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Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims



Phung Duy Quang
 
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ABSTRACT

The aim of this study is to give upper bounds for ruin probabilities of generalized risk processes under interest force with homogenous Markov chain claims. Generalized Lundberg inequalities for ruin probabilities of these processes are derived by the martingale approach.

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  How to cite this article:

Phung Duy Quang , 2014. Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims. Asian Journal of Mathematics & Statistics, 7: 1-11.

DOI: 10.3923/ajms.2014.1.11

URL: https://scialert.net/abstract/?doi=ajms.2014.1.11
 
Received: December 01, 2013; Accepted: January 04, 2014; Published: March 29, 2014



INTRODUCTION

Modern insurance businesses allow insurers to invest their wealth into financial assets. Since a large part of the surplus of insurance businesses comes from investment income, actuaries have been studying ruin problems under risk models with interest force. For example, Sundt and Teugels (1995, 1997) studied the effects of constant rate on the ruin probability under the compound Poisson risk model. Yang (1999) 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 (2002a, b) 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 (2004) 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 study, we study the models considered by Cai and Dickson (2004) to the case homogenous markov chain claims, independent rates of interest and independent premiums. The main difference between the model in our study and the one in Cai and Dickson (2004) is that claims in our model are assumed to follow homogeneous Markov chains. Generalized Lundberg inequalities for ruin probabilities of these processes are derived by the martingale approach.

In this study, we study two style of premium collections. On one hand of the premiums are collected at the beging of each period then the surplus process {Un(1)}n≥1 with initial surplus u can be written as:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(1)

which can be rearranged as:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(2)

On the other hand, if the premiums are collected at the end of each period, then the surplus process {Un(2)}n≥1 with initial surplus u can be written as:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(3)

which is equivalent to:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(4)

where, throughout this study, we denote:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

and:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

if a>b.

We assume that:

Assumption 1: Uo(1) = Uo(2) = u>0
Assumption 2: X = {Xn}n≥0 is sequence of independent and identically distributed non-negative continuous random variables with the same distribution function F(x) = P(X0≤x)
Assumption 3: {In}n≥0 is sequence of independent and identically distributed non-negative continuous random variables with the same distribution function G(t) = P(I0≤t)
Assumption 4: {Yn}n≥0 is a homogeneous Markov chain such that for any n, Yn takes values in a countable set of non-negative numbers E = {y1, y2,…, yn,…} with Y0 = yi∈E and:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Where:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Assumption 5:
X, Y and I are assumed to be independent

We define the finite time and ultimate ruin probabilities in model (1) with assumption 1 to assumption 5, respectively, by:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(5)

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(6)

Similarly, we define the finite time and ultimate ruin probabilities in model (3) with assumption 1 to assumption 5, respectively, by:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(7)

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(8)

In this study, we derive probability inequalities for ψ(1)(u, yi) and ψ(2)(u, yi) by the martingale approach.

UPPER BOUNDS FOR PROBABILITY BY THE MARTINGALE APPROACH

To establish probability inequalities for ruin probabilities of model (1), we first proof the following Lemma.

Lemma 1: Let model (1) satisfy assumptions 1 to 5.

Any yi∈E, if:

M = max{yi; yi∈E}<+∝

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(9)

then, there exists a unique positive constant Ri satisfying:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(10)

Proof: Define:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

We have:

fi(t) = hi(t)-1

Where:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

With:

f(x) = F'(x), g(y) = G'(y)

With:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Then:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

This implies that hi(t) has n-th derivative function on (0, Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims) with Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims. Thus, fi(t) has n-th derivative function on (0, Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims) with Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims and:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Which implies that:

fi(t) is a convex function with fi(0) = 0
(11)

and:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(12)

By P((Y1-X1)(1+I1)-1>0|Y0 = yi)>0, we can find some constant δ>0 such that:

P((Y1-X1)(1+I1)-1>δ>0|Y0 = yi)>0

Then, we can get that:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

This implies that:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(13)

From (11-13) there exists a unique positive constant Ri satisfying (10).

This completes the proof .

Let:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Use Lemma 1, we now obtain a probability inequality for ψ(1)(u, yi) by the martingale approach.

Theorem 1: If model (1) satisfies assumptions 1 to 5, M = max{yi:yi∈E}<+∝ and (9) then for any u>0 and yi∈E:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(14)

Proof: Consider the process {Un(1)} given by (2), we let:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(15)

and Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims. Thus, we have:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

With any nImage for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims1:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

From:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

and Jensen’s inequality implies:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

In addition:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Thus, we have:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Hence, {Sn(1), n = 1, 2…} is a supermartingale with respect to the σ-filtration:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Define Ti(1) = min {n: Vn(1)<0|UO(1) = u, Y0 = yi}, with Vn(1) is given by (15). Hence, Ti(1) is a stopping time and n∧Ti(1) = min(n, Ti(1)) is a finite stopping time.

Therefore, from the optional stopping theorem for supermartingales, we have:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

This implies that:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(16)

From Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims then (16) becomes:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(17)

In addition:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(18)

Combining (17) and (18) imply that:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(19)

Thus, (14) follows by letting Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims in (19).

Similarly, we have Lemma 2.

Lemma 2: Assume that model (3) satisfies assumptions 1 to 5 and E(X1k)<+∞(k = 1, 2).

Any yi∈E, if:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

and:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(20)

Then, there exists a unique positive constant Ri satisfying:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(21)

Proof: Define:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

We have:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

From Y1 is discrete random variables and it takes values in E = {y1, y2,…, yn,…} then:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

with g(y) = G'(y).

We have:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

and:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

This implies that gi(t) has n-th derivative function on (0, +∞) (any n∈N* = N\{0}).

In addition:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

with f(x) = F’(x) satisfying:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

and:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

This implies that h(t) has n-th derivative function on (0, +∞) with n = 1, 2. Thus, fi(t) has n-th derivative function on (0, +∞) with n = 1, 2 and:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Which implies that:

fi(t) is a convex function with fi(0) = 0
(22)

and:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(23)

By P((Y1(1+I1)-1-X1)>0|Y0 = yi)>0, we can find some constant δ>0 such that:

P((Y1(1+I1)-1-X1)>δ>0|Y0 = yi)>0

Then, we can get that:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Imply:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(24)

From (22-24) there exists a unique positive constant Ri satisfying (21).

This completes the proof.

Let:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Use Lemma 2 we now obtain a probability inequality for ψ(2)(u, yi) by the martingale approach.

Theorem 2: If model (3) satisfies assumptions 1 to 5, E(X1k)<+∞(k = 1, 2) and (20) then for any u>0 and yi∈E:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(25)

Proof: Consider the process {Un(2)} given by (4), we let:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(26)

and Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims. Thus, we have:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

with any n≥1:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

From:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

and Jensen’s inequality implies:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

In addition:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Thus, we have:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Hence, {Sn(2), n = 1, 2,…} is a supermartingale with respect to the σ-filtration:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

Define Ti(2) = min{n: Vn(2)<0|UO(2) = u, Y0 = yi}, with Vn(2) is given by (15). Hence, Ti(2) is a stopping time and n∧Ti(2) = min(n, Ti(2)) is a finite stopping time.

Therefore, from the optional stopping theorem for supermartingales, we have:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

This implies that:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(27)

From Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims then (27) becomes:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(28)

In addition:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(29)

Combining (28) and (29) imply that:

Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims
(30)

Thus, (30) follows by letting Image for - Upper Bounds for Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims in (25).

This completes the proof.

CONCLUSION

Our main results in this study, Theorem 1 and Theorem 2 give upper bounds for ψn(1)(u, yi) and ψn(2)(u, yi) by the martingale approach.

REFERENCES

1:  Cai, J., 2002. Discrete time risk models under rates of interest. Probability Eng. Inform. Sci., 16: 309-324.
CrossRef  |  

2:  Cai, J., 2002. Ruin probabilities with dependent rates of interest. J. Applied Probability, 39: 312-323.
Direct Link  |  

3:  Cai, J. and D.C.M. Dickson, 2004. Ruin probabilities with a Markov chain interest model. Insurance, 35: 513-525.
CrossRef  |  

4:  Sundt, B. and J.L. Teugels, 1995. Ruin estimates under interest force. Insurance, 16: 7-22.
CrossRef  |  

5:  Sundt, B. and J.L. Teugels, 1997. The adjustment function in ruin estimates under interest force. Insurance, 19: 85-94.
CrossRef  |  

6:  Yang, H., 1999. Non-exponential bounds for ruin probability with interest effect included. Scand. Actuarial J., 2: 66-79.
CrossRef  |  

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