Integro-Differential Equations for a Jump-Diffusion Risk Process with Dependence between Claim Sizes and Claim Intervals

Show more

1. Introduction

In the risk process that is perturbed by diffusion, the surplus process of an insurance portfolio is given by

(1)

where is the initial surplus, is the positive constant premium income

rate, is the aggregate claims process, in which

is the claim number process (denoting the number of claims up to time t), and the interarrival times is a sequence of positive random variables. is a sequence of nonnegative independent identically distributed (i.i.d.) random variables with distribution function and density function, is a standard Brownian motion that is independent of the aggregate claims process, is a positive constant.

2. Improved Risk Model

In this paper, it is assumed that the claim occurrence process to be of the following type: If a claim is larger than a random variable, then the time until the next claim is exponentially distributed with rate, otherwise it is exponentially distributed with rate. The quantities are assumed to be i.i.d. random variables with distribution function. Assuming that

,

which is the net profit condition.

In the daily operation of insurance company, in addition to the premium income and claim to the operation of spending has a great influence on the outside, and there is also a factor that interest rates should not be neglected. As in [15] , this paper assume that the risk model Equation (1) is invested in a stochastic interest process which is assumed to be a geometric Brownian motion, where r and σ_{2} are positive constants, and is a standard Brownian motion independent of. Let denote the surplus of the insurer at time t under this investment assumption. Thus,

(2)

Denote T to be the ruin time (the first time that the surplus becomes negative), i.e.,

and if.

This article is interested in the expected discounted penalty (Gerber-Shiu) function:

, (3)

where is the indicator function, is the force of interest and is a nonnegative function of and satisfies.

Furthermore, let be the time when the first claim occurs, and random variable being exponentially distributed with rate. Assuming that

,.

For, define

, (4)

such that

, (5)

then,.

3. Integro-Differential Equations for

In this section, a system of integro-differential equations with initial value conditions satisfied by the Gerber-Shiu function is derived.

Define, and

(6)

Lemma 3.1 Let for. For define the hitting time. Then, for, it can be concluded that

(7)

Proof is a reflecting diffusion with generator

,

acting on functions satisfying the reflecting boundary condition.

If

and for t > 0, then, according to Itô’s formula is a local mar-

tingale. Using the separation variable technique, we find that

is a solution, where

,

is a solution of

.

Here

.

Using the initial condition for, we get, consequently

.

Applying the Optional Stopping Theorem, it follows that

,

and thus

.

This ends the proof of Lemma 3.1.

Similarly, the following lemma can also be obtained.

Lemma 3.2 Let for. For define the hitting time. Then, for, it can be concluded that

(8)

Theorem 3.1 Assuming that is second order continuously differentiable functions in u, then satisfies the following integro-differential equation

, (9)

, (10)

with the initial value conditions

,.

Proof Let be the time when the first claim occurs which exponentially distributed with rate. Consider the risk process defined by Equation (2) in an infinitesimal time interval. There are three possible cases in as follows.

1) There are no claims in with probability, thus;

2) There is exactly one claim in with probability. According to different of the claim amount, there are three possible cases in this case as follows.

a) The amount of the claim, i.e., ruin does not occur, and thus;

b) The amount of the claim, i.e., ruin occurs due to the claim;

c) The amount of the claim, i.e., ruin occurs due to oscillation (observe that the probability that this case occurs is zero).

3) There is more than one claim in with probability.

Thus, considering the three cases above and noting that is a strong Markov process, we have

(11)

By Taylor expansion, we have, thus Equation (11) becomes

(12)

Then, by Itô’s formula we have

. (13)

Therefore, by dividing t on both sides of Equation (12), letting, using Equation (13), we obtain Equation (9), and similarly we can obtain Equation (10).

The condition follows from the oscillating nature of the sample paths of. Now, we prove.

For all, let,. Then, by the strong property of, it can be concluded that

According to Lemma 3.1, it can be concluded that

,

Thus, , and correspondingly. Similar results can be derived for.

And for all, let, , according to Lemma 3.2 we obtain, thus.

This ends the proof of Theorem 3.1.

4. Differential Equations for

Let and in Equation (3), correspondingly the expected discounted penalty function turns into the ultimate ruin probability.

Obviously,

,

and

.

Suppose that

, ,

and is exponentially distributed with rate. Then, we get the following theorem.

Theorem 4.1 Assuming that is second order continuously differentiable functions in u, then satisfies the following integro-differential equation

(14)

(15)

with the initial value conditions

Proof According to Equation (9), it can be concluded that

By taking the derivative with respect to u on both sides of the above formula, and after some careful calculations, we obtain Equation (14). And similarly we can prove that Equation (15) holds. This ends the proof of Theorem 4.1.

5. Conclusion

In this paper, we consider a jump-diffusion risk process compounded by a geometric Brownian motion with dependence between claim sizes and claim intervals. We derive the integro-differential equations for the Gerber-Shiu functions and the ultimate ruin probability by using the martingale measure. Further studies are needed for the numerical solution of Equations (9), (10), (14) and (15). The results derived in this paper can be generalized to similar dependence ruin models.

Acknowledgements

This research was supported by the National Natural Science Foundation of China (No. 11601036), the Natural Science Foundation of Shandong (No. ZR2014GQ005) and the Natural Science Foundation of Binzhou University (No. 2016Y14).

References

[1] Gerber, H.U. (1970) An Extension of the Renewal Equation and Its Application in the Collective Theory of Risk. Skandinavisk Aktuarietidskrift, 1970, 205-210.

[2] Dufresne, F. and Gerber, H.U. (1991) Risk Theory for the Compound Poisson Process That Is Perturbed by Diffusion. Insurance: Mathematics and Economics, 10, 51-59.

https://doi.org/10.1016/0167-6687(91)90023-Q

[3] Gerber, H.U. and Landry, B. (1998) On the Discounted Penalty at Ruin in a Jump-Diffusion and the Perpetual Put Option. Insurance: Mathematics and Economics, 22, 263-276.

https://doi.org/10.1016/s0167-6687(98)00014-6

[4] Wang, G. and Wu, R. (2000) Some Distributions for the Classical Risk Process That Is Perturbed by Diffusion. Insurance: Mathematics and Economics, 26, 15-24.

https://doi.org/10.2307/253675

[5] Tsai, C.C.L. and Willmot, G.E. (2002) A Generalized Defective Renewal Equation for the Surplus Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 30, 51-66.

https://doi.org/10.1016/s0167-6687(01)00096-8

[6] Tsai, C.C.L. and Willmot, G.E. (2002) On the Moments of the Surplus Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 31, 327-350.

https://doi.org/10.1016/s0167-6687(02)00159-2

[7] Chiu, S.N. and Yin, C.C. (2003) The Time of Ruin, the Surplus Prior to Ruin and the Deficit at Ruin for the Classical Risk Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 33, 59-66.

https://doi.org/10.1016/s0167-6687(03)00143-4

[8] Boudreault, M., Cossette, H., Landriault, D. and Marceau, E. (2006) On a Risk Model with Dependence between Interclaim Arrivals and Claim Sizes. Scandinavian Actuarial Journal, 5, 265-285.

https://doi.org/10.1080/03461230600992266

[9] Bargès, M., Cossette, H., Loisel, S. and Marceau, E. (2011) On the Moments of the Aggregate Discounted Claims with Dependence Introduced by a FGM Copula. ASTIN Bulletin, 41, 215-238.

[10] Li, J. (2012) Asymptotics in a Time-Dependent Renewal Risk Model with Stochastic Return. Journal of Mathematical Analysis and Applications, 387, 1009-1023.

https://doi.org/10.1016/j.jmaa.2011.10.012

[11] Yong, W. and Xiang, H. (2012) Differential Equations for Ruin Probability in a Special Risk Model with FGM Copula for the Claim Size and the Inter-Claim Time. Journal of Inequalities and Applications, 2012, 156.

https://doi.org/10.1186/1029-242X-2012-156

[12] Fu, K.A. and Ng, C.Y.A. (2014) Asymptotics for the Ruin Probability of a Time-Dependent Renewal Risk Model with Geometric Lévy Process Investment Returns and Dominatedly-Varying-Tailed Claims. Insurance Mathematics & Economics, 56, 80-87.

https://doi.org/10.1016/j.insmatheco.2014.04.001

[13] Zhang, J. and Xiao, Q. (2015 ) Optimal Investment of a Time-Dependent Renewal Risk Model with Stochastic Return. Journal of Inequalities and Applications, 2015,181.

https://doi.org/10.1186/s13660-015-0707-3

[14] Zhao, X.H. and Yin, C.C. (2009) A Jump-Diffusion Risk Model with Dependence between Claim Sizes and Claim Intervals. Acta Mathematica Scientia, 29A, 1657-1664.

[15] Cai, J. and Xu, C. (2005) On the Decomposition of the Ruin Probability for a Jump-Diffusion Surplus Process Compounded by a Geometric Brownian Motion. North American Actuarial Journal, 10, 120-132.

https://doi.org/10.1080/10920277.2006.10596255