability that at least one of them gets ruined within period n ( n ψ n (u, v, α P [(U k (V k ]. (2.5 k1 We also define the finite- time joint survival probability of the cedent and the reinsurer by ( n φ n (u, v, α P [(U k > (V k > ]. (2.6 k1 k1 k1

4 2612 Bui Khoi Dam and Nguyen Quang Chung It is easy to see that ψ n (u, v, α 1 φ n (u, v, α. (2.7 In the remaining part of this section, we will denote the probability space as a triple (Ω, F, P where P is complete probability measure and b x i if a > b. The distribution functions of X 1 and Y 1 are denoted by H(x P (X 1 x and F (y P (Y 1 y, respectively. ia 3 The explicit formula for finite-time joint ruin probability We now give an explicit formula for finite-time joint ruin probability in Theorem 3.1 under following assumptions: Assumption 3.1 The sequence Y {Y n } n 1 is a sequence of i.i.d random variables taking values in a finite set of non-negative numbers G Y {y 1, y 2,..., y N } ( y 1 < y 2 <... < y N and q k P(Y n y k (y k G Y, < q k 1, N q k 1. Assumption 3.2 The sequence X {X n } n 1 is also a sequence of i.i.d random variables taking values in a finite set of non-negative numbers G X {x 1, x 2,..., x M } ( x 1 < x 2 <... < x M and p k P(X n x k (x k G X, < p k 1, M p k 1. k1 Assumption 3.3 X and Y are mutually independent. We set n 1 x 1 x 2...x n 1 u + α (y i x i + αy n, (3.1 Z y 1y 2...y n W y 1y 2...y n n 1 x 1 x 2...x n 1 v + (1 α (y i x i + (1 αy n (3.2 for n 1, 2,... and by convention x. Theorem 3.1. Assuming that the surplus processes given in (2.1 satisfy the assumptions from 3.1 to 3.3. Then, the explicit formula for finite-time joint ruin probability is calculated as follows ψ n (u, v, α 1 m 1 1 m m n1 G 1 G 2 q m1 q m2...q mn... l 1 1 l 2 1 G n l n1 k1 p l1 p l2...p ln (3.3

5 On finite-time ruin probabilities in a risk model 2613 where { } G k max l k : αx lk < Z ym 1 ym 2...ym k x l1 x l2...x lk 1, (1 αx lk < W ym 1 ym 2...ym k x l1 x l2...x lk 1 l k,m for k 1, 2,..., n and by convention x l. Proof. First, considering the probability φ n (u, v, α, we have ( n φ n (u, v, α P ( n P k1 k1 [( u + α [(U k > (V k > ] (Y i X i > ( v + (1 α Since Y 1 takes values in the set G Y, (3.4 implies that φ n (u, v, α m 1 1 ( n P(Y 1 y m1 P ( v + (1 α k1 [( u + α ] (Y i X i >. (Y i X i > (3.4 Y1 (Y i X i > ] y m1. (3.5 Similarly, since Y 2, Y 3,..., Y n take values in the set G Y, (3.5 can be written as φ n (u, v, α m 1 1 { N P(Y 1 y m1 m 2 1 { { N P(Y 2 y m2 Y 1 y m1... ( n (Y 1 y m1 (Y 2 y m2... (Y mn 1 y mn 1 P ( X i > v + (1 α k1 m n1 P(Y n y mn [( u + α (Y i (Y1 (Y i X i > ] y m1 (Y 2 y m2 } }}... (Y mn y mn.... (3.6 Since Y {Y n } n is a sequence of independent random variables, X and Y are mutually independent, we have φ n (u, v, α ( n P k1 [( u + α m 1 1 { N P(Y 1 y m1 m 2 1 (y mi X i > { { N P(Y 2 y m2... ( v + (1 α m n1 P(Y n y mn ]} }} (y mi X i >...

6 2614 Bui Khoi Dam and Nguyen Quang Chung m 1 1 m 1 1 m 2 1 P(Y 1 y m1 m 1 1 m 2 1 where A ym1 y m2...y mn m n1 m n1 n k1 m 2 1 P(Y 2 y m2... m n1 P(Y n y mn P(A ym1 y m2...y mn P(Y 1 y m1 P(Y 2 y m2...p(y n y mn P(A ym1 y m2...y mn { } q m1 q m2...q mn P(A ym1 y m2...y mn [( u+α ( (y mi X i > v+(1 α We now calculate the probability of the event A ym1 y m2...y mn. Since X 1 takes values in the set G X, we have P(A ym1 y m2...y mn M l 1 1 ( n P(X 1 x l1 P X i > ] X1 x l1 M l 1 1 k1 [( u + α (y mi X i > (3.7 ] (y mi X i >. ( v + (1 α (y mi ([ ] P(X 1 x l1 P (u + α(y m1 x l1 > (v + (1 α(y m1 x l1 > [( u + α(y m1 x l1 + α(y m2 X 2 > ] [( + (1 α(y m2 X 2 >... u + α(y m1 x l1 + α ( v + (1 α(y m1 x l1 + (1 α ( v + (1 α(y m1 x l1 n (y mi X i > i2 n X1 (y mi X i > ] x l1 i2 (3.8 Since X {X n } n is a sequence of independent random variables, (3.8 implies that G 1 P(A ym1 y m2...y mn l 1 1 ([( P(X 1 x l1 P u + α(y m1 x l1 + α(y m2 X 2 > ( ] v + (1 α(y m1 x l1 + (1 α(y m2 X 2 >... [( n ( u + α(y m1 x l1 + α (y mi X i > v + (1 α i2

7 On finite-time ruin probabilities in a risk model 2615 (y m1 x l1 + (1 α n ] (y mi X i >. (3.9 i2 Similarly, we have known that X 2, X 3,..., X n take values in the set G X, therefore (3.9 gives us G 1 P(A ym1 y m2...y mn l 1 1 { G 2 P(X 1 x l1 G 1 G 2... l 1 1 l 2 1 G 1 G 2... l 1 1 l 2 1 G n l n1 G n l n1 l 2 1 Combining (2.7, (3.7 and (3.1, we obtain ψ n (u, v, α 1 m 1 1 m 2 1 m n1 { { G n P(X 2 x l2... l n1 P(X 1 x l1 P(X 2 x l2...p(x n x ln } }} P(X n x ln... p l1 p l1...p ln. (3.1 G 1 G 2 q m1 q m2...q mn... l 1 1 l 2 1 G n l n1 p l1 p l2...p ln. Remark 3.1. By using the similar argument as in Theorem 3.1, we get ψ (1 n where (u, α 1 m 1 1 m m n1 H 1 H 2 q m1 q m2...q mn... l 1 1 l 2 1 { } H k max l k : αx lk < Z ym 1 ym 2...ym k x l1 x l2...x lk 1 l k,m H n l n1 p l1 p l2...p ln (3.11 for k 1, 2,..., n and by convention x l. Similarly, an explicit formula for finite- time ruin probability of the reinsurer is ψ (2 n where (v, α 1 m 1 1 m m n1 T 1 T 2 q m1 q m2...q mn... l 1 1 l 2 1 T n l n1 { } T k max l k : (1 αx lk < W ym 1 ym 2...ym k x l1 x l2...x lk 1 l k,m for k 1, 2,..., n and by convention x l. p l1 p l2...p ln (3.12 For a given pair (u, v, the following theorem shows how to determine α so that the joint ruin probability of the cedent and the reinsurer is minimum.

8 2616 Bui Khoi Dam and Nguyen Quang Chung ( Theorem 3.2. When u and v are given and α (, 1, the function ψ n u, v, α attains its minimum value at α u u+v (α (, 1. Proof. First, we consider the joint survival probability φ n (u, v, α, n 1, 2,... ( n [ (Ui φ n (u, v, α P > ( V i > ] ([ 1 ( u P (X i Y i < min α, v ] [ 2 (X i Y i 1 α ( u < min α, v ] [ n n ( u... X i Y i < min 1 α α, v ]. 1 α We set f (α min ( u, ( v α 1 α for α, 1. Then f ( α min ( u α, v 1 α { v 1 α u α α ( ] u, u+v α ( u, 1. u+v (3.13 The function f ( α increases on interval ( u, u+v] and decreases on interval ( u, 1. Hence, α u is the point at which the function f( α attains its u+v u+v maximum value. { We now denote: A αk w Ω : (X i Y i < min ( u, } v α 1 α { and A α k w Ω : (X i Y i < min ( } u v,. α 1 α It is obvious that A αk A α k for k 1,...n. So we obtain or, equivalently, P (A α1 A α2 A αn P (A α 1A α 2 A α n for all α (, 1, φ n (u, v, α φ n (u, v, α for all α (, 1. (3.14 From (2.7 and (3.14, we deduce that the probability ψ n (u, v, α attains minimum at α u. u+v We assume that the optimality criterion is to minimize the joint ruin probability, for the given values of u and v, then α u is said to be the optimal u+v quota share level of the joint ruin probability. If premium incomes and claim amounts form sequences of i.i.d random variables, then it is difficult to establish the explicit formula for the joint ruin probability and ruin probability of each party. Therefore, in the following sections, we give the upper bounds of ψ n (1 (u, α, ψ n (2 (v, α and ψ n (u, v, α by using martingale and inductive methods.

9 On finite-time ruin probabilities in a risk model The martingale method for ruin probabilities Lemma 4.1. If sup(x 1 < +, sup(y 1 < +, E(Y 1 > E(X 1 and P (X 1 Y 1 > > then each of the following equations has a unique positive root E ( e αr(x 1 Y 1 1, (4.1 for any α (α (, 1. E ( e (1 αr(x 1 Y 1 1 (4.2 Proof. We set M 1 sup(x 1 + sup(y 1 and Q 1 (X 1 Y 1. We have e R(X 1 Y 1 e RM 1 for any R [, + ; E(e RM 1 < +. (4.3 Therefore, there exists the expectation value of e R(X 1 Y 1 for any R [, +. That automatically implies that the expectation values E ( e αr(x 1 Y 1 and E ( e (1 αr(x 1 Y 1 on the right hand side of (4.1 and (4.2 exist. According to the hypothesis of Lemma 4.1, the equation (4.4 has a unique positive root E ( e R(X 1 Y 1 1. (4.4 Indeed, if we set g(r E ( e RQ 1 1 for R [, then Differentiating the above function, we get g g(r + R g(r (R lim R R g(. (4.5 lim R Ω e RQ 1 (e RQ 1 1 dp. (4.6 R Let { R n } n 1 ( R n, R + R n >, R n be an arbitrary real-valued sequence as n. The Maclaurin series expansion for e RnQ 1 is given by e RnQ R nq 1 e θ RnQ 1, ( < θ < 1. (4.7 1! Moreover, for all ɛ >, there exists a natural number n such that R n < M 2 for all n 1, where M 2 max{ɛ, R 1, R 2,..., R n }. Thus, we have e RQ 1 (e RnQ 1 1 R n Q1 e (R+θ RnQ 1 Q1 e (R+θM 2 Q 1 M1 e (R+θM 2M 1

10 2618 Bui Khoi Dam and Nguyen Quang Chung for n 1, 2,... and E (M 1 e (R+θM 2M 1 M 1 e (R+θM 2M 1 < +. Applying Lebesgue s Dominated Convergence Theorem, we have e RQ 1 (e RnQ 1 1 e RQ 1 (e RnQ 1 1 lim dp lim dp n R n n R n Ω Ω Ω Q 1 e RQ 1 dp E ( (X 1 Y 1 e R(X 1 Y 1. (4.8 From (4.6 and (4.8 we deduce that g(r is differentiable So, g (R E ( Q 1 e RQ 1 E ( (X 1 Y 1 e R(X 1 Y 1. g ( E (X 1 Y 1 < (4.9 That implies g(r is decreasing at R. Since P (X 1 Y 1 > > there exists δ > so that P (X 1 Y 1 δ >. We have g(r E(e R(X 1 Y 1 1 E(e R(X 1 Y 1 1 (X1 Y 1 δ 1 E(e Rδ 1 (X1 Y 1 δ 1 e Rδ P(X 1 Y 1 δ 1. (4.1 The right-hand side of (4.1 tends to infinity as R +. It implies that lim g(r +. (4.11 R + Combining (4.5,(4.9 and (4.11, we can deduce that the function g(r must intersect the x-axis. In other words, there exists a positive x-intercept of g(r. Let s denote it R (R >. Apparently, R is the root of the equation (4.4. Similarly, we can prove that g(r is twice differentiable. Hence, we also have g (R E ( (X 1 Y 1 2 e R(X 1 Y 1 ; g (R. That means g(r is concave for R [, +. This shows that R > is a unique positive root of g(r. Plugging in R α R and R α α R, we deduce that (4.1 and (4.2 also have 1 α a unique positive root. (for any α (, 1. Remark The coefficient R introduced in [14] and [18] was called the adjustment coefficient.

11 On finite-time ruin probabilities in a risk model We make a convention that R α + if α and R α + if α 1. The following theorem shows that ψ n (1 (u, α and ψ n (2 (v, α have exponential upper bounds. Its proof is similar to the ones in [14] and [18], so the detailed proof process is omitted. Theorem 4.2. Assuming that the processes given in (2.1 satisfy all assumptions in Lemma 4.1. For any α (α (, 1 then ψ (1 n (u, α e u R α (4.12 and n 1, 2,... ψ (2 n (v, α e v R α (4.13 Proof. In order to prove (4.12, we set the stochastic process {Z n } n Z e u R α, Z n e R α ( u+α n (Y i X i e R αu n with n 1, 2,... and the filtration {F n } n 1 where F {, Ω}, F n σ(z 1, Z 2,..., Z n σ(x 1, X 2,..., X n, Y 1, Y 2,..., Y n, n 1, 2,... The stochastic process {Z n } n is a martingale with respect to the filtration {F n } n. Let τ min {n : U n }. Then n τ min (n, τ is a finite stopping time. Thus, by the optional stopping theorem for martingale {Z n } n, we get This implies that E (Z n τ E (Z e u R α. e u R α E (Z n τ E ( Z n τ 1 (τ n E ( Zτ 1 (τ n. (4.14 Combining (4.14 and Z τ 1, we obtain e u R α E (1 τ n P (τ n ψ (1 n (u, α. (4.15 The proof of inequality (4.13 is similar to the one for inequality (4.12. We set L 1 (ɛ { ɛ R ɛ e ur ; ɛ e vr ; ɛ e (u+vr }. (4.16 Clearly, if ɛ L 1 (ɛ then < ɛ < 1 and ɛ e (u+vr is minimal.

12 262 Bui Khoi Dam and Nguyen Quang Chung Corollary 4.3. For any given ɛ L 1 (ɛ there exists α such that ψ n (1 (u, α ɛ and ψ n (2 (v, α ɛ. Particularly, if ɛ e (u+vr then α u. u+v Proof. Since ɛ L 1 (ɛ, by definition, we have ɛ e ur, so Similarly, the condition ɛ e vr ur ln ɛ gives us 1. ( vr ln ɛ. (4.18 Finally, the condition ɛ e (u+vr leads to 1 + vr lnɛ ur lnɛ. (4.19 By (4.17, (4.18 and (4.19 this implies the existence α (, 1 such that From α ur ln ɛ, we obtain e u R α ɛ. According to Theorem 4.2, it implies that Similarly, from 1 + vr ln ɛ 1 + vr lnɛ α ur lnɛ. (4.2 α we have ψ (1 n (u, α ɛ. (4.21 ψ (2 n (v, α ɛ. (4.22 Plugging ɛ e (u+vr into (4.2, we have the unique value α u u+v. This completes the proof. This corollary shows that the upper bounds in (4.21 and (4.22 attain minimum values at α u. However, Theorem 3.2 also shows that α u u+v u+v is a value at which ψ n (u, v, α attains its minimum value. Hence, α u u+v minimises not only the joint ruin probability but also the upper bounds in (4.21 and (4.22. Remark When establishing the upper bounds of ruin probabilities of an insurance company, in [14] and [18], the authors assumed that R > without proving its existence whereas we came up with Lemma 4.1 and proved the existence and uniqueness of R.

13 On finite-time ruin probabilities in a risk model If α 1 then ψ (1 n (u, 1 ψ n (u. [14] and [18] also proved that ψ n (u e ur, this result gave us the exponential upper bound of the insurer s ruin probability in surplus process given in (1.1. Obviously, e u R α e ur. Thus, the upper bound in (4.12 is tighter than the one in [14] and [18]. 3. The upper bound obtained from (4.12 is an increasing function with respect to α while the one in (4.13 is decreasing. So, the two bounds can not attain their minimum values at the same value of α (, 1. Therefore, when considering the interests of both insurance companies, for a given ɛ, Corollary 4.3 shows us how to determine α so that both finite-time ruin probabilities are less than ɛ. 4. By letting n in (4.12 and (4.13, we obtain ψ (1 (u, α e u R α and ψ (2 (v, α e v R α (4.23 where ψ (1 (u, α and ψ (2 (v, α are the ultimate ruin probabilities of the cedent and the reinsurer, respectively. 5. Clearly, we have ψ n (u, v, α e u R α + e v R α. (4.24 In Section 5, we use the inductive method to derive the upper bounds of ψ n (1 (u, α and ψ n (2 (v, α. These bounds are not in exponential form but are tighter than the exponential ones obtained in (4.12 and ( The inductive method for ruin probabilities Throughout this section, without further mention, we will assume that X 1 and Y 1 are continuous random variables. We denote the tail of a distribution function H(x by H(x 1 H(x. The following lemma provides recursive equations for ψ n (1 (u, α and ψ n (2 (v, α. Lemma 5.1. For any given u >, v > and α (, 1, we have: ( ψ n+1(u, (1 1 α H (u + αy df (y α + ψ (2 n+1(v, α + 1 α (u+αy ψ (1 n (u + α(y x, αdh(xdf (y, (5.1 ( 1 H (v + (1 αy df (y 1 α 1 1 α (v+(1 αy ψ n (2 (v + (1 α(y x, αdh(xdf (y. (5.2

14 2622 Bui Khoi Dam and Nguyen Quang Chung n 1, 2,... In particular, ψ (1 1 (u, α ψ (2 1 (v, α ( 1 H α H Proof. For n 1, 2,... ( n+1 ψ n+1(u, (1 α P (U i + P ( n+1 1 α (u+αy P 1 α (αy+u (u + αy ( 1 (v + (1 αy 1 α df (y, (5.3 df (y (5.4 (U i X 1 x, Y 1 y dh(xdf (y ( n+1 P (U i X 1 x, Y 1 y dh(xdf (y ( n+1 (U i X 1 x, Y 1 y dh(xdf (y. If x 1 (u + αy then the cedent s ruin occurs at period n 1. i.e. α P (U 1 X 1 x, Y 1 y 1 (5.5 which implies that P ( n+1 (U i X 1 x, Y 1 y 1. (5.6 If x < 1 (u + αy then the cedent s ruin does not occur at period n1. i.e. α So P ( n+1 P (U 1 X 1 x, Y 1 y (U i X 1 x, Y 1 y P ( n+1 (U i X 1 x, Y 1 y i2 Plugging (5.6 and (5.7 into (5.5, we get ( ψ n+1(u, (1 1 α H (u + αy df (y α ψ (1 n (u + α(y x, α. (5.7

15 On finite-time ruin probabilities in a risk model 2623 On the other hand, ψ (1 + 1 (u, α P (U α (u+αy ψ n (1 (u + α(y x, αdh(xdf (y. P (u + αy 1 αx 1 X 1 x, Y 1 y dh(xdf (y 1 α (u+αy So, (5.1 and (5.3 hold true. Similarly, (5.2 and (5.4 hold. P (u + αy 1 αx 1 X 1 x, Y 1 y dh(xdf (y P (u + αy 1 αx 1 X 1 x, Y 1 y dh(xdf (y 1 α (u+αy ( 1 H (u + αy df (y. α Formulas (5.1 and (5.2 are called recursive equations for ruin probabilities of the cedent and reinsurer, respectively. Theorem 5.2. Assuming that the processes given in (2.1 satisfy the conditions of Lemma 4.1. For any α (α (, 1 then and where γ 1 inf z e Rx dh(x z e R z H(z Proof. For any z, we have Since γ 1 inf z H(z { z γe α R αz ψ (1 n (u, α γe u R α (5.8 ψ (2 n (v, α γe v R α (5.9 and n 1, 2,... e Rx dh(x z e R z H(z e α R αx dh(x e α R αz H(z z inf z z } 1 α Rαz e e α R αx dh(x. e α R αz H(z z e α R αx dh(x e α R αx dh(x (5.1

16 2624 Bui Khoi Dam and Nguyen Quang Chung γe α R αz E (e α R αx 1. (5.11 Replacing z by 1 (u + αy in (5.11 and using (5.3, we have α ψ (1 1 (u, α γe (e α R αx 1 e R α(u+αy df (y ( γe u R α E e α R α(x 1 Y 1 γe u R α. (5.12 Under an inductive hypothesis, we assume that ψ (1 n (u, α γe u R α. (5.13 We prove (5.13 holds for n + 1. Indeed, for x < 1 (u + αy, replacing u by u + α(y x in (5.13, we have α ψ (1 n (u + α(y x, α γe (u+α(y x R α. (5.14 From (5.1, (5.14 and z being replaced by 1 (u + αy in (5.1, we have α ψ (1 n+1(u, α + γe Rα(u+α(y x dh(xdf (y 1 α (u+αy 1 α (u+αy γe R α(u+α(y x dh(xdf (y ( γe R α(u+α(y x dh(xdf (y γe u R α E e α R α(x 1 Y 1 γe u R α (5.15 Then, ψ (1 n (u, α γe u R α holds for all n 1, 2,... The proof of (5.9 is similar to the one for (5.8 where (5.2 and (5.3 are used in its proof. If γ then E(e R X 1 e R z H(z this implies that e R x dh(x e R z H(z E(e R X 1 e R z H(z e Rx dh(x z e R z H(z inf z e Rx dh(x z e R z H(z γ for all z. (5.16 Since E(e R X 1 < + if the equality (5.16 occurs then H(z for all z. This leads to a contradiction. Moreover γ 1 inf z e Rx dh(x z e R z H(z inf z e Rz dh(x z e R z H(z 1.

17 On finite-time ruin probabilities in a risk model 2625 So, < γ 1. We set L 2 (ɛ { ɛ R ɛ γe ur ; ɛ γe vr ; ɛ γe (u+vr } (5.17 and a 1 e (u+vr, a 2 γe (u+vr, b 1 min ( ( e ur, e vr, b2 min γe ur, γe vr. It s obvious that a 2 a 1, b 2 b 1, a 1 b 2 if γ e ur and γ e vr, a 1 > b 2 if γ < e ur or γ < e vr. Therefore, we have L 1 (ɛ L 2 (ɛ [a 1, b 2 ] if γ e ur and γ e vr (5.18 and L 1 (ɛ L 2 (ɛ if γ < e ur or γ < e vr. (5.19 Corollary 5.3. For any given ɛ L 2 (ɛ there exists α such that ψ n (1 (u, α ɛ and ψ n (2 (v, α ɛ. Particularly, if ɛ γe (u+vr then α u. u+v Proof. Since ɛ γe ur, ɛ < γ. The proof is similar to the one for Corollary 4.3. Remark Since, < γ 1, the upper bounds in (5.8 and (5.9 are tighter than the ones (4.12 and ( The upper bounds in (5.8 is an increasing function with respect to α while the one in (5.9 is decreasing. Therefore, there does not exist α (, 1 so that the two upper bounds both attain their minimum values. 3. By letting n in (5.8 and (5.9, we have ψ (1 (u, α γe u R α and ψ (2 (v, α γe v R α (5.2 From (5.8 and (5.9, we obtain the upper bound of the joint ruin probability within period n ψ n (u, v, α γ (e u R α + e v R α, (n 1, 2,... ( Numerical illustration Suppose that Y {Y n } n 1 and X {X n } n 1 satisfy the conditions in Theorem 3.1 where G Y {, 1, 2, 3, 4}, G X {, 1, 2, 3, 4}, u 3.5 and v Distribution functions of Y 1 and X 1 are defined in Tables 1 and 2, respectively:

18 2626 Bui Khoi Dam and Nguyen Quang Chung Y q k P(Y 1 k Table 1: Distribution function of Y 1. X p k P(X 1 k Table 2: Distribution function of X 1. The following table gives some numerical results of ψ n (u, v, α. n α α α Table 3: The different values of ψ n (u, v, α. From Theorem 3.2, we obtain α n α Table 4: The different values of ψ n(u, v, α. From the data in Tables 3 and 4, we see that in the same period, ψ n (u, v, α attains its minimum value at α. This is in a good accordance with Theorem 3.2. Obviously, Y 1 and X 1 satisfy all assumptions in Lemma 4.1 ( E(Y , E(X Therefore, the sequences Y {Y n } n 1 and X {X n } n 1 also satisfy the assumptions in Theorem 4.2. In this case, equation (4.4 can be rewritten as e 4R e 3R e 2R e R e R e 2R e 3R e 4R (6.1 After solving equation (6.1, we obtain R We calculate Table 5 for alpha α.35. We have R α and R α The Table 5 shows that for small periods, the difference between the ruin probability and the upper bound is quite large. Thus, the results obtained in (4.12 and (4.13 are not good for small periods.

19 On finite-time ruin probabilities in a risk model 2627 n ψ (1 n (u, α ψ (2 n (v, α e u R α e v R α Table 5: The values of ruin probabilities and upper bounds (4.12 and ( Conclusion The model studied in this paper can be viewed as an extension of the one studied in [14] and [18] by considering the effect of quota share reinsurance on surplus processes. When X and Y are two sequences of i.i.d random variables taking values in finite sets of non-negative numbers, in Theorem 3.1, we give the explicit formulas for the joint ruin probability and the ruin probabilities of two insurance companies. When X and Y are two sequences of i.i.d random variables, we give the upper bounds of the ruin probabilities by using the martingale method, in Theorem 4.2. We assume that X and Y are two sequences of i.i.d random variables and X 1, Y 1 are continuous random variables. By using the inductive method in Theorem 5.2, we derive the non-exponential upper bounds of the ruin probabilities. These bounds are tighter than the ones obtained by using the martingale method. We have shown that α u minimizes the joint ruin probability and u+v the upper bounds of ruin probabilities of two insurance companies. References [1] J. Cai, Discrete Time Risk Models Under Rates of Interest, Probability in the Engineering and Informational Sciences, 16 (22, no. 3, [2] J. Cai and D. C. M. Dickson, Ruin Probabilities With a Markov Chain Interest Model, Insurance: Mathematics and Economics, 35 (24, no. 3, [3] J. Cai, K. S. Tan, C. Weng and Y. Zhang, Optimal reinsurance under VaR and CTE risk measures, Insurance: Mathematics and Economics, 43 (28, no. 1,

20 2628 Bui Khoi Dam and Nguyen Quang Chung [4] M. L. Centeno, Measuring the effects of reinsurance by the adjustment coefficient, Insurance: Mathematics and Economics, 5 (1986, no. 2, [5] M. L. Centeno, Measuring the effects of reinsurance by the adjustment coefficient in the Sparre Anderson model, Insurance: Mathematics and Economics, 3 (22, no. 1, [6] M. L. Centeno, Excess of loss reinsurance and Gerber s inequality in the Sparre Anderson model, Insurance: Mathematics and Economics, 31 (22, no. 3, [7] D. C. M. Dickson, Insurance Risk and Ruin, Cambridge University Press, 26. [8] N. T. T. Hong, On Finite-Time Ruin Probabilities for General Risk Models, East-West Journal of Mathematics, 15 (213, no. 1, [9] V. K. Kaishev and D. S. Dimitrova, Excess of Loss Reinsurance under Joint Survival Optimality, Insurance: Mathematics and Economics, 39 (26, no. 3, [1] C. Lefèvre and S. Loisel, On Finite-Time Ruin Probabilities for Clasical Risk Models, Scandinavian Actuarial Journal, 28 (28, no. 1, [11] X.Lin, Z. Dongjin and Z. Yanru, Minimizing Upper Bound of Ruin Probability Under Discrete Risk Model with Markov Chain Interest Rate, Communications in Statistics- Theory and Methods, 44 (215, no. 4, [12] J. Ohlin, On a class of measures of dispersion with application to optimal reinsurance, The ASTIN Bulletin, 5 (1969, no. 2, [13] P. Picard and C. Lefèvre, The Probability of Ruin in Finite-Time with Discrete Claim Size Distribution, Scandinavian Actuarial Journal, 1997 (1997, no. 1, [14] S. Ross, Stochastic Processes, New York: John Wiley & Sons, [15] S. Vajda, Minimum variance reinsurance, The ASTIN Bulletin, 2 (1962, no. 2,

21 On finite-time ruin probabilities in a risk model 2629 [16] W. Wei and Y. Hu, Ruin Probabilities for Discrete Time Risk Models With Stochastic Rate of Interest, Statistics & Probability Letters, 78 (28, no. 6, [17] D. Williams, Probability with Martingales, Cambridge University Press, [18] H. Yang, Non-exponential Bounds for Ruin Probability with Interest Effect Included, Scandinavian Actuarial Journal, 1999 (1999, no. 1, [19] L. Zhang, X. Hu and B. Duan, Optimal reinsurance under adjustment coefficient measure in a discrete risk model based on Poisson MA(1 process, Scandinavian Actuarial Journal, 215 (215, no. 5, Received: August 11, 217; Published: October 8, 217