Several models have been proposed for a discrete time2²

We will reserve the use of letter n for the approximation procedures proposed later on.

One central problem in the theory of ruin is to find ψ(u) . For the above model this probability can be calculated using the following relation known as Gerber’s formula [7], (2) ψ(0)=p·μX, (3) ψ(u)=(1−p)ψ(u+1)+p ∑x=1uψ(u+1−x)fX(x)+pF‾X(u), for u=1,2,… , where F‾X(u)=ℙ(Xi>u)=∑x=u+1∞fX(x) u)={\textstyle\sum _{x=u+1}^{\infty }}{f_{X}}(x)$]]>.

An apparently simpler risk model is defined as follows.

In this case, at each unit of time there is always a claim of size Y. If μY denotes the expectation of this claim, the net profit condition now reads μY<1 . It can be shown [4, pp. 467] that this condition implies that ψ(u)<1 , where the time of ruin τ and the ultimate ruin probability ψ(u) are defined as before. One feature that makes this model simple is that all of its elements are discrete, but some studies [2] have been carried out where continuous claims are incorporated.

Under a conditioning argument, it is easy to show that the probability of ruin satisfies the recursive relation (5) ψ(0)=μY, (6) ψ(u)= ∑y=0ufY(y)ψ(u+1−y)+F‾Y(u),u≥1.

Now, given a compound binomial model (1) we can construct a Gerber–Dickson model (4) as follows. Let R1,R2,… be i.i.d. Bernoulli(p) random variables and define Yi=Ri·Xi where Xi∈{1,2,…} , i≥1 , as in model (1). The distribution of these claims is fY(0)=1−p and fY(y)=p·fX(y) , for y≥1 .

Conversely, given model (4) and defining p=1−fY(0) , we can construct a model (1) by letting claims Xi have distribution fX(x)=fY(x)/p , for x≥1 . It can be readily checked that μY=p·μX and that the probability generating function of U(t) in both models coincide. This shows models (1) and (4) are equivalent in the sense that U(t) has the same distribution in both models. As expected, the recursive relations (3) and (6) can be easily obtained one from the other. Thus, results obtained for one model can be translated for the other model. For example, using model (1), authors in [3] find the ruin severity and the surplus just before the ruin, and a discrete version of the popular Gerber–Shiu function is used in [10] and [11] to solve problems on ruin time and ruin severity. A survey of results and models for several discrete time risk models can be found in [12].

In this work we will use the notation of the Gerber–Dickson risk model (4) and for simplicity we will write f(y) , F(y) and μ instead of fY(y) , FY(y) and μY , respectively. Also, as time an other auxiliary variables are considered discrete, we will write, for example, t≥0 instead of t=0,1,… Our main objective is to provide some methods to approximate the ultimate ruin probability for this risk model. The results obtained are the discrete version of those found in [14].

The continuous version of Pollaczeck–Khinchine formula plays a major role in the theory of ruin for the Cramér–Lundberg model. On the contrary, its discrete version is seldom mentioned in the literature on discrete time risk models. In this section we develop this formula and apply it later to find a general method to calculate ultimate ruin probabilities for claims with particular distributions. The construction procedure we use for the discrete case resembles closely the one already known for the continuous case.

Assuming τ<∞ , the non-negative random variable W=|U(τ)| is known as the severity of ruin. It indicates how large the capital drops below zero at the time of ruin. See the top-right plot of Figure 1. The joint probability of ruin and severity not greater than w=0,1,… is denoted by (7) φ(u,w)=ℙ(τ<∞,W≤w∣U(0)=u). In [5] it is shown that, in particular, (8) φ(0,w)= ∑x=0wF‾(x),w≥0. Hence, (9) ℙ(τ<∞,W=w∣U(0)=0)=φ(0,w)−φ(0,w−1)=F‾(w).

This probability will be useful in finding the distribution of the size of the first drop of the risk process below its initial capital u, see Theorem 1 below, which will lead us to the Pollaczeck–Khinchine formula. For every claim distribution, there is an associated distribution which often appears in the calculation of ruin probabilities. This is defined next.

The probability function defined by (10) is also known as the integrated-tail distribution, although this name is best suited to continuous distributions. For example, the equilibrium distribution associated to a Geometric(p) claim distribution with mean μ=1/(1−p) is the same geometric since (11) fe(y)=F‾(y)/μ=(1−p)y+1p/(1−p)=p(1−p)y,y≥0.

As in the continuous time risk models, let us define the surplus process {Z(t):t≥0} by (12) Z(t)=u−U(t)= ∑i=1t(Yi−1). This is a random walk that starts at zero, it has stationary and independent increments and Z(t)→−∞ a.s. as t→∞ under the net profit condition μ<1 . See the bottom-right plot of Figure 1. In terms of this surplus process, ruin occurs when Z(t) reaches level u or above. Thus, the ruin probability can be written as (13) ψ(u)=ℙ(Z(t)≥ufor somet≥1)=ℙmaxt≥1Z(t)≥u,u≥1.

As u≥1 and Z(0)=0 , we can also write (14) ψ(u)=ℙmaxt≥0Z(t)≥u.

We next define the times of records and the severities for the surplus process.

The random variables τ0∗<τ1∗<⋯ represent the stopping times when the surplus process Z(t):t≥0 arrives at a new or the previous maximum, and the severity Yi∗ is the difference between the maxima at τi∗ and τi−1∗ . A graphical example of these record times are shown in the bottom-right plot of Figure 1. In particular, observe that τ1∗ is the first positive time the risk process is less than or equal to its initial capital u, that is, (16) τ1∗=min{t>0:u−U(t)≥0}, 0:u-U(t)\ge 0\},\]]]> and the severity is Y1∗=Z(τ1∗)=u−U(τ1∗) and this is the size of this first drop below level u. Also, since the surplus process has stationary increments, all severities share the same distribution, that is, (17) Yi∗=Z(τi∗)−Z(τi−1∗)∼Z(τ1∗)−Z(0)=Y1∗,i≥1, assuming τi∗<∞ . We will next find out that distribution.

Since the surplus process is a Markov process, the following properties hold: For i≥2 and assuming τi∗<∞ , for 0<s<x , (19) ℙ(τi∗=x∣τi−1∗=s)=ℙ(τi∗−τi−1∗=x−s∣τi−1∗=s)=ℙ(τ1∗=x−s). Also, for k≥1 , (20) ℙ(τk∗<∞∣τk−1∗<∞)=ℙ(τ1∗<∞), (21) ℙ(τk∗=∞∣τk−1∗<∞)=ℙ(τ1∗=∞).

The total number of records of the surplus process {Z(t):t≥0} is defined by the non-negative random variable (22) K=max{k≥1:τk∗<∞}, when the indicated set is not empty, otherwise K:=0 . Note that 0≤K<∞ a.s. since Z(t)→−∞ a.s. under the net profit condition. The distribution of this random variable is established next.

In the following proposition it is established that the ultimate maximum of the surplus process has a compound geometric distribution. This will allow us to write the ruin probability as the tail of this distribution.

For example, suppose claims have a Geometric(p) distribution with mean μ=(1−p)/p . The net profit condition μ<1 implies p>1/2 1/2$]]>. We have seen in equation (11) that the associated equilibrium distribution is again Geometric(p) , and hence the k-th convolution is Negative Binomial with parameters k∈ℕ and p∈(1/2,1) . Straightforward calculations show that the Pollaczeck–Khinchine formula gives the known solution for the probability of ruin, (28) ψ(u)=1−ppu+1,u≥0.

This includes the case u=0 in the same formula. In the following section we will consider claims that have a mixture of some distributions.

Negative binomial mixture (NBM) distributions will be used to approximate the ruin probability when claims have a mixed Poisson (MP) distribution. Although NBM distributions are the analogue of Erlang mixture distributions [18, 19], they cannot be used to approximate any discrete distribution with non-negative support. However, it turns out that they can approximate mixed Poisson distributions. This is stated in [16, Theorem 1], where the authors define NBM distributions by the probability generating function G(z)=limm→∞ ∑k=1mqk,m1−pk,m1−pk,mzrk,m,z<1, where qk,m are positive numbers and their sum over index k equals to 1. This is a rather general definition for a NBM distribution. In this work we will consider a particular case of it.

We will denote by NB(k,p) the negative binomial distribution with parameters k∈ℕ and p∈(0,1) . Its probability function will be written as nb(k,p)(x) and the distribution function by NB(k,p)(x) . More precisely, for integers x≥0 , nb(k,p)(x)=k+x−1xpk(1−p)x,NB(k,p)(x)=1− ∑i=0k−1nb(x+1,1−p)(i). The case k=1 reduces to the Geometric(p) distribution. It will be also useful to recall that the NB(k,p) distribution can be obtained as the distribution of the sum of k independent random variables Geometric(p) -distributed.

It is useful to observe that any NBM distribution can be written as a compound sum of geometric random variables. Indeed, let N be a discrete random variable with the probability function qk=fN(k) , k≥1 , and define SN=∑i=1NXi , where X1,X2,… are i.i.d. r.v.s Geometric(p) -distributed and independent of N. Then, conditioning on the values of N, ℙ(SN=x)= ∑k=1∞qk·ℙ ∑i=1kXi=x= ∑k=1∞qk·nb(k,p)(x),x≥0. Thus, given any NBM(π,p) distribution with π=(fN(1),fN(2),…) , we have the representation (29) SN= ∑i=1NXi∼NBM(π,p). In particular, (30) 𝔼(SN)=𝔼(N)1−pp, (31) FSN(x)= ∑k=1∞fN(k)·NB(k,p)(x),x≥0, and the p.g.f. has the form GSN(r)=GN(GX(r)) . The following is a particular way to write the distribution function of a NBM distribution.

We will show next that the equilibrium distribution associated to a NBM distribution is again a NBM one. For a distribution function F(x) , F‾(x) denotes 1−F(x) .

It can be checked that (33) is a probability function. It is the equilibrium distribution associated to N.

The following proposition states that a compound geometric NBM distribution is again a NBM one. This result is essential to calculate the ruin probability when claims have a NBM distribution.

From (35) and (36), it is not difficult to derive a recursive formula for F‾N∗(k) , namely, (38) F‾N∗(k)=(1−ρ) ∑j=1kfN(j)F‾N∗(k−j)+F‾N(k),k≥1.

The following result establishes a formula to calculate the ruin probability when claims have a NBM distribution.

As an example consider claims with a geometric distribution. This is a NBM distribution with π=(1,0,0,…) . Equations (40–42) yield C‾k=(1−p)/pk+1,k≥0. Substituting in (39) together with ψ(0)=(1−p)/p , we recover the known solution ψ(u)=(1−p)/pu+1 , for u≥0 , mentioned earlier in (28).

This section contains the definition of a mixed Poisson distribution and some of its relations with NBM distributions.

Observe the distribution of X∣(Λ=λ) is required to be Poisson, but the unconditional distribution of X, although discrete, is not necessarily Poisson. In [13], necessary and sufficient conditions are given to determine whether a given distribution belongs to the MP family. This is done via its probability generating function. A large number of examples of these distributions can be found in [9] and a study of their general properties is given in [8]. In particular, it is not difficult to see that 𝔼(X)=𝔼(Λ) . Indeed, conditioning on the values of Λ, 𝔼(X)=∫0∞𝔼(X|Λ=λ)dFΛ(λ)=∫0∞λdFΛ(λ)=𝔼(Λ). Also, the p.g.f. of X can be written as (44) GX(r)=∫0∞e−λ(1−r)dFΛ(λ),r<1. In [17], a recursive formula to evaluate MP probabilities is given. The following proposition establishes a relationship between the Erlang mixture distribution (used as a mixing distribution) and the negative binomial distribution. The former will be denoted by ErlangM(π,β) , with similar meaning for the parameters as in the notation NBM(π,p) used before. In the ensuing calculations the probability function of a Poisson(λ) distribution is denoted by poisson(λ)(x) . The Erlang distribution of parameters k∈ℕ and β>0 0$]]> is denoted by Erlang(k,β) . The distribution function is written as Erlang(k,β)(x) and the density function as erlang(k,β)(x) , that is, erlang(k,β)(x)=(βx)k−1(k−1)!βe−βx,x>0. 0.\]]]> When k=1 one obtains the exponential distribution with parameter β, which is denoted by Exp(β) .