Gerber-Shiu Analysis for a Discrete Risk Model with Delayed Claims and Random Incomes

HUANG Ya, LIU Juan, ZHOU Jie-ming, DENG Ying-chun,

(1- School of Business, Hunan Normal University, Changsha 410081;2- Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education),School of Mathematics and Statistics, Hunan Normal University, Changsha 410081)

Abstract: Ruin theory is the mainly contents of insurance mathematics,as it can supply a very useful early-warning measure for the risk of the insurance company.In this paper,we study a risk model with potentially delayed claims and random premium incomes within the framework of the compound binomial model.Using the technique of generating functions, we derive a recursive formula for the Gerber-Shiu expected discounted penalty function.Specifically, an explicit formula is obtained for the discount-free case.As applications, we derive some useful insurance quantities,including the ruin probability, the density of the deficit at ruin, the joint density of the surplus immediately before ruin and the deficit at ruin, and the density of the claim causing ruin.

Keywords: compound binomial risk model; Gerber-Shiu expected discounted penalty function; delayed claims; random premium; recursive formula

1 Introduction

The classical compound binomial risk model with independent and identically distributed claims has been studied by a lot of literature, such as Gerber[1], Shiu[2],Willmot[3], Dickson[4], Cheng et al[5]and so on.Based on the classical compound binomial risk model, various extensions have been made by many scholars, for example,Reinhard and Snoussi[6]studied the ruin probabilities in a discrete-time semi-Markov risk model which is assumed that the claims are influenced by a homogeneous, irreducible and aperiodic Markov chain with finite state space.As a promotion, Chen et al[7]considered the dividend problems for a discrete semi-Markov risk model and obtained explicit expressions for the total expected discounted dividends until ruin.Cossette et al[8]presented a compound Markov binomial model which introduces the dependency between claim occurrences by a Markov Bernoulli process.Combined with Markov process, such as the Markov modulated risk model and the Markov arrival process (MAP) risk model are introduced in actuarial risk models, related works can be found in Yang et al[9], Li et al[10,11], and Liu et al[12]etc.

In reality, claims may be time-correlated for various reasons, and it is important to study risk model which is able to depict this phenomenon.Indeed, a frame work of delayed claims is built by introducing two kinds of individual claims, namely main claims and by-claims, and allowing possible delay of the occurrence of the by-claims.Among others,in the case of compound binomial model,Yuen and Guo[13]considered a specific dependence structure between the main claim and the by-claim, and obtained the recursive formulas for the finite time ruin probability.Further, based on the same model, Xiao and Guo[14]derived a recursive formula for the joint distribution of the surplus immediately prior to ruin and the deficit at ruin.Li and Wu[15]obtained a recursive formula of the Gerber-Shiu discounted penalty function for the compound binomial risk model with delayed claims.Motivated by the idea of randomized decisions on paying dividends were studied in Tan and Yang[16], Liu and Zhang[17]considered a discrete risk model with delayed claims and randomized dividend strategy.Therefore,the compound binomial risk models with delayed claims in insurance risk theory have attracted a lot of attention in the past few years, and a significant amount of works have been done on this topic.See, for example, Xie and Zou[18], Bao and Liu[19], Wu and Li[20],Yuen et al[21],Zhou et al[22],Xie et al[23]etc.Besides mentioned above,there are some other dependent structures,we can refer to Cossette et al[24],Huang and Li[25],Liu et al[26], Liu and Bao[27,28], Wu et al[29]and so on.

To reflect the cash flows of the insurance company more realistically, we note that since Boucherie et al[30]described the stochastic incomes by adding a compound Poisson process with positive jumps to the classical risk model,risk model with random premium incomes have been one of the major interests in insurance risk theory,recently.Such as,Zhao[31]extended the compound binomial model to the case where the premium income process is a binomial process.Bao and Liu[19]studied a compound binomial model with delayed claims and random premium incomes.Other risk model involving random premium incomes were investigated by Bao[32], Bao and Ye[33], Dong et al[34],Yang and Zhang[35], Yu[36], Zhu et al[37]and the references therein.

In this paper, we aim at the Gerber-Shiu discounted penalty function of a compound binomial risk model with potentially delayed claims and random premiums incomes.This risk model generalizes the model of Li and Wu[15]who also considered a compound binomial risk model with delayed claims in which the main claim induces the by-claim, while we assume the main claim causes a by-claim with a certain probability.In addition,we extend the deterministic premiums incomes in the classical compound binomial risk model to the binomial process.When the main claim produces a by-claim with probability 1 and a certainly premium of 1 is received at the beginning of each time period, the results in this paper will reduce to Li and Wu[15].Hence, this paper generalizes the model of Li and Wu[15].

The remainder of this paper is organized as follows.In section 2, we introduce the compound binomial risk model with delayed claims and random premium incomes.In section 3,we obtain a recursive formula for the Gerber-Shiu discounted penalty function as well as an explicit formula in the discount-free case.In section 4, we study some useful insurance quantities using the explicit formula obtained in section 3.Section 5 concludes this paper.

2 Model

The discrete-time risk model considered in this paper is

where

Here, the nonnegative integers u and t denote the initial surplus and the time period respectively.For every i ∈ N+, ηi, ξi,andare all Bernoulli random variables.P(ηi=1)=p0and P(ηi=0)=1 − p0=q0describe whether or not a premium of 1 is received at the beginning of the ith period, P(ξi= 1) = p and P(ξi= 0) = 1 − p = q describe whether or not a main claim occurs at the end of the ith period,and=1 − θ1describe whether or not the ith main claim causes a by-claim,= 1 − θ2describe that if there is a by-claim at time i,then it may occur simultaneously with its corresponding main claim with probability θ2or delay to the next time period with probability 1−θ2.Additionally,we denote the main claims and the by-claims respectively by {X,Xi;i ∈ N+} and {Y,Yi;i ∈ N+},which are two sequences of i.i.d.positive integer-valued random variables with P(X =k) = fk, EX = µXand P(Y = k) = gk, EY = µYfor every k = 1,2,···.Further assume thatand{Yi;i ∈ N+} are mutually independent.As usual, we setand assume that the positive safety loading condition p0− p(µX+ θ1µY) > 0 holds to guarantee that the ruin does not occur with probability 1.

Let τ = inf{t ∈ N+: U(t) < 0} be the ruin time of model (1), with τ = ∞if ruin does not occur.Then, given τ < ∞, let Uτ−denote the surplus immediately before ruin, and let |Uτ| denote the deficit at ruin.For any nonnegative bounded function ω(υ1,υ2) : N × N+→ N, specially, we set ω(0,υ2) = 0 and any discount factor 0 < υ ≤ 1, the Gerber-Shiu discounted penalty function of model (1) is defined as

Particularly, when υ =1, (3) reduces to the following discount-free form

3 Formulas for the Gerber-Shiu function

As in Yuen and Guo[13], we define an auxiliary process

where Y′has the same probability density as Y.Denote byandthe corresponding Gerber-Shiu functions of the process U′(t).

Considering model (1), there are several different cases at time 1 according to whether or not a premium of 1 is received,whether or not a main claim occurs,whether or not a by-claim is occurs, whether the by-claim occurs simultaneously or occurs at the next period, and whether or not the ruin occurs.Taking into account all these cases comprehensively and using the law of total probability, we have

Similarly, for model (5),

We write

Hereafter, we use the function with a tilde to denote the corresponding generating function, i.e.,

and so on.Multiply (12) and (13) by zu+1(0 < z ≤ 1) and sum over u from 0 to ∞.Then, after some rearrangements and let α(z)=p0+q0z, we obtain

Solving the above two equations with respect towe obtain

For simplicity, let

and

then for every 0

where the last inequality holds because of 0 < υ ≤ 1 and the positive safety loading condition.Moreover, it is obvious that β(υp0q) < 0 and β(1) ≥ 0.Hence, there is a unique root rυ∈ (υp0q,1]to the equation β(z)=0, i.e., we have

Substituting z =rυinto(16)and noting(19),we can solve mυ(0)from(16)and obtain that

Particularly, when υ =1 then r1=1, and

On the other hand, by comparing the coefficients of zu+1on both sides of (16), we have

The above result gives us a recursive formula for mυ(u) with the initial value provided in (20).Next, we shall derive the explicit formula for m(u) when υ =1.

In what follows, we use capital letters to denote the corresponding distribution functions, and use capital letters with a bar to denote the corresponding survival functions, such as

and so on.For υ =1 and any t ≥ 0,summing over u from t to ∞ in(22)and rearranging lead to

Substituting u for t in (25) gives

Equation (27) with respect to m(u) can be solved explicitly by the means of generating function.To this purpose, we denote the density functions of the equilibrium distributions of X and X +Y by ρX(j) and ρ(j) respectively, i.e.,

Then, from equation (27), we can obtain

Therefore,

On the other hand, notice that

is the generating function of the function σ(·) defined as

where

This fact, together with (30), yields

which implies

Theorem 1For each u=1,2,···, we have:

(i) The Gerber-Shiu discounted penalty function mυ(u)for model(1)satisfies the recursive formula (22) with initial value given in (20);

(ii) For the discount-free case (υ = 1), m(0) is given by (21) and m(u) has an explicit expression (34).

Remark 1If p0= 1, θ1= 1 and θ2= θ, all of the above results coincide with the relevant results in [15].

4 Applications

In this section,we shall apply the explicit formula(34)to derive some useful insurance quantities,all of which are special cases of m(u)with different choices on function ω(υ1,υ2).

We first consider the ruin probability ψ(u) defined by

From (4), we know that m(u) reduces to ψ(u) if ω(υ1,υ2) = 1.In this case, recalling(8)–(11), for u ≥ 1, we have

where

For simplicity, we further define

Thus

Similarly, we can obtain

and so on.Then, yields

In addition, given (36), (21) becomes

As a result, concluding the above analysis and recalling Theorem 1(ii), we have the following corollary.

Corollary 1For risk model (1) with u=1,2,···, the ruin probability is

where σ(j), Ψ(0) and A1(u) are given by (31), (44) and (43), respectively.

Remark 2If p0=1, θ1=1 and θ2= θ, relation (44) coincides with (15) of [14](note that a main claim occurs with probability q rather than p in their paper).

We next consider the density function of the deficit at ruin ϕ(y|u) defined by

From (4), we know that m(u) reduces to ϕ(y|u) if ω(υ1,υ2) = 1(υ2=y).In this case,repeating the above procedure, we have:

Corollary 2For risk model (1) with u = 1,2,···, the density function of the deficit at ruin is

where σ(j) is defined in (31),

where

and

Remark 3If p0= 1, θ1= 1 and θ2= θ, relation (48) and (49) are respectively equivalent to (4.13) and (4.14) in [15].

Now, we consider the joint density function of the surplus immediately before ruin and the deficit at ruin ϕ(x,y|u) defined by

From (4), we know that m(u) reduces to ϕ(x,y|u) if ω(υ1,υ2) = 1(υ1=x,υ2=y).Correspondingly, we have:

Corollary 3For risk model (1) with u = 1,2,···, the joint density function of the surplus immediately before ruin and the deficit at ruin is

where σ(j) is defined in (31),

where

and

Remark 4If p0=1, θ1=1 and θ2= θ, relation (53) is equivalent to (14) in [14](note that they considered Uτ−1=Uτ−−1 in their paper).

Furthermore, we consider the density function of the claim causing ruin ϕ∗(s|u)defined by

From (4), we know that m(u) reduces to ϕ∗(s|u) if ω(υ1,υ2) = 1(υ1+υ2=s).Then, we have:

Corollary 4For risk model (1) with u = 1,2,···, the density function of the claim causing ruin is

where σ(j) is defined in (31),

where

and

Remark 5If p0= 1, θ1= 1 and θ2= 1 in relation (57) then ϕ∗(s|0) = p(s −1)hs/q, which is equivalent to the last relation in [38].

5 Conclusion

We aim mainly at the Gerber-Shiu discounted expected penalty function of the extensive compound binomial risk model.One of the extension points is supposing that the premium incomes are stochastic which is different than the linear hypothesis.Another extension point is generalizing the original assumption that each main claim induce a by-claim to each main claim produce a by-claim with a certain probability.A recursive formula of the Gerber-Shiu discounted penalty function is obtained.For the discount-free case, an explicit formula is given.Utilizing such an explicit expression,we derive some useful insurance quantities, including the ruin probability, the density of the deficit at ruin, the joint density of the surplus immediately before ruin and the deficit at ruin, and the density of the claim causing ruin.

AcknowledgementsThe authors would like to thank anonymous referees for their helpful comments and suggestions,which improved an earlier version of the paper.