Ruin probabilities for time-correlated claims in the compound binomial model

Insurance: Mathematics and Economics 29 (2001) 47–57

Ruin probabilities for time-correlated claims in the compound binomial model K.C. Yuen a,∗ , J.Y. Guo b a

Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong b Department of Mathematics, Nankai University, Tianjin, China Received 1 September 2000; received in revised form 1 February 2001; accepted 16 February 2001

Abstract In this paper we consider the ruin probability for a risk process with time-correlated claims in the compound binomial model. It is assumed that every main claim will produce a by-claim but the occurrence of the by-claim may be delayed. Recursive formulas for the finite time ruin probabilities are obtained and explicit expressions for ultimate ruin probabilities are given in two special cases. © 2001 Elsevier Science B.V. All rights reserved. MSC: 0167-6687 Keywords: By-claim; Compound binomial; Gambler’s ruin; Main claim; Generating function; Ruin probability; Survival probability

1. Introduction In the classical ruin theory, the stationary and independent increment assumptions on the surplus process of an insurance company plays an important role. However, this assumption is very restrictive. Many authors have been studying models with various kinds of dependencies among claim amounts and claim numbers in life and non-life insurances. Among others, Gerber (1982) studied the ruin problem when the annual gains form a linear time series which includes the autoregressive model and the moving average model; Goovaerts and Dhaene (1996) gave a compound Poisson approximation for a portfolio of dependent risks; Ambagaspitiya (1998, 1999) and Partrat (1994) considered the distributions and the moment generating functions of the sum of the correlated aggregate claims where the correlation exists among the claim number processes of disjoint classes of insurance business; Reinhard (1984) and Asmussen (1989) paid their attention to the Markov-modulated models which exhibit another kind of correlation. In this paper we consider the ruin probabilities for time-correlated claims in the compound binomial model which has been studied by various authors, for example, Gerber (1988), Shiu (1989), Willmot (1993) and Dickson (1994). We extend the model to the case involving two classes of correlated risks. Specifically, the occurrence of one claim with a certain distribution of severity induces the occurrence of another claim with a different distribution of severity. However, the time of occurrence of the induced claim may be delayed to a later time period. This phenomenon ∗

Corresponding author. Tel.: +852-2859-1915; fax: +852-2858-9041. E-mail address: [email protected] (K.C. Yuen). 0167-6687/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 ( 0 1 ) 0 0 0 7 1 - 3

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may happen in reality. For a catastrophe such as an earthquake or a rain-storm, it is very likely that there exist other insurance claims after the immediate ones. Another possible interpretation of our model is that the induced claim may be treated as a random portion of the total claims taking some units of time to be settled. This kind of risk models has also been studied in the actuarial literature. For instance, Waters and Papatriandafylou (1985) used martingale techniques to derive upper bounds for ruin probabilities allowing for delay in claims settlement.

2. The model Here, we consider a discrete-time model which involves two types of insurance claims; namely the main claims and the by-claims. Denote the discrete time units by k = 0, 1, 2, . . . . It is assumed that each main claim induces a by-claim. In any time period, the probability of having a main claim is q, 0 < q < 1, and thus the probability of no main claim is p = 1 − q. The occurrences of main claims in different time periods are independent. The by-claim and its associated main claim may occur simultaneously with probability θ, or the occurrence of the by-claim may be delayed to the next time period with probability 1 − θ . All claim amounts are independent, positive and integer valued. The main claim amounts X1 , X2 , . . . , are independent and identically distributed. Put X = X1 . Then the common probability function of the Xi s is given
by fm = Pr(X = m) for
∞m = 1, 2, . . . . The corresponding m and µ = probability generating function and mean are f˜(z) = ∞ f z m X m=1 n=1 nfn , respectively. Let Y1 , Y2 , . . . be the independent and identically distributed by-claim amounts and put Y = Y1 . Denote their common probability function by gn = Pr(Y = n) for n = 1, 2, . . . , and write the probability generating function and mean as g(z) ˜ =
∞ n and µ , respectively. It is easy to see that this set-up collapses to that of the compound binomial model g z n Y n=1 when there are no by-claims. Assume that the premium rate is 1 per period. With an initial surplus of a non-negative integer u, the surplus process is defined as Sk = u + k − UkX − UkY

(2.1)

for k = 0, 1, 2, . . . , where UkX and UkY are the total amount of main claims and the total amount of by-claims, respectively, in the first k time periods. Following Shiu (1989) and Willmot (1993), we work with the survival probability for notational convenience. The finite-time survival probability is defined to be φ(u, k) = Pr(Sj ≥ 0; j = 0, 1, 2, . . . , k).

(2.2)

The finite-time ruin probability is then ψ(u, k) = 1 − φ(u, k). Suppose that a by-claim occurs in the time period k as a result of the main claim in period k − 1. A practical point to consider is whether the insurer should have established a reserve, presumably E(Y ) plus a loading, for this by-claim at the end of period k − 1. This scenario would lead to a concept of ruin for which ruin at time k − 1 would mean that the insurer had negative cash at that time or had insufficient cash to set up reserves required at that time. This scenario would be appropriate, for example, if claim payments were delayed and outstanding claims reserves were set up at the end of each time period. Our model assumes the simpler case in which ruin occurs when the insurer’s cash is negative. Let Uk be the sum of UkX and UkY . It is obvious that E(U1 ), the expectation of U1 , equals qµX + qθ µY . It should be noted that there may be three kinds of claims in any time period; the main claim, the by-claim induced by the current main claim, and the by-claim induced by the previously occurred main claim. Thus, E(Un+1 ) = E(Un ) + (qµX + qθ µY ) + q(1 − θ)µY . By induction, we get E(Un+1 ) = (n + 1)(qµX + qθ µY ) + nq(1 − θ)µY = nq(µX + µY ) + qµX + qθ µY .

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Therefore, we further assume that q(µX + µY ) < 1.

(2.3)

This assumption ensures that the premium rate exceeds the net claim rate and hence the safety loading is positive. Then the ultimate survival probability is φ(u) = lim φ(u, k), k→∞

and the ultimate ruin probability is ψ(u) = 1 − φ(u).

3. Recursive formulas To evaluate the finite-time ruin probabilities, we need to study the claim occurrences in two scenarios. The first is that if a main claim occurs in a certain period, its associated by-claim also occurs in the same period. Hence there will be no by-claim in the next time period and the surplus process really gets renewed. The second is simply the complement of the first scenario. In other words, if there exists a main claim, its associated by-claim will occur one period later. Conditional on the second scenario, that is, the main claim occurred in the previous period and its associated by-claim will occur at the end of the current period, we define the corresponding process as S1k = u + k − UkX − UkY − Y

(3.1)

for k = 1, 2, . . . , with S10 = u. This conditional surplus process is similar to (2.1) except for the subtraction of the by-claim random variable Y . Denote the corresponding k-period conditional survival probability of process (3.1) by φ1 (u, k). Then, by the law of total probability, it follows that   φ(u − m − n, k − 1)fm gn + q(1 − θ) φ1 (u − m, k − 1)fm , φ(u − 1, k) = pφ(u, k − 1) + qθ m≤u

m+n≤u

(3.2) where φ1 (u − 1, k) = p





φ(u − n, k − 1)gn + qθ

n≤u

+q(1 − θ )

φ(u − m − n − l, k − 1)fm gn gl

m+n+l≤u



φ1 (u − m − n, k − 1)fm gn

(3.3)

m+n≤u

for u ≥ 1, and k ≥ 1. Obviously φ(u, 0) = φ1 (u, 0) = 1 for all u ≥ 0. Similar to the work of Willmot (1993), we use the technique of generating functions to solve for φ of (3.2). Define the generating functions as ˜ k) = φ(z,

∞  u=0

φ(u, k)zu

and

φ˜ 1 (z, k) =

∞ 

φ1 (u, k)zu .

u=0

Multiplying both (3.2) and (3.3) by zu and summing over u from 1 to ∞ yield ˜ k) = p(φ(z, ˜ k − 1) − φ(0, k − 1)) + qθ φ(z, ˜ k − 1)f˜(z)g(z) zφ(z, ˜ + q(1 − θ)φ˜ 1 (z, k − 1)f˜(z),

(3.4)

˜ k − 1)g(z) ˜ k − 1)f˜(z)g˜ 2 (z) + q(1 − θ)φ˜ 1 (z, k − 1)f˜(z)g(z). ˜ + qθ φ(z, ˜ zφ˜ 1 (z, k) = p φ(z,

(3.5)

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Consider the bivariate generating functions ¯ t) = φ(z,

∞ 

˜ k)t k , φ(z,

φ¯ 1 (z, t) =

k=0

∞ 

φ˜ 1 (z, k)t k ,

and

φ¯ 0 (t) =

k=0

∞ 

φ(0, k)t k .

k=0

Parallel to the derivations of (3.4) and (3.5), we obtain ¯ t) − φ(z, ˜ 0)) = pt(φ(z, ¯ t) − φ¯ 0 (t)) + qθt f˜(z)g(z) ¯ t) + q(1 − θ)t f˜(z)φ¯ 1 (z, t), z(φ(z, ˜ φ(z,

(3.6)

¯ t) + qθt f˜(z)g˜ 2 (z)φ(z, ¯ t) + q(1 − θ)t f˜(z)g(z) z(φ¯ 1 (z, t) − φ˜ 1 (z, 0)) = ptg(z) ˜ φ(z, ˜ φ¯ 1 (z, t) ¯ t) + qθ t f˜(z)g(z) ¯ t) + q(1 − θ)t f˜(z)φ¯ 1 (z, t)). = g(z)(pt ˜ φ(z, ˜ φ(z,

(3.7)

˜ 0) = φ˜ 1 (z, 0) = (1 − z)−1 Note that the right-hand side of (3.7) appears almost the same as that of (3.6). Since φ(z, by definition, (3.6) and (3.7) can be rewritten as z ¯ t) + q(1 − θ)t f˜(z)φ¯ 1 (z, t) − ptφ¯ 0 (t), = (pt + qθ t f˜(z)g(z)) ˜ φ(z, 1−z z z ¯ t) − zφ¯ 1 (z, t) − = g(z)(z ˜ φ(z, + ptφ¯ 0 (t)). 1−z 1−z

¯ t) − zφ(z,

Combining the two equations, we get   ˜(z) ˜(z)g(z) z q(1 − θ) f f ˜ ¯ t)[z − t (p + q f˜(z)g(z))] φ(z, ˜ = + t (1 − g(z)) ˜ − ptφ¯ 0 (t) 1 − q(1 − θ)t . 1−z 1−z z (3.8) Let UkW be the total amount of claims in the first k periods in the compound binomial model with individual claim k . Denote the ˜ k) = (p + q f˜(z)g(z)) amount W = X + Y . Then the probability generating function of UkW is h(z, ˜ W probability function and the distribution function of Uk by h(i, k) and H (i, k), respectively. We will see that these functions are found to be useful in the development of recursive formulas for the finite-time survival probabilities. ˜ 1) = p + q f˜(z)g(z). ˜ 1) and writing (z − It is easy to see that h(z, ˜ Dividing both sides of (3.8) by z − t h(z, −1 ˜ t h(z, 1)) in terms of a power series in t, we have, for k = 1, 2, . . . , ˜ k) = zk φ(z,

k−1  ˜ k) h(z, ˜ k − 1) q(1 − θ) − p φ(0, k − 1 − j )h(z, ˜ j )zk−1−j + f˜(z)(1 − g(z)) ˜ h(z, 1−z 1−z j =0

+pq(1 − θ )f˜(z)g(z) ˜

k−2 

˜ j )zk−2−j φ(0, k − 2 − j )h(z,

(3.9)

j =0

˜ k) = h(z, ˜ k + 1) − p h(z, ˜ k). Since the probability by matching the coefficients of t k . It is clear that q f˜(z)g(z) ˜ h(z, generating function h˜ can be written in terms of H and h, (3.9) is equivalent to ˜ k) = zk φ(z,

∞  i=0

zi H (i, k) − p

k−1  j =0

φ(0, k − 1 − j )

∞ 

zi+k−1−j h(i, j ) +

i=0

q(1 − θ) ˜ ˜ f (z)h(z, k − 1) 1−z

k−2

−

 1−θ ˜ ˜ j + 1) − p h(z, ˜ j )). ˜ k − 1)) + p(1 − θ) zk−2−j φ(0, k − 2 − j )(h(z, (h(z, k) − p h(z, 1−z j =0

(3.10)

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51

˜ k − 1)(p + q f˜(z)) with H1 (i, k) as the corresponding distribution function. InterFurther define h˜ 1 (z, k) = h(z, changing the order of summation and rearranging terms in (3.10), we finally obtain ˜ k) = zk φ(z,

∞ 

zi H (i, k) − p

k−1 

i=0

i=0

−p

k−1 

zi

φ(0, k − 1 − j )h(i + j + 1 − k, j )

j =k−1−i

∞  k−1  1−θ ˜ 1−θ ˜ zi φ(0, k − 1 − j )h(i + j + 1 − k, j ) + h1 (z, k) − h(z, k) 1−z 1−z i=k

j =0

+p(1 − θ )

k−1 

˜ j ) − p 2 (1 − θ) zk−1−j φ(0, k − 1 − j )h(z,

j =1

=θ

∞ 

˜ j) zk−2−j φ(0, k − 2 − j )h(z,

j =0

zi H (i, k) − p

i=0

−p

k−2 

k−1 

k−1 

zi

i=0

φ(0, k − 1 − j )h(i + j + 1 − k, j )

j =k−1−i

∞  ∞ k−1   zi φ(0, k − 1 − j )h(i + j + 1 − k, j ) + (1 − θ) zi H1 (i, k) i=k

j =0

i=0

 k−1  +p(1 − θ )  zi i=0

k−1 

φ(0, k − 1 − j )h(i + j + 1 − k, j ) − h(0, 0)φ(0, k − 1)zk−1

j =k−1−i

 ∞  k−1  + zi φ(0, k − 1 − j )h(i + j + 1 − k, j )

 −p 2 (1 − θ ) 

i=k k−2 

+

k−2 

zi

i=0

j =1

φ(0, k − 2 − j )h(i + j + 2 − k, j )

j =k−2−i ∞ 

zi

i=k−1

k−2 

 φ(0, k − 2 − j )h(i + j + 2 − k, j ) .

(3.11)

j =0

On the other hand, by definition ˜ k) = z φ(z, k

∞ 

φ(i, k)z

i+k

=

∞ 

zi φ(i − k, k).

(3.12)

i=k

i=0

Then, comparing the coefficients of zi in the last summation of (3.12) to those on the right-hand side of (3.11), we obtain the following recursive formulas: φ(0, k) = θ H (k, k) + (1 − θ )H1 (k, k) − pθ

k−1 

φ(0, k − 1 − j )h(j + 1, j )

j =0

−p 2 (1 − θ )

k−2  j =0

φ(0, k − 2 − j )h(j + 2, j ),

(3.13)

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Table 1 Values of φ(u, k) with f˜(z) = g(z) ˜ =z k 1

2

3

4

5

6

θ = 0.2

φ(0, k) φ(3, k) φ(5, k)

0.940 1 1

0.868 1 1

0.845 1 1

0.818 0.998 1

0.807 0.996 1

0.793 0.995 0.999

θ = 0.8

φ(0, k) φ(3, k) φ(5, k)

0.760 1 1

0.742 1 1

0.682 1 1

0.677 0.994 1

0.651 0.993 1

0.649 0.987 0.999

φ(i − k, k) = θ H (i, k) + (1 − θ )H1 (i, k) − pθ

k−1 

φ(0, k − 1 − j )h(i + j + 1 − k, j )

j =0

−p 2 (1 − θ )

k−2 

φ(0, k − 2 − j )h(i + j + 2 − k, j )

(3.14)

j =0

for 1 ≤ k ≤ i. It is apparent that (3.13) and (3.14) give φ(i, 0) = 1, i ≥ 1 which are consistent with the definition of φ. For k ≥ 1, φ(0, k) can be calculated recursively through (3.13). Given these values, (3.14) can then be used to evaluate φ(u, k) for u ≥ 1. As an illustration, we obtain the following values of φ in terms of the probability functions of X and Y through the recursive formulas: φ(0, 1) = p + q(1 − θ )f1 ,

φ(0, 2) = p2 + pqf1 g1 + pq(1 − θ)f1 + pq(1 − θ)f2 ,

φ(1, 1) = p + q(1 − θ )(f1 + f2 ) + qθf1 g1 , φ(1, 2) = p2 + pq(f1 g2 + f2 g1 ) + pq(1 − θ )(f1 + f2 + f3 ) + q(1 − θ)f1 g1 (p + qf1 ). Note that H , H1 and h in (3.13) and (3.14) can be computed using the probability generating functions, h˜ and h˜ 1 . To calculate numerical values of φ(u, k) of (2.2), we need to specify p, θ and the distributions of X and Y . Assume that p = 0.7 and θ = 0.2, 0.8. We consider the survival probabilities up to k = 6 time periods and arbitrarily choose u = 0, 3, 5. Table 1 displays the values of φ with f˜(z) = g(z) ˜ = z while Table 2 exhibits the values of φ with f˜ = z(2 − z)−1 and g(z) ˜ = z. As expected, both tables show that the larger the initial surplus, the higher the survival probabilities and that the probabilities for θ = 0.8 is less than or equal to those for θ = 0.2. Furthermore, the survival probabilities decrease as k increases with other values of parameters being fixed. As k tends to infinity, the value of φ(u, k) approaches to the ultimate survival probability, φ(u). Table 2 ˜ =z Values of φ(u, k) with f˜(z) = z(2 − z)−1 and g(z) k 1

2

3

4

5

6

θ = 0.2

φ(0, k) φ(3, k) φ(5, k)

0.820 0.978 0.994

0.721 0.945 0.983

0.652 0.912 0.968

0.603 0.881 0.952

0.565 0.854 0.935

0.534 0.829 0.919

θ = 0.8

φ(0, k) φ(3, k) φ(5, k)

0.730 0.966 0.992

0.627 0.923 0.976

0.556 0.883 0.957

0.509 0.847 0.937

0.473 0.816 0.917

0.445 0.788 0.899

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53

4. Ultimate ruin probabilities for some special cases In this section, we discuss two special cases in which explicit expressions for the ultimate survival probabilities can be derived. We first study the case when θ = 1. Actually, this case is very similar to the one proposed by Willmot (1993). In our set-up, there is a by-claim, Y, occurring simultaneously with the main claim, X. Hence, the only difference is that we use W = X + Y as our claim amount random variable while Willmot simply considers X. Since his results for the ultimate survival probabilities can be easily extended to our first case, we simply present the expression for the initial surplus being zero φ(0) =

1 − q(µX + µY ) . 1−q

To obtain φ(u), u ≥ 1, specific forms of f˜ and g˜ are required. Explicit expressions for φ(u) in some special cases are given in Willmot (1993). We now turn to the second case in which θ = 0. This implies that the by-claim definitely occurs one period after the occurrence of its main claim. The second case further assumes that f1 = g1 = 1. Then the model is almost the same as the classical gambler’s ruin model except that the two dollars are paid out separately, one each in the current and the next time period. Thus, the gambler has an opportunity to earn one dollar before paying out the second dollar. Intuitively, the ultimate ruin probabilities for this special case should be smaller than those for the classical model. Instead of using the recursive formulas (3.13) and (3.14), we apply the techniques of Willmot (1993) to derive the ultimate ruin probabilities. From (3.8), it is easily seen that for any |t| < 1, there exists a unique solution z(t) with |z(t)| < 1 of the equation ˜ 1) = 0. z − t (p + q f˜(z)g(z)) ˜ = z − t h(z,

(4.1)

After substituting θ = 0 and f˜(z) = g(z) ˜ = z into (3.8), (4.1) implies that ∞  k=0





z(t) qt z(t) qz(t) φ(0, k)t = 1+ = 1+ . pt(1 − z(t)) 1 − qtz(t) pt(1 − z(t)) p k

(4.2)

Moreover, given (4.1) and for an analytic function l with l(0) = 0, Lagrange’s expansion yields the following power series in t: l(z(t)) =

∞ n n−1  t d ˜ 1))n ]s=0 . [l (s)(h(s, n! ds n−1

(4.3)

n=1

˜ n) = [h(s, ˜ 1)]n and Since h(s, h(n − 1, n) =

1 dn−1 ˜ h(s, n)|s=0 , (n − 1)! ds n−1

(4.3) gives z(t) =

∞ n  t n=1

n

h(n − 1, n)

and

n−1  ∞   z(t) tn = . H (i, n) 1 − z(t) n n=1

i=0

(4.4)

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Plugging (4.4) into (4.2), we obtain  k 

∞ ∞  z(t) q 1  tk k + − z(t) φ(0, k)t = H (i, k + 1) p k + 1 p 2 t 1 − z(t) k=0 k=0 i=0 

∞  k ∞ 1 tk q   q  h(k, k + 1) k = H (i, k + 1) + 2 − 2 t . p p k+1 p k+1 k=0

i=0

k=0

Then comparing the coefficients of t k yields k

 1 q h(k, k + 1). φ(0, k) = 2 H (i, k + 1) − 2 p (k + 1) p (k + 1)

(4.5)

i=0

To derive an expression for the ultimate survival probability of φ(0), we need to find the limit of (4.5). From the result of Willmot (1993), the limit of the first term in (4.5) has the form k

 1 1 − 2q lim 2 H (i, k + 1) = . k→∞ p (k + 1) p2 i=0

As for the second term in (4.5), h(k, k + 1) = 0 if k is odd. By Stirling’s formula and the fact that pq ≤ 41 , we see that for k = 2m, m = 1, 2, . . . , (2m)! 1 1 m p m+1 q m ≤ p4−m h(k, k + 1) = C2m+1 . k+1 2m + 1 m!(m + 1)! Clearly, the right side converges to zero as m tends to infinity. Thus, φ(0) = lim φ(0, k) = k→∞

¯ We now define φ(z) = (4.6),

1 − 2q . p2


∞

u=0 φ(u)z

¯ ¯ t) = φ(z) = lim(t − 1)φ(z, t↑1

(4.6) u.

Then, using a Tauberian theorem for power series, we get from (3.8) and

p(1−qz) −p(1−qz) lim(t−1)φ¯ 0 (t) = z − (p+qz2 ) t↑1 z−(p+qz2 )



1 − 2q p2

=

∞ 

1−

u=0

q u+2 p u+2



zu .

Hence φ(u) = 1 −

u+2 q p

and

ψ(u) =

u+2 q . p

(4.7)

Note that the ruin probabilities for the classical gambler’s ruin model are (qp−1 )u+1 . In the present special case, condition (2.3) is equivalent to 2q < 1. This reveals the fact that the ruin probabilities of ψ(u) in (4.7) are smaller than those for the classical gambler’s ruin model.

5. An extension We now extend the proposed model to the case that the by-claim may occur concurrently with the main claim or one period after the main claim or two periods after the main claim. Specifically, the occurrence of

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55

each by-claim may be delayed for 0, 1 or 2 time periods with probability θ, θ (1 − θ), or (1 − θ)2 , respectively. In accordance with (3.1)–(3.3), we need to consider five types of survival probabilities in the extended model: the survival probability φ of the full model; the survival probability φ1 conditional on the fact that a by-claim will occur in the second time period; the survival probability φ2 conditional on the fact that a by-claim will occur in the third time period; the survival probability φ3 conditional on the fact that by-claim will occur in each of the second and third time periods; and the survival probability φ4 conditional on the fact that two by-claims will occur in the second time period. These classifications result in the following set of equations: 

φ(u − 1, k) = pφ(u, k − 1) + qθ +q(1 − θ )2 φ1 (u − 1, k) = p









φ4 (u − m, k − 1)fm + q(1 − θ)2





φ2 (u − m − n, k − 1)fm gn ,

φ1 (u − m − n, k − 1)fm gn

m+n≤u

φ1 (u − n, k − 1)gn + qθ

+qθ (1 − θ )



m≤u

m≤u

n≤u

φ4 (u − 1, k) = p



φ(u − m − n − l, k − 1)fm gn gl

φ1 (u − m − n, k − 1)fm gn + q(1 − θ)2

φ2 (u − 1, k) = pφ1 (u, k − 1) + qθ





m+n+l≤u

m+n≤u

+qθ (1 − θ )

φ1 (u − m, k − 1)fm

φ2 (u − m, k − 1)fm ,

φ(u − n, k − 1)gn + qθ

+qθ (1 − θ )

 m≤u

m+n≤u

m≤u

n≤u

φ3 (u − 1, k) = p

φ(u − m − n, k − 1)fm gn + qθ(1 − θ)





φ3 (u − m, k − 1)fm ,

m≤u

φ1 (u − m − n − l, k − 1)fm gn gl

m+n+l≤u

φ4 (u−m−n, k−1)fm gn +q(1 − θ)2

m+n≤u

φ(u − n − l, k − 1)gn gl + qθ

n+l≤u





φ3 (u − m − n, k − 1)fm gn ,

m+n+l≤u

φ(u − m − n − l − d, k − 1)fm gn gl gd

m+n+l+d≤u

+qθ (1 − θ )



φ1 (u − m − n − l, k − 1)fm gn gl

m+n+l≤u

+q(1 − θ )2



φ2 (u − m − n − l, k − 1)fm gn gl

(5.1)

m+n+l≤u

for u ≥ 1, and k ≥ 1. Although these equations look very complex, the approach presented in Section 3 can still be used to solve for φ. ˜ h, H , h˜ 1 , and H1 defined in Section 3. Denote the probability function associated with h˜ 1 and H1 by Recall h, h1 . We now define another two probability generating functions ˜ k − 1) h˜ 2 (z, k) = (p + q f˜2 (z))h(z,

and

˜ k − 1). h˜ 3 (z, k) = (p + q g(z)) ˜ h(z,

Their probability functions (distribution functions) are denoted by h2 (i, k) and h3 (i, k) (H2 (i, k) and H3 (i, k)). Then, parallel to the derivation of (3.13) and (3.14), we obtain

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K.C. Yuen, J.Y. Guo / Insurance: Mathematics and Economics 29 (2001) 47–57

φ(u, k) = p(1 − θ )2

k−3 

φ1 (0, k − 3 − i)



θ(θ − 2)h1 (u + i + 3, i + 2) − pθ(θ − 2)h(u + i + 3, i + 1)

i=0

−p(1 − θ )2 h1 (u + i + 3, i + 1) + p 2 (1 − θ)2 h(u + i + 3, i) +

k−3 



 φ(0, k − 3 − i) p 2 θ (1 − θ)(2θ − 3)h(u + i + 3, i + 1) − p 3 (1 − θ)3 h(u + i + 3, i)

i=0

+pθ 2 (θ − 2)h(u + i + 3, i + 2) − p(1 − θ)2 φ1 (0, k − 2)h1 (u + 2, 1) −pθ φ(0, k − 2)h(u + 2, 1) + θ 2 (2 − θ)H (u + k, k) + θ(1 − θ)(3 − 2θ)H1 (u + k, k) +2p(1 − θ )3 H1 (u + k, k − 1) − p(1 − θ)3 H (u + k, k − 2) + q(1 − θ)3 H2 (u + k, k − 1) for u ≥ 0 and k ≥ 1, where φ1 (0, k) = pg1 φ(0, k − 1) + p(1 − θ )2

k−3 

φ1 (0, k − 3 − i)



θ(θ − 2)h(i + 3, i + 2)

i=0

−p(2θ 2 − 4θ + 1)h(i + 3, i + 1) + p 2 (1 − θ)2 h(i + 3, i)



 p φ(0, k − 3 − i) θ 2 (θ − 2)h3 (i + 3, i + 3) + p 3 (1 − θ)3 h(i + 3, i) q k−3

+

i=0

+pθ (1 − θ )(2θ − 3)h3 (i + 3, i + 2) − p 2 (1 − θ)3 h3 (i + 3, i + 1) −p 2 θ (1 − θ )(2θ − 3)h(i + 3, i + 1) − pθ 2 (θ − 2)h(i + 3, i + 2)  p + φ(0, k − 2) pθ h(2, 1) − p(1 − θ)h3 (2, 1) − θh3 (2, 2) q 2

pθ p 2 (θ −2)+θ(1−θ)(3−2θ) H (k, k) − φ(0, k − 1)h3 (1, 1) − p(1 − θ) φ1 (0, k−2)h(2, 1)+ q q

p p + p(1 − θ )3 + θ (1 − θ )(2θ − 3) H (k, k − 1) + θ(1 − θ)(3 − 2θ)H3 (k, k) q q (1 − θ)3 p 2 θ2 (θ − 2)H3 (k, k + 1) − H (k, k − 2) q q p2 + (1 − θ )3 H3 (k, k − 1) + (1 − θ)3 H1 (k, k) − p(1 − θ)3 H1 (k, k − 1). q −

Along the same line of (5.1), we can generalize the model to the case involving d periods of delay for the occurrence of the by-claim with d > 2. Undoubtedly, such a generalization makes the derivation of the ruin probabilities tremendously complicated and tedious.

Acknowledgements The authors are grateful to the referees for their helpful comments and suggestions. This research was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7202/99H).

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