On a compound Markov binomial risk model with time-correlated claims

Transcription

1 Mathematica Aeterna, Vol. 5, 2015, no. 3, On a compound Markov binomial risk model with time-correlated claims Zhenhua Bao School of Mathematics, Liaoning Normal University, Dalian , China hhbao@126.com He Liu School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology, Baotou , China younglust@163.com Abstract In this paper we consider an extension to the compound Markov binomial risk model in which two kinds of dependent claims are introduced. For the proposed risk model, the generating functions of two conditional expected discounted penalty functions are obtained. Based on these results, we derive a recursive formula for the conditional expected discounted penalty function when there is no claim at time 0. The relationship between the two conditional expected discounted penalty functions is then investigated. Mathematics Subject Classification: 62P05, 91B30 Keywords: Markov binomial risk model, Delayed claims, Expected discounted penalty function, Recursive equation, Generating function 1 Introduction As a discrete analogue of classical compound Poisson risk model, the compound binomial risk model was first proposed by [1] and further studied by a fair amount of researchers, see [2] and the references therein for details. [3] extended the compound binomial model to the case involving two types of correlated risks. The occurrence of one claim may induce the occurrence of another claim with different distribution of severity. However, the time of occurrence of the induced claim may be delayed to the next period with a certain

2 432 Zhenhua Bao and He Liu probability. They obtained recursive formula for the finite time ruin probabilities and explicit expressions for ultimate ruin probabilities in some special cases. [4] further investigated the joint distribution of the surplus prior to ruin and deficit at ruin. [5, 6] extended the compound binomial risk model to the compound Markov binomial model which introduces time-dependence in the aggregate claim amount increments governed by a Markov process. Motivated by the above mentioned literature, in the present paper we consider the compound Markov binomial risk model with time correlated claims. 2 The Model Let {I t, t N + } be a stationary homogeneous Markov chain with state space {0, 1} and transition probability matrix p00 p πq 1 πq P = =, p 10 p 11 1 q1 π π +1 πq wherep ij = PrI t+1 = j I t = ifort Nandi,j {0,1}. Initialprobabilities are denoted by PrI 0 = 1 = q = 1 PrI 0 = 0 for 0 < q < 1, and π is the dependence parameter with 0 π < 1. Note that {I t, t N} is sometimes called a Markov-Bernoulli sequence. Now we consider a risk model which involves two kinds of insurance claims, namely the main claims and the by-claims. The main claim amounts {X i, i N + } are i.i.d. positive and integer valued r.v. s with common probability function p.f. f X, cumulative distribution function c.d.f. F X and mean µ X. The total amount of main claims up to time k N is defined as U X t = t X i I i, i=1 with U X 0 = 0. The occurrence r.v. I i and individual claim amount r.v. X i are independent in each time period. Suppose that each mainclaim may induce a by-claim. The by-claim and its associated main claim may occur simultaneously with probability θ 0 θ 1, or the occurrence of the by-claim may be delayed to the next time period with probability 1 θ. The by-claim amounts {Y i, i N + } are assumed to be i.i.d. positive integer valued r.v. s having common p.f. f Y, c.d.f. F Y and mean µ Y. The independence between {X i, i N + } and {Y i, i N + } is also assumed. The premium rate is assumed to be equal to 1. The surplus process of an insurance company is defined as U t = u+t Ut X Ut Y, t N, 1

3 On the compound Markov binomial risk model with time-correlated claims 433 where U 0 = u N corresponds to the initial surplus and U Y t is the total amount of by-claims up to time t with U Y 0 = 0. Since the sequence {I t, t N} is stationary, we have PrI t = 1 = q for t N +. From [3] we know that E[U t+1 ] = tqµ X +µ Y + qµ X + qθµ Y. Therefore, we assume that qµ X +µ Y < 1, 2 which ensures that the surplus process U t goes almost surely to infinity as t. Let T = min{t N +, U t < 0} be the time of ruin with T = if ruin does not occur. Note that if ruin occurs, U T is the deficit at ruin and U T 1 is the surplus one period prior to ruin. Denote by mu i = E [ v T wu T 1, U T 1 {T< } U 0 = u,i 0 = i ], i = 0,1, the conditional Gerber-Shiu discounted penalty function, where 1 {A} is the indicator function of event A and wi,j : N N + N is a bounded function. The unconditional Gerber-Shiu function is defined as then it is clear that mu = E [ v T wu T 1, U T 1 {T< } U 0 = u ], mu = 1 qmu 0+qmu Main Results In this section, we aim to derive recursive equations for the conditional expected discounted penalty function mu i, i = 0,1. Similar to the one in [3], we study the claim occurrences in two scenarios. The first is a main claim and its associated by-claim occur concurrently, then the surplus process gets renewed. The second case is that a main claim occurs but its associated byclaim will be delayed to the next time period. Conditional on the second scenario, we define a complementary surplus process U 1t = u+t U X t U Y t Y, t N +, 4 with U 10 = u. Denote the corresponding conditional penalty function of process 4 by m 1 u i, i = 0,1.

4 434 Zhenhua Bao and He Liu 3.1 The generating function of the conditional penalty function We first show that explicit expressions for the generating functions of the conditional expected discounted penalty functions can be obtained. Considering what will happen at time 1, we have where mu 0 = vp 00 mu+1 0 +vp 01 σu+1 θw1 u+1+θw 3 u+1, 5 mu 1 = vp 10 mu+1 0 +vp 11 σu+1 θw1 u+1+θw 3 u+1, 6 m 1 u 1 = vp 10 σ1 u+w 2 u+1 +vp 11 σ2 u+1 θw 3 u+1+θw 4 u+1, 7 u+1 u+1 σu = 1 θ m 1 u+1 k 1f X k+θ mu+1 k 1f X Y k, σ 1 u = k=1 u+1 mu+1 k 0f Y k, k=1 u+1 u+1 σ 2 u = 1 θ m 1 u+1 k 1f X Y k+θ mu+1 k 1f X Y Y k, and w 1 i = w 3 i = k=i+1 k=i+1 k=2 wi 1,k if X k, w 2 i = wi 1,k if X Y k, w 4 i = k=2 k=i+1 k=3 k=i+1 wi 1,k if Y k, wi 1,k if X Y Y k. Throughout the rest of the paper, we will denote the generating function of a function by adding a hat on the corresponding letter. Multiplying 5 by u+1 and summing over u from 0 to, we get ˆm 0 = vp 00 ˆm 0 m0 0 +vp01 1 θ ˆm 1 1ˆf X +ŵ 1 +vp 01 θ ˆm 1ˆf X Y +ŵ 3, which is equivalent to v p 00 ˆm 0 p 01 θˆf X Y ˆm 1 = p 01 1 θˆf X ˆm 1 1+p 01 ŵ 13 p 00 m0 0, 8

5 On the compound Markov binomial risk model with time-correlated claims 435 where w ij k = 1 θw i k+θw j k for k N + and 1 i < j 4. and For the other two processes 6 and 7, we have analogously p 10ˆm 0+ v p 11θˆf X Y ˆm 1 = p 11 1 θˆf X ˆm 1 1+p 11 ŵ 13 p 10 m0 0, 9 p 10 ˆfY ˆm 0 p 11 θˆf X Y Y ˆm 1 = v p 111 θˆf X Y ˆm 1 1+p 10 ŵ 2 +p 11 ŵ In what follows we denote the determinant of a matrix A by A, denote its transposed matrix and adjoint matrix by A T and A, respectively. Let p 00 p 01 θˆf X Y p 01 1 θˆf X A = p 10 p 11 θˆf X Y p 11 1 θˆf X, p 10 ˆfY p 11 θˆf X Y Y p 11 1 θˆf X Y p 01 ŵ 13 p 00 m0 0 B = p 11 ŵ 13 p 10 m0 0, p 10 ŵ 2 +p 11 ŵ 34 and ˆm = ˆm 0, ˆm 1, ˆm 1 1 T, then we can rewrite the Eq.s 8-10 in the following matrix form Q ˆm = vb, 11 where Q = I va and I is the identity matrix. Solving the linear system of equations 11 gives ˆm = v Q Q B. 12 Lemma 3.1. For 0 < v 1, the generalied Lundberg s equation has a real root in the interval vp 00,v], say ρ = ρv. Proof. After careful calculations we obtain Q 2 = 0, 13 Q 2 = vp 00 + v vπ p 11 ˆf X Y. 14

6 436 Zhenhua Bao and He Liu By noting that Qvp 00 vp 00 2 = p 01p 10 p 00 vˆf X Y vp 00 < 0, Qv v 2 = v p 01 +π p 11 ˆf X Y v = vp 01 1 ˆf X Y v 0, we conclude that there exists a real number ρ = ρv in vp 00,v] such that 1 ρ 2 Qρ = 0. To ultimately invert ˆm, we first need an explicit expression for m0 0. Since Qρ = 0, we know that there exists at least one nontrivial solution of the following equation [Qρ] T X = Solving the homogeneous linear system of equations 15, we get a non-ero particular solution X 0 as X 0 = vp 10 ˆfY ρ, θρ vp 00 ˆf Y ρ, 1 θρ vp 00 T. Therefore, we obtain from 11 that X T Bρ = 0, which implies 0 { } θp11 ρ vp 00 +vp 01 p 10 ˆf Y ρŵ 13 ρ +1 θρ vp 00 p 10 ŵ 2 ρ+p 11 ŵ 34 ρ m0 0 =. 16 p 10 θρ+v1 θp00 ˆfY ρ Moreover, substituting ρ vp 00 = v ρ p 11ρ vπˆf X Y ρ, into 16, we have the following result for the conditional expected discounted penalty function m0 0. Theorem 3.2. For u = 0, it holds that m0 0 = where hρ = ρp 11 vπ. 1 p 10 ρ θρ+v1 θp 00 { ρ θhρ+v1 θp 01 p 10 ŵ13 ρ +v1 θhρˆf X ρ p 10 ŵ 2 ρ+p 11 ŵ 34 ρ }, 17 On the other hand, one can easily compute that the adjoint matrix Q of Q has the form Q = vp 00 I +vc, 18

7 On the compound Markov binomial risk model with time-correlated claims 437 where p 00 p 11 ˆfX Y p 01 θˆf X Y p 01 1 θˆf X C = 1 θhˆf X Y 1 θhˆf X. p 10 p 10 ˆfY θhˆf X Y Y θhˆf X Y Now we are ready to give explicit expressions for the generating functions of the conditional expected discounted penalty functions. Let γj = w 13 j v1 θp 11 +v1 θ j f X Y j i+1w 13 i i=1 j f X j i+1 p 10 w 2 i+p 11 w 34 i, j N +, i=1 then ˆγ = j=1 j γj has the form ˆγ = Substituting 18 into 12 we get ˆfX Y 1 v1 θp 11 ŵ 13 +v1 θ ˆf X p 10 ŵ 2 +p 11 ŵ 34. [ ˆm 0 = v2 p 01ˆγ+m0 0 vπ+1 θp 01 p 10 ˆf X Y p 00 ], 19 Q and ˆm 1 = v Q [ hˆγ+p 10 m0 0 where the constant m0 0 is determined by 17. v1 θ hˆf X Y ], 3.2 Recursive equation for mu 0 In this subsection, we aim to derive a defective renewal equation for mu 0. Setting = ρ in 19 yields p 00 m0 0 = p ˆfX Y ρ 01ˆγρ+v π +1 θp 01 p 10 m ρ

8 438 Zhenhua Bao and He Liu Thus, substituting 20 into 19 yieds [ ρ vp 11 ˆfX Y ˆf X Y ρ +v 2 π ˆfX Y = vp 01 ˆγ ˆγρ +v 2 π +1 θp 01 p 10 m0 0 ˆfX Y which is equivalent to ˆm 0 = v ρhˆf X Y hρˆf X Y ρ ρ ρ ˆf ] X Y ρ ˆm 0 ρ ˆf X Y ρ, ρ ˆm 0+vp 01 ˆγ ˆγρ ρ +v 2 ρˆf X Y π +1 θp 01 p ˆf X Y ρ 10 m ρ ρ For j N +, we define ξj = 1 v 2 π α ρ ρ f X Yj+v 1 ˆf X Y ρ hρβ ρ j, ρ1 ρ where v 2 π1 ρ+vhρ 1 ˆf X Y ρ α ρ =, ρ1 ρ / β ρ j = ρ i f X Y i+j ρ i FX Y i, i=0 i=0 with F X Y i = k=i+1 f X Yk. Then it is easily seen that β ρ j is a p.f., and hence ξj is also a p.f. Moreover, direct calculations show that Therefore, we can rewrite 21 as v ρhˆf X Y hρˆf X Y ρ ρ ρ ˆm 0 = α ρˆξˆm 0+vp01 ˆγ ˆγρ ρ = α ρˆξ. +v 2 ρˆf X Y π +1 θp 01 p ˆf X Y ρ 10 m ρ ρ The following theorem shows that mu 0 satisfies a recursive equation.

9 On the compound Markov binomial risk model with time-correlated claims 439 Theorem 3.3. For u N, we have u mu 0 = α ρ mu i 0ξi+1+vp 01 i=0 +v 2 π +1 θp 01 p 10 m0 0 where the constant m0 0 is determined by 17. Proof. It is not different to see that ˆξˆm 0 = i=u+1 i=u+2 j j mj i 0ξi, j=1 i=1 ˆγ ˆγρ = ρ j=1 j ρ i γi+j, i=0 ρ i u 1 γi ρ i u 2 f X Y i, 23 and ρˆf X Y ˆf X Y ρ ρ = ˆfX Y ρ ˆf X Y ρ ˆf X Y = j ρ i f X Y i+j. j=1 i=1 Therefore, we can rewrite 22 as j j+1 mj 0 = α ρ j+1 mj i 0ξi+1 j=0 j=0 j=0 i=0 +vp 01 j+1 ρ i j 1 γi i=j+1 + v2 π +1 θp 01 p 10 m0 0 j+1 ρ i j 1 f X Y i, ρ j=0 which yields 23 by comparing the coefficients of j The evaluation of mu 1 It is known from 5 that i=j+2 mu 0 vp 00 mu+1 0 vp 01 = σu+w 13 u+1. 24

10 440 Zhenhua Bao and He Liu Substituting 24 into 6 yields mu 1 = p 11 p 01 mu 0 vπ p 01 mu+1 0, 25 which is the relationship between the expected discounted penalty function in the ordinary renewal case and the delayed renewal case. We remark that in the classical compound Markov binomial risk model, the relationship between the expected discounted penalty function in the ordinary renewal case and the delayed renewal case has been discussed by [7]. ACKNOWLEDGEMENTS. This research was supported by the Program for Liaoning Excellent Talents in University LR References [1] H.U. Gerber. Mathematical fun with the compound binomial process. Astin Bull., , [2] S. Cheng, H.U. Gerber, E.S.W. Shiu. Discounted probabilities and ruin theory in the compound binomial model. Insurance Math. Econom., , [3] K.C. Yuen, J. Guo. Ruin probabilities for time-correlated claims in the compound binomial model. Insurance Math. Econom., , [4] Y. Xiao, J. Guo. The compound binomial risk model with time-correlated claims. Insurance Math. Econom., , [5] H. Cossette, D. Landriault, É. Marceau. Ruin probabilities in the compound Markov binomial model. Scand. Actuar. J., 2003, [6] H.Cossette, D.Landriault,ÉMarceau.Exactexpressionsand upper bound for ruin probabilities in the compound Markov binomial model. Insurance Math. Econom., , [7] K.C. Yuen, J. Guo. Some results on the compound Markov binomial model. Scand. Actuar. J., 2006, Received: May, 2015