1 Introduction
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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)
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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\(\eta_{0}=\sup\{\eta>0,m_{2}(\eta) < \infty\}\), \(\gamma=\sup \{\eta>0,\sup_{t>0} \int_{0}^{t} \theta(\eta e^{-rs})\,\mathrm{d}s<\infty \}\).
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 |