Ruin probabilities for a perturbed risk model with stochastic premiums and constant interest force
- Jianhua Cheng1,
- Yanwei Gao1Email author and
- Dehui Wang1
https://doi.org/10.1186/s13660-016-1135-8
© Cheng et al. 2016
Received: 20 January 2016
Accepted: 23 June 2016
Published: 9 September 2016
Abstract
In this paper, we consider a perturbed compound Poisson risk model with stochastic premiums and constant interest force. We obtain the upper bound and Lundberg-Cramér approximation for the infinite-time ruin probability, and consider the asymptotic formula for the finite-time ruin probability when the claim size is heavy-tailed. We show that the model in our paper has similar results to the classical risk process and some existing generalized models.
Keywords
MSC
1 Introduction
As an alternative, many papers assume that the premium income is no longer a linear function of time t. For example, Boikov [2] generalized the classical risk model to the case where the premium was modeled as another compound Poisson process, he derived the integral equations and exponential bounds for non-ruin probability. Melnikov [3] and Wang et al. [4] also focused on this kind of risk model. In addition, perturbed risk models have been discussed by many people since the pioneering work of Dufresne and Gerber [5]. See, for example, Furer and Schmidli [6], Schmidli [7] and the references therein.
Recently, more and more actuaries have been paying an increasing amounts of attention to the study of model with interest rate or investment return due to the practical importance. For example, Sundt and Teugels [8] and Cai and Dickson [9] studied the compound Poisson risk model with a constant interest rate force. Paulsen and Gjessing [10] and Kalashnikov and Norberg [11] considered the classical risk model with stochastic investment. For a perturbed risk process with investment, see Cai and Yang [12] and Zhu et al. [13]. Melnikov [3], Wang et al. [4] and Wei et al. [14] focused on risk models with stochastic premiums and when all capital of the insurer was invested in stock.
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\(\{N_{1}(t),t \geq0\}\) and \(\{N_{2}(t),t \geq0\}\) are Poisson processes with intensities \(\lambda_{1}\) and \(\lambda_{2}\), respectively;
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\(\{X_{i},i \geq1\}\) and \(\{Y_{i},i \geq1\}\) are two sequences of i.i.d random variables with the same distributions \(F(x)\) and \(G(y)\), respectively;
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\(\{X_{i},i \geq1\}\), \(\{Y_{i},i \geq1\}\), \(\{N_{1}(t),t \geq0\}\), \(\{N_{2}(t),t \geq0\}\) and \(\{B(t),t \geq0\}\) are mutually independent;
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the positive safety loading condition holds true, i.e.,$$ c+\lambda_{1}EX>\lambda_{2}EY. $$(1.3)
In the rest of this paper, we consider the upper bounds and the Lundberg-Cramér approximation for the infinite-time ruin probability, and we obtain the asymptotic formula for the finite-time ruin probability when the claim size is heavy-tailed. The results are shown in Section 2 and the proofs are given in Section 3.
2 Main results
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\(m_{1}(\eta)=E (e^{\eta X} )= \int_{0}^{\infty}e^{\eta y}\,\mathrm{d}F(x)\), \(m_{2}(\eta)=E (e^{\eta Y} )= \int_{0}^{\infty}e^{\eta y}\, \mathrm{d}G(y)\);
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\(\theta(z)= \frac{1}{2}\sigma^{2}z^{2}-cz+\lambda _{1}(m_{1}(-z)-1)+\lambda_{2}(m_{2}(z)-1)\);
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\(\eta_{0}=\sup\{\eta>0,m_{2}(\eta) < \infty\}\), \(\gamma=\sup \{\eta>0,\sup_{t>0} \int_{0}^{t} \theta(\eta e^{-rs})\,\mathrm{d}s<\infty \}\).
Theorem 2.1
Remark 2.1
From (1.3), by the convex property of \(\theta(z)\) and noting the fact that \(\theta'(0)=\lambda_{2}EY-c-\lambda_{1}EX<0\), \(\theta(0)=0\), and \(\theta(z)\rightarrow\infty\) as \(z\rightarrow \eta_{0}\), we know that there must exist a unique positive number \(z_{0}\) such that \(\theta(z_{0})=0\). Since \(\theta(z)<0\) for \(z< z_{0}\), then for any \(0<\eta\leq z_{0}\), \(\int_{0}^{t} \theta(\eta e^{-r s})\,\mathrm{d}s<0\). As a result, \(\sup_{t>0} \int_{0}^{t} \theta(\eta e^{-r s})\,\mathrm{d}s \leq0\) for any \(0<\eta\leq z_{0}\). Therefore, taking \(\eta=z_{0}\) in (2.1), we can get \(\psi(u) \leq e^{-z_{0}u}\). The right-hand side is exactly the upper bound for the risk model without investment (see Melnikov [3]), and \(z_{0}\) is the corresponding adjustment coefficient. The inequality shows that the ruin probability with interest is smaller than the one without interest.
Remark 2.2
In the result, γ has a complex and tedious expression. Actually, if assuming that \(\eta_{0}>0\), we have \(\gamma=\eta_{0}\).
Inspired by Remark 2.1, we can obtain a tighter bound for the ruin probability as follows.
Theorem 2.2
Remark 2.3
Note that \(\tilde{\theta}(z_{0})=\theta(z_{0})-ruz_{0}=-ruz_{0}<0\), and that the function \(\tilde{\theta}(z)\) is convex and converges to ∞ as \(z\rightarrow\eta_{0}\), it is easy to check that \(\tilde {\theta}(z)=0\) has a unique solution greater than \(z_{0}\). In addition, since \(\tilde{\theta}(\tilde{\eta}_{0}(u))=0\), and \(\tilde {\theta}(z)=\infty\) for all \(z >\eta_{0}\), it follows that \(\tilde{\eta}_{0}(u) \leq\eta_{0}=\gamma\). Thus, we find a suitable number \(\tilde{\eta}_{0}(u)\) so that we can get a best estimation for the upper bound of the ruin probability.
Example 2.1
Upper bounds of ruin probability for different η , u , and r
η | 0.1216 | 0.1299 | 0.1325 | 0.1381 | 0.1431 | |
---|---|---|---|---|---|---|
r = 0.08 | u = 10 | 0.2964 | 0.2844 | 0.2859 | 0.2964 | 0.3165 |
u = 20 | 0.0879 | 0.0776 | 0.0759 | 0.0745 | 0.0757 | |
r = 0.1 | u = 10 | 0.2964 | 0.2815 | 0.2807 | 0.2848 | 0.2929 |
u = 20 | 0.0879 | 0.0768 | 0.0729 | 0.0716 | 0.0707 |
In this example, \(\eta=0.1216\) solves the equation \(\theta (z)=0\), then the upper bounds for the model without investment could be obtained (see Remark 2.1). However, we can find tighter bounds since \(r>0\). For example, when \(r=0.08\) and \(u=10\), the result for \(\eta=0.1299\) is better than that for \(\eta=0.1216\), furthermore, better than those for other values of η in Table 1. Actually, it corresponds to the best estimate of the ruin probability in this case, i.e., \(\tilde{\eta}_{0}(u)=0.1299\). Similarly, 0.1325, 0.1381, and 0.1431 are best choices of η for the cases that \(r=0.1\) and \(u=10\), \(r=0.08\) and \(u=20\), \(r=0.1\) and \(u=20\), respectively.
For the Lundberg-Crámer approximation, we have the following theorem.
Theorem 2.3
Remark 2.4
The number γ here is the so-called adjustment coefficient or the Lundberg exponent. It follows that \(\lim_{u\rightarrow\infty}{\frac{-\ln\psi (u)}{u}}=\gamma\) from (2.3) and (2.4), showing asymptotic behavior of \(\psi(u)\).
When the claim size is heavy-tailed, i.e., \(\eta_{0}=0\), asymptotic formula for the ruin probability will be considered usually. We recall several important classes of heavy-tailed distributions first.
It is well known that \(\mathcal{R}_{-\alpha} \subset\mathcal{S} \subset\mathcal{L}\). For more details of heavy-tailed distributions and their applications, see Embrechts et al. [15].
Now, we consider asymptotic formula for the finite-time ruin probability in model (1.2). We write \(f(x) \sim g(x)\) if \(\lim_{x\rightarrow\infty}{\frac{f(x)}{g(x)}}=1\) throughout this paper.
Theorem 2.4
Furthermore, we have the following result.
Theorem 2.5
Remark 2.5
These theorems generalize the results for the models in Tang [16] where \(c=0\) and \(\sigma=0\), Jiang and Yan [17] where \(\lambda_{1}=0\), and extend the investigation for the model in Wei et al. [14] where \(\sigma=0\). Meanwhile, the conclusions are consistent with that of Veraverbeke [18], who pointed out that the diffusion term could be asymptotically negligible when the claims are subexponentially distributed.
3 Proofs of the main results
It is easy to see that \(\{(V(t),t),t \geq0\}\) is a Markov process, let \(\{\mathcal{F}_{t},t \geq0\}\) be the natural filtration of \(\{V(t),t \geq0\}\), i.e., \(\mathcal {F}_{t}=\sigma (V(s),0 \leq s \leq t)\), then T is an \(\mathcal{F}_{t}\)-stopping time. We can construct a martingale by Dynkin’s formula, which indicates the relationship between martingale and the infinitesimal generator of the Markov process, then we derive the upper bound for ruin probability via a martingale approach.
Lemma 3.1
Assume that \(\eta_{0}>0\), then for any \(0<\eta<\gamma\), \(M_{t}=\exp \{- \int_{0}^{t} \theta(\eta e^{-rs})\,\mathrm{d}s-\eta V(t) \}\) is an \(\mathcal{F}_{t}\)-martingale.
Proof
Proof of Theorem 2.1
Proof of Theorem 2.2
Proof of Theorem 2.3
Proof of Theorem 2.4
Proof of Theorem 2.5
Declarations
Acknowledgements
This work is supported by National Natural Science Foundation of China (Nos. 11271155, 11371168, J1310022, 11501241), Natural Science Foundation of Jilin Province (20130101066JC, 20150520053JH), and Science and Technology Research Program of Education Department in Jilin Province for the 12th Five-Year Plan (440020031139).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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