On the probability of ruin in the compound Poisson risk model with potentially delayed claims

AbstractIn this paper, we consider the compound Poisson risk model involving two types of dependent claims, namely main claims and by-claims. The by-claim is induced by the main claim with a certain probability and the occurrence of a by-claim may be delayed depending on associated main claim amount. Using Rouché’s theorem, both of the survival probability with zero initial surplus and the Laplace transform of the survival probability are obtained from an integro-differential equations system. Then, using the Laplace transform, we derive a defective renewal equation satisfied by the survival probability. An exact representation for the solution of this equation is derived through an associated compound geometric distribution. For exponential claim sizes, we present an explicit formula for the survival probability. We also illustrate the influence of model parameters in the dependent risk model on the survival probability by numerical examples.



Introduction
In fact, insurance claims may be delayed due to various reasons. This phenomenon 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. Since the work by [9], risk models with this special feature have been discussed by many authors. For example, " Boogaert and Haezendonck (1989) studied the mathematical properties of a liability process with settling delay within the framework of an economics environment" (See [3]). [13] studied a compound binomial model with delayed claims and obtained recursive formulas for finite time survival probabilities. [10] also studied this risk model. They derived an upper bound for the ruin probability. [18] studied an extension to the risk model proposed in [13]. [14] studied a risk model with delayed claims, in which the time of delay for the occurrence of a by-claim is assumed to be exponentially distributed. [8] presented a sample path large deviation principle for the delayed claims risk model presented in [14]. [11] first considered the risk model with delayed claims and a constant dividend barrier in a financial market driven by a time-homogeneous Markov chain. The expected discounted dividend payments prior to ruin were derived. [12] considered the compound Poisson risk model with delayed claims and proved that the ruin probability for this risk model decreases as the probability of the delay of by-claims increases. In addition, the dependence between other components in risk models was also studied. For instance, [1] considered the case in which the distribution of a claim interval is controlled by the previous claim size through a mechanism that if the claim size exceeds a random level the next claim interval will follow one type of distribution, if not, it will follow another type of distribution. Their model depicts common sense that when a certain kind of catastrophe is big enough, people will pay more attention to it and so the time until the next occurrence is longer. [2] and [4] considered a particular dependence structure among the inter-claim time and subsequent claim size. Furthermore, [16] have studied risk models with dependence between inter-claim times and claim sizes.
Note that all risk models described in the paragraph above rely on the assumption that each main claim induces a by-claim with certainty. Motivated by the fact that each main claim induces a by-claim randomly and the probability of delay of each claim is not independent of claim amounts, in this paper, we consider the compound Poisson risk model with two types of individual claims where the two types of claim have different distributions of severity. In this risk model, there will be a main claim, at every jump time of the number process. Let the aggregate main claims process be a compound Poisson process and let {N (t); t ≥ 0} be the corresponding Poisson claim number process, with intensity λ. Its jump times are denoted by {T i } i≥1 with T 0 = 0. The main claim amounts {Y i } i≥1 are assumed to be independent and identically distributed (i.i.d.) positive random variables with common distribution F. Each main claim induces another type of claim called a by-claim with probability q. Let {X i } i≥1 be the by-claim amounts, assumed to be i.i.d. positive random variables with common distribution G. The main claim amounts and by-claim amounts are independent and their means are denoted by μ F and μ G . Moreover, we assume the claim occurrence process to be of the following type: there will be a main claim Y i at every epoch T i of the Poisson process. It is assumed that each main claim induces a by-claim with probability q and the main claim does not induce a by-claim with probability 1 − q. Moreover, if the main claim induces a by-claim and the main claim amount Y i is less than a threshold B i , the by-claim X i and its associated main claim Y i occur simultaneously; if the main claim induces a by-claim but the main claim amount Y i is larger than or equal to the threshold B i , the occurrence of the by-claim X i is delayed to T i+1 . If the occurrence of the by-claim X i is delayed to T i+1 , then the delayed by-claim X i and main claim Y i+1 occur simultaneously. The quantities {B i } i≥1 are assumed to be i.i.d. random variables with common distribution B.
In this setup, the surplus process U (t) of this risk model is defined as where u is the initial capital, c is the constant rate per unit time at which the premiums are received, and R(t) is the sum of all by-claims X i that occurred before time t.
One of the key quantities in the risk model is the survival probability, or non-ruin probability, denoted by (u) as a function of u ≥ 0, which is the probability that the surplus of the insurer is always above zero, namely, The corresponding ruin probability is then (u) = 1 − (u), which is the probability that the surplus of the insurer is below zero at some time. The survival probability (ruin probability) can be used to provide an early warning system for the guidance of an insurance project (see [6]).
The expectation of the aggregate claims at time t is given by Thus in order to ensure that the premium rate exceeds the net claim rate and guarantee that ruin does not occur almost surely, we assume the following positive safely loading condition holds, i.e., We point out that when q = 1, our model reduces to the continuous time risk model proposed by [17]. Moreover, when B is the constant, our model reduces to the delayed claims risk model of [12]. Therefore, the work of this paper can also be seen as a complement to the work of [12], [17] and extends their results by taking the randomness of delayed claim into account.
The rest of this paper is structured as follows. In Sect. 2, we derive an integro-differential equations system for survival probabilities. Then, by employing Rouché's theorem, both of the survival probability with zero initial surplus and the Laplace transform of the survival probability are obtained. By using Laplace transforms, we derive a defective renewal equation for the survival probability in Sect. 3. An exact representation for the solution of this equation is also derived through an associated compound geometric distribution. For exponential claim size distribution, in Sect. 4 we present the explicit formulas for survival probabilities. In Sect. 5, we illustrate the influence of the model parameters in the dependent risk model on the survival probability by numerical examples.

System of integro-differential equations and Laplace transforms
In this section, we first derive the integro-differential equations for the survival probabilities. We then study the Laplace transforms for the survival probability (u), based on the integro-differential equations.
With other things being the same, we consider a slight change in the risk model (1.1). Instead of having one main claim Y 1 and a by-claim X 1 with probability qPr(Y 1 < B 1 ) at the first epoch T 1 , another by-claim X is added at the first epoch T 1 , i.e., by-claim X and main claim Y 1 occur at T 1 simultaneously. Hence, the corresponding surplus process U 1 (t) of this auxiliary risk model is defined as where X denotes the extra by-claim amount, This surplus process is similar to (1.1) except for the subtraction of the by-claim random variable X . Denote the corresponding survival probability for this auxiliary model by 1 (u) which is very useful in the derivation of (u).

Theorem 2.1
For any u ≥ 0, (u) and 1 (u) satisfy the following system of integro-differential equations: Proof Consider what will happen at the first epoch T 1 . We need to study the claim occurrences in three scenarios.
(I) Obviously there will be a main claim Y 1 at the first epoch T 1 . The main claim does not induce a by-claim, then the surplus process gets renewed expect for the initial value. (II) There will be a main claim Y 1 at the first epoch T 1 . The main claim will induce a by-claim X 1 and the main claim size Y 1 < B 1 , then the by-claim X 1 also occurs at the first epoch T 1 , the surplus process U (t) will renew itself with different initial reserve. The probability of this event is qPr(Y 1 < B 1 ). (III) There will be a main claim Y 1 at the first epoch T 1 . The main claim will induce a by-claim X 1 and the main claim size Y 1 ≥ B 1 , the occurrence of the by-claim X 1 will be delayed to T 2 , i.e., the delayed by-claim X 1 and main claim Y 2 occur simultaneously. In this case, U (t) will not renew itself but transfer to the auxiliary model described in the paragraph above. The probability of this event is qPr( Taking what happened at the first epoch T 1 into account, we can set up the following equation for (u): With the auxiliary model, similar analysis gives where * denotes the operation of convolution. Setting s = u + ct in (2.4), (2.5) and differentiating with respect to u, we get the system of integro-differential equations (2.2) and (2.3). This completes the proof of this theorem.
In the following, we will give the Laplace transforms of the survival probabilities (u) and 1 (u) from the integro-differential equations.
Let χ 1 (y) = B(y)F (y) and χ 2 (y) = (1 − B(y))F (y). For Re s ≥ 0, we definẽ We also define the Laplace transforms of (u) and 1 (u) as Taking Laplace transforms of (3.3) and (3.4) and making some simplifications, we obtain which can further be simplified tõ In order to obtain˜ (s), for the further sake of deriving (u), we only need to find (0) and 1 (0). Denoting the denominator on the right-hand side of (2.6) as cs .
Finally, it is easy to see that ρ(δ) → 0 as δ → 0 + from the expression of l δ (s). Since lim δ→0 + l δ (s) = l(s), we can conclude that l(s) = 0 has no root with positive real part on the right half plane.
It follows from everything above that equation (s) = 0 has exactly one positive real root σ = λ/c on the right half complex plane. This completes the proof.
Since˜ (s) is analytic function for Re s ≥ 0, σ must also be a zero of the numerator of (2.6). This yields the relation between (0) and 1 (0), namely . (2.8) Combining (2.7) and (2.8), we can obtain the solutions for (0) and 1 (0) in the following theorem.

Theorem 2.3 The survival probabilities with zero initial surplus are given by
. (2.10) so we obtain which is the well-known formula for the non-ruin probability with zero initial capital in the classic compound Poisson risk model. 2. Let q = 1 and B be a constant, i.e., Pr(Y ≥ B) = θ , thenχ 1 (s) = θb(s) andχ 2 (s) = (1 − θ)b(s). In this case, each main claim and its associated by-claim occur simultaneously with probability 1 − θ , or the occurrence of the by-claim may be delayed with probability θ . Actually, the risk model given by (1.1) is the compound Poisson risk model with delayed claims studied by [12]. Then Eq. (2.6) can be simplified as˜ . (2.13) This equation is consistent with Eq. 4.1 in [12]. Similarly, since lim u→+∞ (u) = 1 , by the Final-value theorem of Laplace transforms, we have lim s→0 s˜ (s) = 1, i.e., , so we obtain , (2.14) which is the formula for the survival probability with zero initial capital in the compound Poisson risk model with delayed claims.

Defective renewal equations for survival probabilities
In this section, our goal is to show that the survival probabilities also satisfy a defective renewal equation in the dependent risk model with delayed claims. To identify the form of this defective renewal equation, we first analyse the Laplace transform of (u).
As in [5], we define an operator r of a real-valued function f , with respect to a complex number r , to be It is clear that the Laplace transform of f ,f (s), can be expressed as s f (0), and that for distinct r 1 and r 2 , If r 1 = r 2 = r , The properties for this operator can be found in [5].

Lemma 3.1 The Laplace transform˜ (s) of the survival probability satisfies
Proof Substituting (2.8) into (2.6), we can get which leads to (3.1). This completes the proof.
Using Lemma 3.1, we are now in a position to derive the defective renewal equation for (u).

Theorem 3.2 (u) satisfies the following defective renewal equation
Proof Inverting the Laplace transform in (3.1), one finds which corresponds to (3.2). For (3.2) to be a defective renewal equation, it remains to show that λ(μ F + qμ G ) < c. The inequality is the positive safely loading condition (1.2). Thus, we complete the proof. Now, we define an associated compound geometric distribution function K (u) = 1 − K (u) by
Remark 3. 4 We point out that when q = 1, the compound Poisson risk model with potentially delayed claims is equivalent to the dependent risk model studied by [17]. To see that, letting q = 1 in (3.2), we have which is constant with defective renewal equation (5.2) derived by [17].

Explicit results for exponential claim size distributions
We now consider the case where both the claim sizes are exponentially distributed, with distribution functions F ∼ Exp(ν) and G ∼ Exp(ω), respectively, where ν = 1 μ F and ω = 1 μ G . Then we havẽ For the special case B ∼ Exp(μ) we obtaiñ So we can deriveχ and σ in (2.8) is the unique solution s with Re s > 0 of By solving this equation, we obtain four roots: The positive relative security loading condition (1.2) implies that only σ 2 has a positive real part. Hence, σ = λ/c is the only zero with positive real part on the right half plane. This proves the correctness of Lemma 2.2. From (2.9) and (2.10), it is easy to obtain the survival probabilities with zero initial surplus: Substituting the expressions ofb 1 (s),b 2 (s),χ 1 (s),χ 2 (s), (0) and 1 (0) into (2.6) and taking inverse Laplace transform, we can derive explicit expression for survival probability (u), where which is constant with the expression (6.1) obtained in [17]. 2. When q = 0, each main claim does not induce a by-claim. In this case, the survival probability is given by

Numerical examples
In this section, we illustrate the applications of the results in the previous sections and the influence of the model parameters in the dependent risk model on survival probabilities by numerical examples.  Figure 1 shows the survival probabilities in Example 5.1, for u ∈ [0, 5] and q = 0, 0.25, 0.5, 0.75, 1. One can see from the graph that with fixed u, these survival probabilities decrease as q increases.  Figure 2 shows the survival probabilities in Example 5.2, for u ∈ [0, 4] and μ = 0.5, 1, 1.5, 2, 2.5. We see from Fig. 2 that the coefficient μ of threshold B has a significant effect on the non-ruin probabilities. Figure 2 also shows the fact that the non-ruin probability is an increasing function of μ.

Concluding remarks
In this paper, we study the compound Poisson risk model with potentially delayed claims. In this risk model, there will be a main claim Y i at every epoch T i of the Poisson process and the main claim Y i will induce a by-claim X i with probability q. Moreover, the occurrence of the by-claim X i may be delayed to T i+1 depending on main claim size Y i and the threshold B i . The quantities {B i } i≥1 are assumed to be i.i.d. non-negative random variables. We show how to apply the Laplace transform to this dependent risk model. A defective renewal equation for the survival probability is obtained and an exact representation for the solution of this equation is derived through an associated compound geometric distribution. The explicit results for survival probabilities are derived when the claims are exponentially distributed. Some examples and numerical illustrations are also given.