UPPER BOUNDS FOR RUIN PROBABILITIES IN DISCRETE TIME RISK MODELS WITHDEPENDENT RANDOM VARIABLES

Bui Khoi DamHanoi University of Technology

andNguyen Huy Hoang

National Economics University, Vietnam

Abstract.In this paper, we study ruin probabilities of risk models, in discretetime, for insurance companies. Our results show that the ruinprobabilities still have the usual exponential form when assuming thatthe claim size and the premium are dependent random variables. Inaddition, more general models of ruin probabilities where the effects ofinterest, i.e. sequence of non-negative random variables, areconsidered. The proof follows the Martingale’s method. Our resultsconsiderably extended the ones by Cai, J. ([4], 2002), Yang, H. ( [14] ,1999).Keywords: risk models, ruin probabilities, m-dependent random variablessequence, Lundberg’s inequality, interest, supermartingale.

1. IntroductionIn classical risk models of financial insurance, the claim size

and premium are assumed to be consequences ofidentically independent distributed random variables. Thus, theproperty of the insurance company at the nth period is the followingsequence of random variables:

, (1.1)

where , u is the initial capital of the company.

We denote

,

Then the ruin probabilities at the nth period can be defined asfollows:

,

And the ruin probabilities (in infinite time period)

(1.2) Assume that there exists satisfying that

, Then the ruin probabilities satisfy the Lundberg’s inequality

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. More details can be found in [14], theorem 1.3. We call the model (1.1) the risk model without effects of interest. However, in reality, the claim size or premium is effected by interest;therefore this model is generalized so as to be more applicable, considering effects of interest, when the interest is either defined byidentically distributed non-negative random variables of or a constant. In that case, (1.1) is generalized as follows:

(1.3)

or

If we denote

,

the ruin probabilities subsequently can be defined by

(1.4)The influence of interest, being a constant, to ruin probabilities

has been investigated by Sundt and Teugels in ([10], 1995; [11], 1997), using thePoisson’s risk model. It was then continued by Rolski et al ([8],1999),Asmussen([1], 2000), where it being considered in continuous time. Thespecial case of the model (1.3) where the sequence of interests

are constants was then considered by Yang ([14], 1999).Additionally, Cai ([2], 2002) considered the model (1.3) with beinga sequence of non-negative identically independent distributed randomvariables.

However, the assumption that the claim size, premium and interestare sequences of independent random variables is not very suitable inreality. Some authors like Muller and Pflug ([7], 2001) has considered riskmodels with the claim size being sequences of dependent randomvariables. Cai ([4], 2002) considered model (1.3), assuming that is a sequence of dependent random variables.

In this paper we consider models (1.1) and (1.3) with more generalassumptions that both sequences of random variables, the one of claimsizes and the one of premiums are sequences of m-dependent random variables. In model (1.3) the sequence of interests

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is considered to be of random variables (dependent in general), non-negative and identically distributed. We have considerably extended theresults of above mention authors. Namely, we have proved thatLundburg’s inequality can be used for evaluating the upper bound ofruin probabilities in the risk models, whatever the interest is takeninto account or not. (Theorem 2.1 and 3.1).

First of all, we introduce the concepts of sequence m-dependentrandom variables. Definition 1.1

Sequence of random variables is called m -dependent, if algebras

is independent of

Example 1.1

A sequence of independent random variables is called o -dependent.

Let be a sequence of random variables, independent andidentically distributed.If we define

then is a sequence of m – dependent random variables.

If is a sequence of 1- dependent random variables thensubsequences and are sequences of independentrandom variables.

We first consider model (1.1) assuming that and aresequences of m-dependent random variables. 2. Risk models without considering effects of interest This section is mainly concerned with the following theorem. Theorem 2.1

In risk model (1.1) if sequences of premiums and claim sizes are assumedto be mutually independent and be sequences of non-negative random, m – dependent, identicallydistributed Suppose there exists R > 0 such that

,

Then the risk probabilities (1.2) satisfy Lundberg’s inequality as follows

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where .

Proof.

Observe that

=

(2.1)

To be clear, we first prove for the case where and are sequences of random 1-dependent variables. Then, two subsequences

and are of random independent variables. Wedenote

Consider two sequences of random variables

and

We have

,

Thus,

,

Hence,

4

. (2.2)

Let , it is easily seen that are

martingale. In fact,

(Note that )

Therefore, are martingale (thus, are submartingale as well).

Next we evaluate the following probability .

We have

Using the maximal algorithm for martingale (see [13], page 492)yields

Thus, we obtain

(2.3) Similarly,

(2.4)

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By (2.2), (2.3) and (2.4) we obtain

.

Hence,

.

Similarly we can prove the case where and are m-dependent. In fact, consider (m+1) sequences of particular sums

.......................................................

(Note that in each particular sum its elements areindependent random variables).Similar to the above 1– dependent case, we have

Thus,

,

Therefore,

, (2.5)

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We next evaluate probabilities

Which leads to the similar result to the 1 – dependent case

(2.6)

Similarly we have

(2.7)

From (2.5), (2.6) and (2.7) it leads to

Hence,

In the next sections we consider risk models where the effects ofinterest are taken into account. 3. Risk models having effects of interest

In this section we consider model (1.3) assuming that the sequencesof premiums and of claim sizes are sequences of non-negative random, identically distributed, with limited expectation, m –dependent and independent of each other variables; In each period, theseamounts of money are influenced by the interests, which is the sequence

of non-negative, identically distributed, with limited

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expectation and independent of and . Then, theproperty of the insurance company at period n is defined as follows

or

,

Hence,

;

Therefore, if we denote

,

then risk probabilities are defined as

The main result of this section is captured in the following theorem. Theorem 3.1

In risk model (1.3), if the sequences of premiums and of claim sizes aresequences of non-negative random, identically distributed, with limited expectation, m –dependent and independent of each other variables. is the sequence of non-negative,

identically distributed, with limited expectation and independent of and .

Suppose there exists satisfying the conditions

Then the risk probabilities are defined as follows

, where

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In order to prove this theorem we need first to prove the followinglemmas. Lemma 3.1(see [5] page 201)

For the probability space let , be mutually

independent random vectors and be a Borel’s function over where . If

and

Then g is a Borel function over with . In other words, under the

assumptions, we can compute as if U was a constant vector.

Lemma 3.2

If premiums and claim sizes are sequences of non-negative random,identically distributed, with limited expectation, m – dependent and independent of each othervariables. is the sequence of non-negative, identically distributed, with limited

expectation and independent of and . Suppose that there exists R > 0satisfying the condition

Let then is a supermartingale

and

, where .

Proof.

To begin with, we prove that is supermartingale, i.e.

a.s.

In fact,

Since

Thus,

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We prove that

by using lemma 3.1 for the case

Note that in this case U, V are mutually independent.Let

Consider the function

Let

,

By using Jensen’s inequality we have

Thus,

According to the properties of conditional expectation we obtain

Therefore, is a supermartingale (non-negative).

Next, it is easily seen that

10

.

(Because the properties of the non-negativity supermartingale and theassumption that )

In short, lemma 3.2 is proven. Next we shall prove theorem 3.1.

Proof of theorem 3.1.

Without loss of generality suppose that and aresequences of 1-dependent random variables. Let’s denote

Consider two consequences

and

11

where

We prove that and are supermartingales.

(Note that in each particular sum its elements areindependent random variables) In fact,

…

Since

We have

Now consider

From lemma 3.2 we have

Therefore,

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In short, is supermartingale.

Similarly, is also supermartingale.

The next part of the proof is similar to the proof in theorem 2.1. Observing that

,

we obtain

,

Therefore,

(3.1)

Next we evaluate the probability .

We have

(Since is a non-negative supermartingale and since

)

It implies that

(3.2)Similarly we have

(3.3)

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From (3.1), (3.2) and (3.3) follows

.

In short,

.

By application of the same proof technique for the case andare sequences of m-dependent random variables, we obtain the

final result

Corollary

If interests , and premiums and claim sizes are sequences of non-negative, identically distributed, with limited expectations, m-dependent andmutually independent random variables and there exists R > 0 satisfying

then

, where

Proof

This directly follows from theorem 2.

References[1] . Asmussen, S. (2000). Ruin probabilities. World Scientific, Singapore.

[2] . Buhlman, H. (1970). Mathematical Methods in Risk Theory. Berlin -Heidelberg - New York: Springer.

[3] . Cai, J. (2002). Discrete time risk models under rates of interest.Probability in the Engineering and Informational Sciences. 16, 309 - 324.[4] . Cai, J. (2002). Ruin Probabilities with Dependent Rates ofInterest. J. Appl. Probab. 39, n0 .2, 312 - 323.

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[5] . Chow, Y. S. and Teicher, H . (1978). Probability Theory. Berlin -Heidelberg - New York: Springer – Verlag.

[6] . Dam, B.K. (2005) . Renewal sequenses of Dependent randomvariables. Vietnam J. Math. 1, 73 - 83.[7] . Muller, A. and Pfug, G. (2001). Asymptotic ruin probabilities forrisk processes with dependent increments. Insurance: Mathematics and Economics.728, 1 – 12.[8] . Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J.L. (1999).Stochastic Processes for Insurance and Finance. John Wiley, Chichester.

[9] . Ross, S.(2000). Introduction to Probability models (SeventhEdition). Academic Press.

[10] . Sundt, B. and Teugels, J.L. (1995). Ruin estimates under interestforce. Insurance: Mathematics and Economics. 16, 7 – 22.[11] . Sundt, B. and Teugels, J.L. (1997). The adjustment function inruin estimates under interest force. Insurance: Mathematics and Economics. 19,85 – 94.[12] . Schmidt, K.D. (1995). Lectures on Risk Theory. Technische UniversitatDresden

[13] . Shiryaev, A. N. (1996). Probability, Second Edition, Springer-Verlag.

[14] . Yang, H. (1999). Non-exponetial bounds for ruin probability withinterest effect included. Scandinavian Actuarial Journal. 1, 66 – 79.

Bui Khoi DamEmail: bkdam-fam@mail.hut.edu.vn ; bkdamvn@yahoo.comNguyen Huy HoangEmail: hoangtoancb@neu.edu.vn ; hoangtoancb@gmail.com;Mobile: 090 414 4835

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