1. Introduction

Claude Lefèvre and Stéphane Loisel[1] studied the problem of ruin in the classical compound binomial and compound Poisson risk models. Their primary purpose is to extend those models which is an exact formula derived by Picard and Lefèvre[9] for the probability of (non-ruin) ruin within finite time. Hong N.T.T. (see [7]) recently built an exact formula for finite time ruin (non-ruin) probability for model:

With are positive integer number.

However, Claude Lefèvre and Stéphane Loisel[1] did not provide an exact formula for ruin probability of generalized risk processes under constant interest force with sequences of random variables such that these sequences are usually assumed to be positive integer – valued random variables, with surplus process written as

(1.1)

where is initial surplus (), is constant interest (), and are positive integer numbers, and take values in a finite set of positive integer numbers; and are assumed to be independent.

With these assumptions, the aim of this paper is to build an exact formula for finite time ruin (non-ruin) probability of model (1.1). In our study, we extended the result of Hong N. T. T for model (1.1) with any r > 0. This is the first time that gives an exact formula for ruin (non-ruin) probability for model (1.1) whose exact formula for finite time ruin (non-ruin) probability are derived by using technique of classical probability.

The paper is organized as follows; in Section 2, we build

an exact formula for ruin (non-ruin) probability for model (1.1) with and being independent and identically distributed positive integer – valued random variables, and are assumed to be independent. An extended result in Section 2 with and being homogeneous Markov chains is given in Section 3. A numerical example is give to illustrate these results in Section 4. Finally, we conclude our paper in Section 5.

2. Computing Ruin Probability of Generalized Risk Processes under Constant Interest Force with Sequences of Independent and Identically Distributed Random Variables

Let model (1.1). We assume that:

Assumption 2.1. u and t are positive integer numbers.

Assumption 2.2 is a sequence of independent and identically distributed random variables, take values in a finite set of positive integer numbers

with ,

.

Assumption 2.3 is a sequence of independent and identically distributed random variables, take values in a finite set of positive integer numbers with ,

.

Assumption 2.4 and are assumed to be independent.

From (1.1), we have:

(2.1)

Supposing that the ruin time is defined by, where .

We define the finite time ruin (non-ruin) probabilities of model (1.1) with Assumption 2.1 to Assumption 2.4, respectively, by

(2.2)

(2.3)

Throughout this paper, we denote if

To establish a formula for , we first proof the following Lemma.

Lemma 2.1. Any u and are positive integer numbers.

With being a positive integer number and satisfies:

(2.4)

then

(2.5)

Proof.

From (2.4), we have

Implies

Hence (2.5) holds.

This completes the proof.

Next, we give an exact formula for finite time non-ruin (ruin) probability of model (1.1).

Theorem 2.1. Let model (1.1) satisfy Assumption 2.1 to Assumption 2.4, then finite time non-ruin probability of model (1.1) is defined by

(2.6)

where

,

,

…

,

In addition, is integer part of the .

Proof.

Firstly, we have

(2.7)

By Assumption 2.2, we let with being positive integer numbers and satisfying: .

Let

.

Since is a sequence of independent random variables then

(2.8)

Hence, (2.7) is given

(2.9)

where

(2.10)

By Assumption 2.3, we let with being positive integer numbers andsatisfying: .

Let

In addition, is integer part of ,

By using Lemma 2.1, , , …, are integer numbers.

Thus, (2.10) is written as

(2.11)

As Assumption 2.3, we let with is positive integer number then . Combining Assumption 2.3, (2.10) and formulas define , we have .

Therefore, (2.11) can be rearranged as

(2.12)

For the reason that is a sequence of independent random variables then

In the other hand, system of events in (2.12) is incompatible then

(2.13)

Next, we consider

and

By using, and are independent, if and hold then and are independent events.

In addition, system of events in (2.9) is incompatible.

Therefore, combining (2.8) and (2.13), we have

(2.14)

This completes the proof.

Corollary 2.1. Let model (1.1) satisfy Assumption 2.1 to Assumption 2.4, then finite time ruin probability of model (1.1) is defined by

(2.15)

Remark 2.1. Formula (2.6) (or (2.15)) gives a method to compute exactly finite time non-ruin (ruin) probability of model (1.1) which and are sequences of independent and identically distributed random variables, and they take values in a finite set of positive integer numbers.

3. Computing Ruin Probability of Generalized Risk Processes under Constant Interest Force with Homogeneous Markov Chains

Let model (1.1). We assume that:

Assumption 3.1. are positive integer numbers.

Assumption 3.2. is a homogeneous Markov chain, take values in a finite set of positive integer numbers with where . In addition,

.

Assumption 3.3. is a homogeneous Markov chain, take values in a finite set of positive integer numbers with where . In addition,

Assumption 3.4. and are assumed to be independent.

Supposing that the ruin time is defined by where .

We define the finite time ruin (non-ruin) probability of model (1.1) using Assumption 3.1 to Assumption 3.4, respectively, by

(3.1)

(3.2)

Similar to Theorem 2.1, we have

Theorem 3.1. Let model (1.1) satisfy Assumption 3.1 to Assumption 3.4, then finite time non-ruin probability of model (1.1) is defined by

(3.3)

where, is defined in the same way with Theorem 2.1.

Proof.

We proof similarly as Theorem 2.1, where,

(2.8) replaced by

In the other hand, we have

In addition, (2.13) substituted by

By using the same method to prove Theorem 2.1, we have formula (3.3).

This completes the proof.

Corollary 3.1. Let model (1.1) satisfy Assumption 3.1 to Assumption 3.4, then finite time ruin probability of model (1.1) is defined by

(3.4)

Remark 3.1. Formula (3.3) (or (3.4)) gives a method to compute exactly finite time non-ruin (ruin) probability of model (1.1) which and are homogeneous Markov chains and they take values in a finite set of positive integer numbers.

4. Numerical Illustration

4.1. Numerical Illustration for

Let be a sequence of independent and identically distributed random variables, take values in a finite set of positive integer numbers with having a distribution:

Let be a sequence of independent and identically distributed random variables, take values in a finite set of positive integer numbers with having a distribution:

By using the C progaram, the is calculated with the assumptions above of random variables .

Table 4.1 shows for a range of value of u.

Table 4.1. Ruin probabilities of model (1.1) with Assumption 2.1- 2.4 and r = 0,15

4.2. Numerical Illustration for

Let be a homogeneous Markov chain, take values in a finite set of positive integer numbers

,with having a distribution

In addition, matrix is given by

Let be a homogeneous Markov chain, take values in a finite set of positive integer numbers

, with having a distribution

In addition, matrix is given by

By using the C progaram, the is calculated with the assumptions above of random variables and matrixs .

Table 4.2 shows for a range of value of u.

Table 4.2. Ruin probabilities of model (1.1) with Assumption 3.1-3.4 and r = 0,15

5. Conclusions

By using technique of classical probability with , claims, premiums all are positive integer numbers and is a positive number, this paper constructed an exact formula for ruin (non-ruin) probability of model (1.1), where sequences of claims and premiums are independent and identically distributed random variables or homogeneous Markov chains. Our main results in this paper not only prove Theorem 2.1 and Theorem 3.1 but also give numerical examples to illustrate for Theorem 2.1 and Theorem 3.1. These results proof for the suitability of theoretical results and practical examples. It also mean that:

When initial is increasing then , are decreasing,

With being unchanged, when is increasing then , are increasing.

ACKNOWLEDGMENTS

The authors are thankful to the referee for providing valuable suggestions to improve the quality of the pape.

In addition, the author would like to thank my advisor, Professor Bui Khoi Dam for his patient guidance, encouragement and advice.