Some Results of Ruin Probability for the Classical Risk Process †

The computation of ruin probability is an important problem in the collective risk theory. It has applications in the fields of insurance, actuarial science, and economics. Many mathematical models have been introduced to simulate business activities and ruin probability is studied based on these models. Two of these models are the classical risk model and the Cox model. In the classical model, the counting process is a Poisson process and in the Cox model, the counting process is a Cox process. Thorin (1973) studied the ruin probability based on the classical model with the assumption that random sequence followed the Γ distribution with density function f(x) = x 1 β −1 β 1 β Γ(1/β) e − x β , x > 0, where β > 1. This paper studies the ruin probability of the classical model where the random sequence follows the Γ distribution with density function f(x) = α n Γ(n) xn−1e−αx, x > 0, where α > 0 and n ≥ 2 is a positive integer. An intermediate general result is given and a complete solution is provided for n = 2. Simulation studies for the case of n = 2 is also provided.


Introduction
To study the probability of ruin, many models have been proposed and investigated.These models include the classical and the Cox model.Although the applications of ruin probability are not restricted to the insurance type problem, the idea can be explained easily by an insurance business model.An insurance company's capital at time t can be expressed as R(t) = u + C(t) − O(t), where R(t) is the company's capital at time t, u is the initial capital, C(t) is the income function, and O(t) is the expense function.The income function C(t) can be expressed as a relatively simple function and often taken to be C(t) = ct, where c > 0 is the income coefficient.The expense function O(t), however, is a random process because of the nature of the insurance claims.One way to describe O(t) is to define: where Z i is the size of the payment and N (t) is a counting process.The classical model assumes that N (t) is a Poisson process and the Cox model assumes that N (t)is a Cox process.Thorin (1973) studied the classical model where Z i has Γ distribution with density function where β > 1.In this paper, we study the classical model when Z i follows theΓ distribution with density function where α > 0 and n ≥ 2 is a positive integer.Thorin's case models situations where the probability distribution of ten random payments Z i is monotonically strictly decrease.The case we study here can be applied to situations when the probability distribution of the random payments Z i places higher mass to values near the center.It is meaningful to study this case because many data sets in practice have mount shape distributions.
The organization of the paper is as follows: In section 2, we give the main theorems and section 3 presents the simulation study.Section 4 contains the concluding remarks.

Main Results
We define a risk process as Z k , 1 is a sequence of independent random variables with identical distributions.We further assume that the common distribution function is F with mean µ and variance σ 2 , and F (0) = 0.The counting process N (t) and{Z k } ∞ 1 are independent, and c is a positive constant.N (t) can be interpreted as the number of claims of an insurance company in the time interval (0, t].At each jump point of N (t), the insurance company has to pay out a stochastic amount of money.On the other hand, the company receives c units of revenue per unit time.
Let Ψ(u) = P {u + X(t) < 0 for some t > 0}.Thenfor u ≥ 0, Ψ(u) can be interpreted as the ruin probability of an insurance company which has initial capital u when facing the risk process X(t).Φ(u) = 1 − Ψ(u) is the non-ruin probability.Notice that Ψ(u) = 1 for u < 0.
From now on, we assume that N (t) is a Poisson process with standing λ > 0, and therefore, E(N (t)) = λt.Let Then We can see that and is called the safety loading.It is the ratio of the expected profit to the expected payments.ρ = 1, for example, indicates that the company's expected profit is of the same size as its expected payments.Larger value of ρ indicates a company is in a "save" state.Thus ρ can be considered as an index to measure the safety (non-ruin) of an insurance company.We also assume that ρ > 0.
This condition can be interpreted as the premiums received per unit time exceed the expected claim payments per unit time.
If Z i has exponential distribution, then (for example, see Grandell(1991)).If Z i has Γ distribution with density function where β > 1, then the ruin probability is where R is the positive solution of the equation for r < 1/β (for example, see Thorin (1973),or Grandell (1991), p.14).Now we consider the case that Z i has Γ distribution with density function where α > 0 and n is a positive integer.We state a well-known result as a lemma: Lemma 1 For a classical risk model, if N (t) has intensity λ and Z i has distribution function F (z), then Here, D = d du is the differential operator.(Interested readers please see equation (3) in Grandell (1991), p.4.) Theorem 1 For a classical risk model, if N (t) has intensity λ and Z i has Γ distribution with density function where α > 0 and n ≥ 1 is an integer, then the non-ruin probability Φ(u) is a solution of the ordinary differential equation: Proof.From Lemma 1, we have Substituting dF (z) = f (z)dz, we have The change of variables z = u − w leads to Let A(u; 0) = Φ(u) and let . Then h(w, η) and ∂h(w,η) ∂η are continuous.Let Then by Lang (1979), p.119, Theorem 5 of Chapter V, we can exchange the order of differentiation and integration, so we have And by using similar methods, for 2 ≤ k ≤ n, we have So for 1 ≤ k ≤ n, we have From ( 3) and (4), we have From ( 5) and ( 6), we have After simplification, we have The above equation is solved in Grandell (1991), page 6. 2 The following theorem gives a complete solution of differential equation (1) for n = 2.
Theorem 2 For a classical risk model, if N (t) has intensity λ and Z i has Γ distribution with density function then the non-ruin probability Φ(u) is : where Proof.By Theorem 1, Φ(u) is a solution of the ordinary differential equation: The characteristic equation can be rewritten as and has solutions And (7) has general solution Since ν 1 and ν 2 are the solutions of equation ( 9), we have Thus, let u → ∞, we have: From ( 3), ( 10) and (11), we have After simplification, we have From (8), we have The solution is Thus we have 2 Theorem 2 has an interesting equivalent version.It is stated as follows: Theorem 3 For a classical risk model, if N (t) has intensity λ = 1 and Z i has Γ distribution with density function then the non-ruin probability Φ(u) is: where It is obvious that Theorem 2 implies Theorem 3 (by lettingλ = 1, α = 1).Now we show that Theorem 3 implies Theorem 2 as follows: Suppose that Theorem 3 holds.For a classical risk model, let N (t) be a Poisson process with intensity λ, and assume all Z i 's have Γ distribution with identical density function and Zi = αZ i , for all i.Then Ñ (t) is a Poisson process with intensity 1, all Zi s have identical density function Zi , Here, C and U are the c and u in Theorem 3, respectively.(Recall that u is the initial capital of an insurance company and the company receives c units of revenue per unit time.)P is the non-ruin probability Φ(u) in Theorem 3 calculated by using equation (12); NR is the number of the non-ruin risk processes; F is the relative frequency of non-ruin, and E is the difference of F and P. E can be interpreted as the error of simulation results (assuming our theory is correct.)The small E values in our study confirm our theoretical results from a different perspective.
For the estimation of the ruin probability Ψ(u) , we can use the Lundberg inequality where R is the Lundberg exponent, that is, R is the positive solution of the equation (For example, see Grandell, p.11.)For the risk process in Theorem 3, equation ( 16) becomes is the unique solution of equation (17) satisfying 0 < R < 1.For c = 2.1, 2.2 and 2.4, we have R = 0.03191, 0.06125 and 0.11338 respectively.Ψ(u) is strictly decreasing in R for fixed u > 0, it is suffice to examine R = 0.03191.So if u > 500, then by (15), the ruin probability Ψ(u) ≤ e −0.03191×500 = 1.1771 × 10 −7 is sufficiently small.So if inf{t : R(t) > 500} < inf{t : R(t) < 0}, we classify the company as "non-ruin".Otherwise, if inf{t : R(t) < 0} < inf{t : R(t) > 500}, we consider the company as "ruin".In other words, in our example, "non-ruin" means that the time for the total capital exceeding 500 comes prior to the time the total capital being less than 0. This assumption is reasonable because once the total capital exceeds 500, the chance of "ruin" is slim.
We consider the hypothesis test problem

Concluding Remarks
The computation of a ruin probability is in many cases difficult.This paper gives a way of computing the ruin probability in a special case.Although the result of Theorem 1 is for gamma distributions with n being positive integer, we are currently only able to give a complete solution for the case of n =2.We have obtained solutions for the characteristic equations corresponding to equation (1) in Theorem 1 for n =3 and 4. The computation of the ruin probabilities for these cases is ongoing.