1. Introduction

For over a century, there has been a major interest in actuarial science. Since a large portion of the surplus of insurance business from investment income, actuaries have been studying ruin problems under risk models with rates of interest. For example, Teugels and Sundt[10],[11] studied the effects of constant rate on the ruin probability under the compound Poisson risk model. Yang[13] established both exponential and non – exponential upper bounds for ruin probabilities in a risk model with constant interest force and independent premiums and claims. Cai[3],[4] investigated the ruin probabilities in two risk models, with independent premiums and claims and used a first – order autoregressive process to model the rates of in interest. Cai and Dickson[5] obtained Lundberg inequalities for ruin probabilities in two discrete- time risk process with a Markov chain interest model and independent premiums and claims.

In this paper, we study the models considered by Cai and Dickson[5] to the case homogenous Markov chain claims and homogenous Markov chain rates of interest and independent premiums. The main difference between the model in our paper and the one in Cai and Dickson[5] is that claims and rates of interest in our model are assumed to follow homogeneous Markov chains.

We let be premiums, be claims, be interests and they define on probability space To establish probability inequalities for ruin probabilities of these models, we study two styles of premium collections. On the one hand of the premiums are collected at the beginning of each period then the surplus process with initial surplus can be written as

(1.1)

which can be rearranged as

(1.2)

On the other hand, if the premiums are collected at the end of each period, then the surplus process with initial surplus can be written as

(1.3)

which is equivalent to

(1.4)

where throughout this paper, we denote and if .

We assume that:

Assumption 1.1.

Assumption 1.2. is a sequence of independent and identically distributed non – negative continuous random variables with the same distributive function .

Assumption 1.3. is a homogeneous Markov chain, take values in a finite set of non - negative numbers with and where

Assumption 1.4. is homogeneous Markov chain, take values in a finite set of non - negative numbers with and where

Assumption 1.5. and are assumed to be independent.

We define the finite time and ultimate ruin probabilities of model (1.1) with assumption 1.1 to assumption 1.5, respectively, by

(1.5)

(1.6)

Similarly, we define the finite time and ultimate ruin probabilities of model (1.3) with assumption 1.1 to assumption 1.5, respectively, by

(1.7)

(1.8)

In this paper, we derive probability inequalities for and . The paper is organized as follows; in Section 2, we give recursive and integral equations for and . In Section 3 we derive probability inequalities for and by an inductive approach. A numerical example is given to illustrate these results in Section 4. Finally, we conclude our paper in Section 5.

2. Integral Equation for Ruin Probabilities

We first give recursive equations for and an integral equation for .

Theorem 2.1. If model (1.1) satisfies the assumptions 1.1 to 1.5 then for n = 1, 2, …

(2.1)

and

(2.2)

Proof.

Give

Let

,

.

From (1.1), we have and

(2.3)

In addition,

(2.4)

Let be independent copies of , , respectively with

Thus, (2.4) and (1.2) imply that

(2.5)

On the other hand, (1.5) implies

Thus, we have

(2.6)

From (2.3), we have

.

From (2.5), we have

Therefore, (2.6) is written as

(2.7)

Thus, the integaral equation for in Theorem 2.1 follows immediately from the dominated convergence theorem by letting in (2.7).

This completes the proof

Similarly, the following recursive equation for and integral equation for are hold.

Theorem 2.2. If model (1.3) satisfies assumptions 1.1 to 1.5 then, for n = 1, 2, …

(2.8)

and

(2.9)

Next, we establish probability inequalities for ruin probabilities of model (1.1) and model (1.3).

3. Probability Inequalities for Ruin Probabilities

To establish probability inequalities for ruin probabilities of model (1.1), we first proof the following Lemma.

Lemma 3.1. Let model (1.1) satisfy assumptions 1.1 to 1.5 and .

If, any,

and

(3.1)

then there exists a unique positive constant satisfying:

(3.2)

Proof.

Define

We have

From is discrete random variables and it takes values in then has -th derivative function on (any ).

In addition, with satisfying :

and .

This implies that has -th derivative function on with . Thus, has -th derivative function on with and

.

This implies that

(3.3)

and

(3.4)

By , we can find some constant such that

Then, we can get that

Imply

(3.5)

From (3.3), (3.4) and (3.5) there exists a unique positive constant satisfying (3.2).

This completes the proof .

Let:

Using Lemma 3.1 and Theorem 2.1, we obtain a probability inequality for by an inductive approach.

Theorem 3.1. If model (1.1) satisfies assumptions 1.1 to 1.5, and (3.1) then

for any , and

(3.6)

where

Proof.

Firstly, we have

.

For any , we have

(3.7)

(3.8)

Then, for any , and , we can write

(3.9)

Thus, combining (3.8) and (3.9), we have

(3.10)

Applying an inductive hypothesis, we assume for any , and ,

(3.11)

Then (3.10) implies that (3.11) holds with n = 1.

For , and , we have

where

,

and

For any

then

.

That then

(3.12)

Therefore, by Lemma 3.1, (2.1), (3.7) and (3.12), we get

Thus

Consequently, for any (3.11) holds. Therefore, (3.6) follows by letting in (3.11).

This completes the proof

Remark 3.1. Let . From and , we have

Therefore, upper bound for ruin probability in (3.6) is better than .

Similar to Lemma 3.1, we have Lemma 3.2.

Lemma 3.2. Assume that model (1.3) satisfies assumptions 1.1 to 1.5 and . If any and ,

and

(3.13)

then there exists a unique positive constant satisfying:

Let

Next, we use Lemma 3.2 and Theorem 2.2 to give a probability inequality for by an inductive approach.

Theorem 3.2. If model (1.3) satisfies assumptions 1.1 to 1.5, and (3.13) then, for any and

(3.14)

where

Proof.

Similarly with Theorem 3.1, we have and any

(3.15)

(3.16)

Then, for any , and

Hence

(3.17)

Under an inductive hypothesis, we assume that

(3.18)

Then, (3.17) implies that (3.18) holds with n = 1.

For , and , we have

where

,

and

For any

then

We get then

(3.19)

Therefore, by Lemma 3.2, (2.8), (3.15) and (3.19), we get

Thus

Consequently, for any n =1, 2, … (3.18) holds. Therefore, (3.14) follows by letting in (3.18). 

Remark 3.2.

Let

From and , we have

Hence, upper bound for ruin probability in (3.14) is better than .

4. A Numerical Illustration

In this section we give a numerical example to illustrate the bounds of derived in Section 3.

Let be a sequence of independent and identically distributed non-negative continuous random variables with the same distributive function .

Let be a homogeneous Markov chain such that for any , take values in with having a distribution:

and matrix is given by

Let be a homogeneous Markov chain such that for any , take value in with having a distribution:

and matrix is given by

Then, we have

Therefore

(4.1)

In the other hand,

(4.2)

and

(4.3)

Combining (4.1), (4.2) and (4.3) imply that Lemma 3.1 holds.

Next, we solve equation (3.2).

Firstly, we have

where

()

and

Respective equation (3.2) for , by

(4.4)

(4.5)

Using Maple, we find respective root of (3.2) for , by

Hence, .

We can apply the result of Theorem 3.1 for

(4.6)

where

Table 1 shows values upper bounds of for a range of value of u

Table 1. Upper bounds      of

5. Conclusions

Our main results in this paper are Theorem 2.1 and Theorem 2.2 giving recursive equations for and and integral equations for and ; Theorem 3.1 and Theorem 3.2 giving probability inequalities for and by an inductive approach. In addition, a numerical example is given illustrating Theorem 3.1.

ACKNOWLEDGMENTS

The author is thankful to the referee for providing valuable suggestions to improve the quality of the paper.

In addition, the author would like to express his sincere gratitude to Professor Bui Khoi Dam for many scientific suggestions during the preparation of this paper.