Ruin Probabilities in Risk Based on a Generalized FGM Dependence Structure
Abstract: In this paper, we consider a discrete time insurance risk model, in which insurance and financial risks jointly follow a bivariate generalized FGM distribution. Assuming that every convex combination of the marginal distributions of insurance and financial risks belongs to strongly regular variation class, we derive some asymptotic equivalence formulas for these probabilities with both finite and infinite time horizons, all in the form of linear combinations of the tail probabilities of the insurance and financial risks.

1. 引言

${U}_{0}=x$ , ${U}_{n}=x\underset{i=1}{\overset{n}{\prod }}{B}_{i}+\underset{i=1}{\overset{n}{\sum }}\left({C}_{i}-{A}_{i}\right)\underset{j=1+i}{\overset{n}{\prod }}{B}_{j}.$

${\stackrel{⌢}{U}}_{n}={U}_{n}\underset{j=1}{\overset{n}{\prod }}{Y}_{j}=\left(x\underset{i=1}{\overset{n}{\prod }}{B}_{i}-\underset{i=1}{\overset{n}{\sum }}{X}_{i}\underset{j=1+i}{\overset{n}{\prod }}{B}_{j}\right)\underset{j=1}{\overset{n}{\prod }}{Y}_{j}=x-\underset{i=1}{\overset{n}{\sum }}{X}_{i}\underset{j=1}{\overset{i}{\prod }}{Y}_{j}.$ (1.1)

$\psi \left(x,n\right)=\mathrm{Pr}\left(x-{S}_{n}<0\right)=\mathrm{Pr}\left({S}_{n}>x\right)$ (1.2)

${M}_{n}=\underset{1\le m\le n}{\mathrm{max}}{S}_{m}>x$ ，则有限时间n的破产概率表示为：

$\psi \left(x,n\right)=\mathrm{Pr}\left(\underset{1\le m\le n}{\mathrm{inf}}\left(x-{S}_{m}\right)<0\right)=\mathrm{Pr}\left(\underset{1\le m\le n}{\mathrm{max}}{S}_{m}>x\right)=\mathrm{Pr}\left({M}_{n}>x\right).$ (1.3)

$\psi \left(x\right)=\underset{n\to \infty }{\mathrm{lim}}\psi \left(x,n\right)=\underset{n\to \infty }{\mathrm{lim}}\mathrm{Pr}\left(x-{S}_{n}<0\right)=\mathrm{Pr}\left({S}_{\infty }>x\right)$ (1.4)

$\psi \left(x\right)=\underset{n\to \infty }{\mathrm{lim}}\psi \left(x,n\right)=\underset{n\to \infty }{\mathrm{lim}}\mathrm{Pr}\left(\underset{1\le m\le n}{\mathrm{max}}{S}_{m}>x\right)=\mathrm{Pr}\left({M}_{\infty }>x\right).$ (1.5)

2.准备工作和引理

2.1. 重尾分布族

2.2. 广义FGM分布

$\Pi \left(x,y\right)=F\left(x\right)G\left(y\right)\left[1+\theta A\left(F\left(x\right)\right)B\left(G\left(y\right)\right)\right]$ (2.1)

$C\left(u,v\right)=uv\left[1+\theta A\left(u\right)B\left(v\right)\right]$ (2.2)

2.3. 若干引理

$\stackrel{¯}{{F}_{1}\ast \cdots \ast {F}_{n}}\in \mathcal{S}\left(\alpha \right)~\underset{i=1}{\overset{n}{\sum }}\left(\underset{j=1,j\ne i}{\overset{n}{\prod }}{\stackrel{^}{F}}_{j}\left(\alpha \right)\right){\stackrel{¯}{F}}_{i}\left(x\right).$ (2.3)

$\mathrm{Pr}\left(\underset{i=1}{\overset{n}{\prod }}{X}_{i}>x\right)~\underset{i=1}{\overset{n}{\sum }}\left(\underset{j=1,j\ne i}{\overset{n}{\prod }}E{X}_{j}^{\alpha }\right){\stackrel{¯}{F}}_{i}\left(x\right)$ (2.4)

$\mathrm{Pr}\left({X}_{1}{X}_{2}>x\right)~E{X}_{1}^{\alpha }{\stackrel{¯}{F}}_{2}\left(x\right)+E{X}_{2}^{\alpha }{\stackrel{¯}{F}}_{1}\left(x\right).$

3. 主要结论及其证明

$bF+\left(1-b\right)G\in {\mathcal{R}}_{-\alpha }^{*},$

1) $F\in {\mathcal{R}}_{-\alpha }^{*}$$\stackrel{¯}{G}\left(x\right)=o\left(\stackrel{¯}{F}\left(x\right)\right)$ ;

2) $F\in {\mathcal{R}}_{-\alpha }^{*}$$G\in {\mathcal{R}}_{-\alpha }$$\stackrel{¯}{G}\left(x\right)=O\left(\stackrel{¯}{F}\left(x\right)\right)$ ;

3) $F\in {\mathcal{R}}_{-\alpha }^{*}$$G\in {\mathcal{R}}_{-\alpha }^{*}$ 且函数 $a\left(x\right)=\frac{\stackrel{¯}{F}\left({e}^{x}\right)}{\stackrel{¯}{G}\left({e}^{x}\right)},$ 对任意的 $y>0,$$a\left(xy\right)\asymp a\left(x\right)$

${F}_{A}\left(x\right)\triangleq F\left(x\right)\left[1-A\left(F\left(x\right)\right)\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}{G}_{B}\left(y\right)\triangleq G\left(y\right)\left[1-B\left(G\left(y\right)\right)\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}x,y\ge 0$ (3.1)

$\stackrel{¯}{H}\left(x\right)~\left[\left(1+{\eta }_{A}\theta \right)E{Y}^{\alpha }-{\eta }_{A}\theta E{Y}_{B}^{\alpha }\right]\stackrel{¯}{F}\left(x\right)+\left[\left(1+{\eta }_{B}\theta \right)E{X}^{\alpha }-{\eta }_{B}\theta E{X}_{A}^{\alpha }\right]\stackrel{¯}{G}\left(x\right).$ (3.2)

$A\left(F\left(x\right)\right)=1-\frac{{F}_{A}\left(x\right)}{F\left(x\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}B\left(G\left(y\right)\right)=1-\frac{{G}_{B}\left(y\right)}{G\left(y\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}x,y\ge 0$

$\Pi \left(\text{d}s,\text{d}t\right)=\left(1+\theta \right)F\left(\text{d}s\right)G\left(\text{d}t\right)-\theta {F}_{A}\left(\text{d}s\right)G\left(\text{d}t\right)-\theta F\left(\text{d}s\right){G}_{B}\left(\text{d}t\right)+\theta {F}_{A}\left(\text{d}s\right){F}_{B}\left(\text{d}t\right)$ (3.3)

$\begin{array}{c}\stackrel{¯}{H}\left(x\right)=\mathrm{Pr}\left(XY>x\right)\\ =\int {\int }_{st>x,s>0,t>0}\prod \left(\text{d}s,\text{d}t\right)\\ =\left(1+\theta \right){\stackrel{¯}{H}}^{*}\left(x\right)-\theta \mathrm{Pr}\left({X}_{A}^{*}{Y}^{*}>x\right)-\theta \mathrm{Pr}\left({X}^{*}{Y}_{B}^{*}>x\right)+\theta \mathrm{Pr}\left({X}_{A}^{*}{Y}_{B}^{*}>x\right)\end{array}$ (3.4)

$\begin{array}{c}\stackrel{¯}{H}\left(x\right)~\left(1+\theta \right)\left[E{Y}^{\alpha }\stackrel{¯}{F}\left(x\right)+E{X}^{\alpha }\stackrel{¯}{G}\left(x\right)\right]-\theta \left[E{Y}^{\alpha }\mathrm{Pr}\left({X}_{A}^{*}>x\right)+E{X}_{A}^{\alpha }\stackrel{¯}{G}\left(x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\theta \left[E{Y}_{B}^{\alpha }\stackrel{¯}{F}\left(x\right)+E{X}^{\alpha }\mathrm{Pr}\left({Y}_{B}^{*}>x\right)\right]+\theta \left[E{Y}_{B}^{\alpha }\mathrm{Pr}\left({X}_{A}^{*}>x\right)+E{X}_{A}^{\alpha }\mathrm{Pr}\left({Y}_{B}^{*}>x\right)\right]\end{array}$ (3.5)

${\stackrel{¯}{F}}_{A}\left(x\right)~\left(1-{\eta }_{A}\right)\stackrel{¯}{F}\left(x\right)$ , ${\stackrel{¯}{G}}_{B}\left(x\right)~\left(1-{\eta }_{B}\right)\stackrel{¯}{G}\left(x\right)$ (3.6)

$\stackrel{¯}{H}\left(x\right)~\left[\left(1+{\eta }_{A}\theta \right)E{Y}^{\alpha }-{\eta }_{A}\theta E{Y}_{B}^{\alpha }\right]\stackrel{¯}{F}\left(x\right)+\left[\left(1+{\eta }_{B}\theta \right)E{X}^{\alpha }-{\eta }_{B}\theta E{X}_{A}^{\alpha }\right]\stackrel{¯}{G}\left( x \right)$

$\psi \left(x,n\right)=\mathrm{Pr}\left({M}_{n}>x\right)~{A}_{n}\stackrel{¯}{F}\left(x\right)+{B}_{n}\stackrel{¯}{G}\left(x\right)$ (3.7)

${B}_{n}=\underset{i=1}{\overset{n}{\sum }}{\left(E{Y}^{\alpha }\right)}^{i-1}\left[\left(1+{\eta }_{B}\theta \right)E{\left({M}_{n-i}+{X}_{n-i+1}\right)}_{+}^{\alpha }-{\eta }_{B}\theta E{\left({M}_{n-i}+{X}_{A,n-i+1}\right)}_{+}^{\alpha }\right].$

$n=1$ 时，显然， $\psi \left(x,1\right)=\mathrm{Pr}\left({M}_{1}>x\right)=\mathrm{Pr}\left({X}_{1,+}{Y}_{1}>x\right)=\stackrel{¯}{H}\left(x\right)$ ，即 $n=1$ 成立;

$\mathrm{Pr}\left({X}_{n}+{M}_{n-1}>x\right)~\left(1+{A}_{n-1}\right)\stackrel{¯}{F}\left(x\right)+{B}_{n-1}\stackrel{¯}{G}\left(x\right)$ (3.8)

$\begin{array}{c}\psi \left(x,n\right)=\mathrm{Pr}\left({M}_{n}>x\right)=\mathrm{Pr}\left({\left({X}_{n}+{M}_{n-1}\right)}_{+}{Y}_{n}>x\right)\\ =\left(1+\theta \right)\mathrm{Pr}\left({\left({X}_{n}^{*}+{M}_{n-1}\right)}_{+}{Y}^{*}>x\right)-\theta \mathrm{Pr}\left({\left({X}_{A,n}^{*}+{M}_{n-1}\right)}_{+}{Y}^{*}>x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\theta \mathrm{Pr}\left({\left({X}_{n}^{*}+{M}_{n-1}\right)}_{+}{Y}_{B}^{*}>x\right)+\theta \mathrm{Pr}\left({\left({X}_{A,n}^{*}+{M}_{n-1}\right)}_{+}{Y}_{B}^{*}>x\right)\end{array}$ (3.9)

$\begin{array}{c}\psi \left(x,n\right)~\left(1+\theta \right)\left[E{\left({X}_{n}+{M}_{n-1}\right)}_{+}^{\alpha }\stackrel{¯}{G}\left(x\right)+E{Y}^{\alpha }\mathrm{Pr}\left({\left({X}_{n}^{*}+{M}_{n-1}\right)}_{+}>x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\theta \left[E{\left({X}_{A,n}+{M}_{n-1}\right)}_{+}^{\alpha }\stackrel{¯}{G}\left(x\right)+E{Y}^{\alpha }\mathrm{Pr}\left({\left({X}_{A,n}^{*}+{M}_{n-1}\right)}_{+}>x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\theta \left[E{\left({X}_{n}+{M}_{n-1}\right)}_{+}^{\alpha }\mathrm{Pr}\left({Y}_{B}^{*}>x\right)+E{Y}_{B}^{\alpha }\mathrm{Pr}\left({\left({X}_{n}^{*}+{M}_{n-1}\right)}_{+}>x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\theta \left[E{\left({X}_{A,n}+{M}_{n-1}\right)}_{+}^{\alpha }\mathrm{Pr}\left({Y}_{B}^{*}>x\right)+E{Y}_{B}^{\alpha }\mathrm{Pr}\left({\left({X}_{A,n}^{*}+{M}_{n-1}\right)}_{+}>x\right)\right]\end{array}$

$\begin{array}{l}\psi \left(x,n\right)~\left[\left(1+\theta {\eta }_{A}\right)E{Y}^{\alpha }-\theta {\eta }_{A}E{Y}_{B}^{\alpha }+E{Y}^{\alpha }{A}_{n-1}\right]\stackrel{¯}{F}\left(x\right)\\ \text{}+\left[\left(1+\theta {\eta }_{B}\right)E{\left({X}_{n}+{M}_{n-1}\right)}_{+}^{\alpha }-\theta {\eta }_{B}E{\left({X}_{A,n}+{M}_{n-1}\right)}_{+}^{\alpha }+E{Y}^{\alpha }{B}_{n-1}\right]\stackrel{¯}{G}\left( x \right)\end{array}$

$\psi \left(x,n\right)=\mathrm{Pr}\left({M}_{n}>x\right)~{A}_{n}\stackrel{¯}{F}\left(x\right)+{B}_{n}\stackrel{¯}{G}\left(x\right)\text{.}$

$\psi \left(x,n\right)=\mathrm{Pr}\left({S}_{n}>x\right)~{A}_{n}\stackrel{¯}{F}\left(x\right)+{C}_{n}\stackrel{¯}{G}\left(x\right)$ (3.10)

${C}_{n}=\underset{i=1}{\overset{n}{\sum }}{\left(E{Y}^{\alpha }\right)}^{i-1}\left[\left(1+{\eta }_{B}\theta \right)E{\left({S}_{n-i}+{X}_{n-i+1}\right)}_{+}^{\alpha }-{\eta }_{B}\theta E{\left({S}_{n-i}+{X}_{A,n-i+1}\right)}_{+}^{\alpha }\right].$

$\psi \left(x,n\right)=\mathrm{Pr}\left({M}_{n}>x\right)~{{A}^{\prime }}_{n}\stackrel{¯}{F}\left(x\right)+{{B}^{\prime }}_{n}\stackrel{¯}{G}\left(x\right)$ (3.11)

$\psi \left(x,n\right)=\mathrm{Pr}\left({S}_{n}>x\right)~{{A}^{\prime }}_{n}\stackrel{¯}{F}\left(x\right)+{{C}^{\prime }}_{n}\stackrel{¯}{G}\left(x\right)$ (3.12)

${B}_{n}^{\text{'}}=\underset{i=1}{\overset{n}{\sum }}{\left(E{Y}^{\alpha }\right)}^{i-1}\left[\left(1-\theta \right)E{\left({M}_{n-i}+{X}_{n-i+1}\right)}_{+}^{\alpha }+\theta E{\left({M}_{n-i}+{X}_{A,n-i+1}\right)}_{+}^{\alpha }\right],$ ,

${C}_{n}^{\text{'}}=\underset{i=1}{\overset{n}{\sum }}{\left(E{Y}^{\alpha }\right)}^{i-1}\left[\left(1-\theta \right)E{\left({S}_{n-i}+{X}_{n-i+1}\right)}_{+}^{\alpha }+\theta E{\left({S}_{n-i}+{X}_{A,n-i+1}\right)}_{+}^{\alpha }\right].$

$\psi \left(x\right)=\mathrm{Pr}\left({M}_{\infty }>x\right)~{A}_{\infty }\stackrel{¯}{F}\left(x\right)\text{\hspace{0.17em}}+{B}_{\infty }\stackrel{¯}{G}\left(x\right)$ (3.13)

${B}_{\infty }=\frac{1}{1-E{Y}^{\alpha }}\left[\left(1+{\eta }_{B}\theta \right)E{\left({M}_{\infty }+X\right)}_{+}^{\alpha }-{\eta }_{B}\theta E{\left({M}_{\infty }+{X}_{A}\right)}_{+}^{\alpha }\right].$

$\psi \left(x\right)=\mathrm{Pr}\left({S}_{n}>x\right)~{A}_{\infty }\stackrel{¯}{F}\left(x\right)+{C}_{\infty }\stackrel{¯}{G}\left(x\right)$ (3.14)

${C}_{\infty }=\frac{1}{1-E{Y}^{\alpha }}\left[\left(1+{\eta }_{B}\theta \right)E{\left({S}_{\infty }+X\right)}_{+}^{\alpha }-{\eta }_{B}\theta E{\left({S}_{\infty }+{X}_{A}\right)}_{+}^{\alpha }\right].$

[1] Tang, Q. and Tsitsiashvili, G. (2003) Precise Estimates for the Ruin Probability in Finite Horizon in a Discrete-Time Model with Heavy-Tailed Insurance and Financial Risks. Stochastic Processes and Their Applications, 108, 299-325. http://dx.doi.org/10.1016/j.spa.2003.07.001

[2] Liu, F. (2013) Heavy-Tailed Distribution in the Presence of Dependence in Insurance and Finance. Doctoral Dissertation, University of Liverpool, Liverpool. http://repository.liv.ac.uk/14419/

[3] Vervaat, W. (1979) On a Stochastic Difference Equation and a Representation of Non-Negative Infinitely Divisible Random Variables. Advances in Applied Probability, 11, 750-783. http://www.jstor.org/stable/1426858

[4] Tang, Q. and Tsitsiashvili, G. (2004) Finite- and Infinite-Time Ruin Probabilities in the Presence of Stochastic Returns on Investments. Advances in Applied Probability, 36, 1278-1299. http://dx.doi.org/10.1239/aap/1103662967

[5] Chen, Y.Q. and Xie, X.S. (2005) The Finite Time Ruin Probability with the Same Heavy-Tailed Insurance and Financial Risks. Acta Mathematicae Applicatae Sinica, 21, 153-156. http://link.springer.com/article/10.1007%2Fs10255-005-0226-y

[6] Zhang, Y., Shen, X. and Weng, C. (2009) Approximation of the Tail Probability of Randomly Weighted Sums and Applications. Stochastic Processes and Their Applications, 119, 655-675. http://dx.doi.org/10.1016/j.spa.2008.03.004

[7] Chen, Y. (2011) The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks. Journal of Applied Probability, 48, 1035-1048. http://dx.doi.org/10.1239/jap/1324046017

[8] Chen, Y., Liu, J. and Liu, F. (2015) Ruin with Insurance and Financial Risks Following the Least Risky FGM Dependence Structure. Insurance: Mathematics and Economics, 62, 98-106. http://dx.doi.org/10.1016/j.insmatheco.2015.03.007

[9] Yi, L., Chen, Y. and Su, C. (2011) Approximation of the Tail Probability of Randomly Weighted Sums of Dependent Random Variables with Dominated Variation. Journal of Mathematical Analysis and Applications, 376, 365-372. http://dx.doi.org/10.1016/j.jmaa.2010.10.020

[10] Jiang, J. and Tang, Q. (2011) The Product of Two Dependent Random Variables with Regularly Varying or Rapidly Varying Tails. Statistics & Probability Letters, 81, 957-961. http://dx.doi.org/10.1016/j.spl.2011.01.015

[11] Li, J. and Tang, Q. (2015) Interplay of Insurance and Financial Risks in a Discrete-Time Model with Strongly Regular Variation. Bernoulli, 21, 1800-1823. http://arxiv.org/pdf/1507.07673.pdf

[12] 杨洋, 王开永. 保险风险管理中的破产渐近分析[M]. 北京: 科学出版社, 2013.

[13] Embrechts, P., Klüppelberg, C. and Mikosch, T. (2013) Modelling Extremal Events: For Insurance and Finance. Vol. 33, Springer Science & Business Media, Berlin.

[14] Nelsen, R.B. (2007) An Introduction to Copulas. Springer Science & Business Media, Berlin.

Cite this paper: Wen, L. (2016) Ruin Probabilities in Risk Based on a Generalized FGM Dependence Structure. Open Access Library Journal, 3, 1-8. doi: 10.4236/oalib.1102680.
References

[1]   Tang, Q. and Tsitsiashvili, G. (2003) Precise Estimates for the Ruin Probability in Finite Horizon in a Discrete-Time Model with Heavy-Tailed Insurance and Financial Risks. Stochastic Processes and Their Applications, 108, 299-325.
http://dx.doi.org/10.1016/j.spa.2003.07.001

[2]   Liu, F. (2013) Heavy-Tailed Distribution in the Presence of Dependence in Insurance and Finance. Doctoral Dissertation, University of Liverpool, Liverpool.
http://repository.liv.ac.uk/14419/

[3]   Vervaat, W. (1979) On a Stochastic Difference Equation and a Representation of Non-Negative Infinitely Divisible Random Variables. Advances in Applied Probability, 11, 750-783.
http://www.jstor.org/stable/1426858

[4]   Tang, Q. and Tsitsiashvili, G. (2004) Finite- and Infinite-Time Ruin Probabilities in the Presence of Stochastic Returns on Investments. Advances in Applied Probability, 36, 1278-1299.
http://dx.doi.org/10.1239/aap/1103662967

[5]   Chen, Y.Q. and Xie, X.S. (2005) The Finite Time Ruin Probability with the Same Heavy-Tailed Insurance and Financial Risks. Acta Mathematicae Applicatae Sinica, 21, 153-156.

[6]   Zhang, Y., Shen, X. and Weng, C. (2009) Approximation of the Tail Probability of Randomly Weighted Sums and Applications. Stochastic Processes and Their Applications, 119, 655-675.
http://dx.doi.org/10.1016/j.spa.2008.03.004

[7]   Chen, Y. (2011) The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks. Journal of Applied Probability, 48, 1035-1048.
http://dx.doi.org/10.1239/jap/1324046017

[8]   Chen, Y., Liu, J. and Liu, F. (2015) Ruin with Insurance and Financial Risks Following the Least Risky FGM Dependence Structure. Insurance: Mathematics and Economics, 62, 98-106.
http://dx.doi.org/10.1016/j.insmatheco.2015.03.007

[9]   Yi, L., Chen, Y. and Su, C. (2011) Approximation of the Tail Probability of Randomly Weighted Sums of Dependent Random Variables with Dominated Variation. Journal of Mathematical Analysis and Applications, 376, 365-372.
http://dx.doi.org/10.1016/j.jmaa.2010.10.020

[10]   Jiang, J. and Tang, Q. (2011) The Product of Two Dependent Random Variables with Regularly Varying or Rapidly Varying Tails. Statistics & Probability Letters, 81, 957-961.
http://dx.doi.org/10.1016/j.spl.2011.01.015

[11]   Li, J. and Tang, Q. (2015) Interplay of Insurance and Financial Risks in a Discrete-Time Model with Strongly Regular Variation. Bernoulli, 21, 1800-1823.
http://arxiv.org/pdf/1507.07673.pdf

[12]   杨洋, 王开永. 保险风险管理中的破产渐近分析[M]. 北京: 科学出版社, 2013.

[13]   Embrechts, P., Klüppelberg, C. and Mikosch, T. (2013) Modelling Extremal Events: For Insurance and Finance. Vol. 33, Springer Science & Business Media, Berlin.

[14]   Nelsen, R.B. (2007) An Introduction to Copulas. Springer Science & Business Media, Berlin.

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