• Previous Article
    Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling
  • DCDS-B Home
  • This Issue
  • Next Article
    Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics
July  2019, 24(7): 3037-3050. doi: 10.3934/dcdsb.2018298

The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence

Department of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China

* Corresponding author: Wuyuan Jiang

Received  February 2018 Revised  June 2018 Published  October 2018

We consider a compound Poisson risk process perturbed by a Brownian motion through using a potential measure where the claim sizes depend on inter-claim times via the Farlie-Gumbel-Morgenstern copula. We derive an integro-differential equation with certain boundary conditions for the distribution of the maximum surplus before ruin. This distribution can be calculated through the probability that the surplus process attains a given level from the initial surplus without first falling below zero. The explicit expressions for this distribution are derived when the claim amounts are exponentially distributed.

Citation: Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298
References:
[1]

H. AlbrecherC. Constantinescu and S. Loisel, Explicit ruin formulas for models with dependence among risks, Insurance: Mathematics and Economics, 48 (2011), 265-270.  doi: 10.1016/j.insmatheco.2010.11.007.  Google Scholar

[2]

H. Albrecher and J. Teugels, Exponential behavior in the presence of dependence in risk theory, Journal of Applied Probability, 43 (2006), 257-273.  doi: 10.1239/jap/1143936258.  Google Scholar

[3]

S. Asmussen, Stationary distributions for fluid flow models with or without Brownian noise, Communications in Statististics-Stochastic Models, 11 (1995), 21-49.  doi: 10.1080/15326349508807330.  Google Scholar

[4]

A. N. Borodin and P. Salminen, Handbook of Brownian Motion-Facts and Formulae, 2nd edition, Birkhäuser-Verlag, New York, 2002. doi: 10.1007/978-3-0348-8163-0.  Google Scholar

[5]

M. BoudreaultH. CossetteD. Landriault and E. Marceau, On a risk model with dependence between interclaim arrivals and claim sizes, Scandinavian Actuarial Journal, 2006 (2006), 265-285.  doi: 10.1080/03461230600992266.  Google Scholar

[6]

E. C. K. Cheung, A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium, Insurance: Mathematics and Economics, 48 (2011), 384-397.  doi: 10.1016/j.insmatheco.2011.01.006.  Google Scholar

[7]

E. C. K. Cheung and D. Landriault, A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model, Insurance: Mathematics and Economics, 46 (2010), 127-134.  doi: 10.1016/j.insmatheco.2009.07.009.  Google Scholar

[8]

E. C. K. Cheung and D. Landriault, Perturbed MAP risk models with dividend barrier strategies, Journal of Applied Probability, 46 (2009), 521-541.  doi: 10.1239/jap/1245676104.  Google Scholar

[9]

E. C. K. CheungD. LandriaultG. E. Willmot and J. K. Woo, Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models, Insurance: Mathematics and Economics, 46 (2010), 117-126.  doi: 10.1016/j.insmatheco.2009.05.009.  Google Scholar

[10]

H. CossetteE. Marceau and F. Marri, On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula, Insurance: Mathematics and Economics, 43 (2008), 444-455.  doi: 10.1016/j.insmatheco.2008.08.009.  Google Scholar

[11]

H. CossetteE. Marceau and F. Marri, Analysis of ruin measures for the classical compound Poisson risk model with dependence, Scandinavian Actuarial Journal, 2010 (2010), 221-245.  doi: 10.1080/03461230903211992.  Google Scholar

[12]

M. Denuit, J. Dhaene, M. J. Goovaerts and R. Kaas, Actuarial Theory for Dependent Risks-Measures, Orders and Models, Wiley, New York, 2005. Google Scholar

[13]

H. U. Gerber, An extension of the renewal equation and its application in the collective theory of risk, Skandinavisk Aktuarietidskrift, (1970), 205-210.   Google Scholar

[14]

H. U. Gerber and B. Landry, On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance: Mathematics and Economics, 22 (1998), 263-276.  doi: 10.1016/S0167-6687(98)00014-6.  Google Scholar

[15]

W. Y. Jiang and Z. J. Yang, Dividend payments in a risk model perturbed by diffusion with multiple thresholds, Stochastic Analysis and Applications, 31 (2013), 1097-1113.  doi: 10.1080/07362994.2013.819784.  Google Scholar

[16]

W. Y. Jiang and Z. J. Yang, The maximum surplus before ruin for dependent risk models through Farlie-Gumbel-Morgenstern copula, Scandinavian Actuarial Journal, 2016 (2016), 385-397.  doi: 10.1080/03461238.2014.936972.  Google Scholar

[17]

S. Li, The distribution of the dividend payments in the compound Poisson risk models perturbed by diffusion, Scandinavian Actuarial Journal, 2006 (2006), 73-85.  doi: 10.1080/03461230600589237.  Google Scholar

[18]

S. Li, The time of recovery and the maximum severity of ruin in a Sparre Andersen model, North American Actuarial Journal, 12 (2008), 413-427.  doi: 10.1080/10920277.2008.10597533.  Google Scholar

[19]

S. Li and D. C. M. Dickson, The maximum surplus before ruin in an Erlang$(n)$ risk process and related problems, Insurance: Mathematics and Economics, 38 (2006), 529-539.  doi: 10.1016/j.insmatheco.2005.11.005.  Google Scholar

[20]

S. Li and J. Garrido, On ruin for Erlang(n) risk process, Insurance: Mathematics and Economics, 34 (2004), 391-408.  doi: 10.1016/j.insmatheco.2004.01.002.  Google Scholar

[21]

S. Li and Y. Lu, On the maximum severity of ruin in the compound Poisson model with a threshold dividend strategy, Scandinavian Actuarial Journal, 2010 (2010), 136-147.  doi: 10.1080/03461230902850162.  Google Scholar

[22]

E. O. Mihalyko and C. Mihalyko, Mathematical investigation of the Gerber-Shiu function in the case of dependent inter-claim time and claim size, Insurance: Mathematics and Economics, 48 (2011), 378-383.  doi: 10.1016/j.insmatheco.2011.01.005.  Google Scholar

[23]

C. C. L. Tsai and G. E. Willmot, A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 30 (2002), 51-66.  doi: 10.1016/S0167-6687(01)00096-8.  Google Scholar

[24]

Z. M. Zhang and H. Yang, Gerber-Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times, Journal of Computational and Applied Mathematics, 235 (2011), 1189-1204.  doi: 10.1016/j.cam.2010.08.003.  Google Scholar

show all references

References:
[1]

H. AlbrecherC. Constantinescu and S. Loisel, Explicit ruin formulas for models with dependence among risks, Insurance: Mathematics and Economics, 48 (2011), 265-270.  doi: 10.1016/j.insmatheco.2010.11.007.  Google Scholar

[2]

H. Albrecher and J. Teugels, Exponential behavior in the presence of dependence in risk theory, Journal of Applied Probability, 43 (2006), 257-273.  doi: 10.1239/jap/1143936258.  Google Scholar

[3]

S. Asmussen, Stationary distributions for fluid flow models with or without Brownian noise, Communications in Statististics-Stochastic Models, 11 (1995), 21-49.  doi: 10.1080/15326349508807330.  Google Scholar

[4]

A. N. Borodin and P. Salminen, Handbook of Brownian Motion-Facts and Formulae, 2nd edition, Birkhäuser-Verlag, New York, 2002. doi: 10.1007/978-3-0348-8163-0.  Google Scholar

[5]

M. BoudreaultH. CossetteD. Landriault and E. Marceau, On a risk model with dependence between interclaim arrivals and claim sizes, Scandinavian Actuarial Journal, 2006 (2006), 265-285.  doi: 10.1080/03461230600992266.  Google Scholar

[6]

E. C. K. Cheung, A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium, Insurance: Mathematics and Economics, 48 (2011), 384-397.  doi: 10.1016/j.insmatheco.2011.01.006.  Google Scholar

[7]

E. C. K. Cheung and D. Landriault, A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model, Insurance: Mathematics and Economics, 46 (2010), 127-134.  doi: 10.1016/j.insmatheco.2009.07.009.  Google Scholar

[8]

E. C. K. Cheung and D. Landriault, Perturbed MAP risk models with dividend barrier strategies, Journal of Applied Probability, 46 (2009), 521-541.  doi: 10.1239/jap/1245676104.  Google Scholar

[9]

E. C. K. CheungD. LandriaultG. E. Willmot and J. K. Woo, Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models, Insurance: Mathematics and Economics, 46 (2010), 117-126.  doi: 10.1016/j.insmatheco.2009.05.009.  Google Scholar

[10]

H. CossetteE. Marceau and F. Marri, On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula, Insurance: Mathematics and Economics, 43 (2008), 444-455.  doi: 10.1016/j.insmatheco.2008.08.009.  Google Scholar

[11]

H. CossetteE. Marceau and F. Marri, Analysis of ruin measures for the classical compound Poisson risk model with dependence, Scandinavian Actuarial Journal, 2010 (2010), 221-245.  doi: 10.1080/03461230903211992.  Google Scholar

[12]

M. Denuit, J. Dhaene, M. J. Goovaerts and R. Kaas, Actuarial Theory for Dependent Risks-Measures, Orders and Models, Wiley, New York, 2005. Google Scholar

[13]

H. U. Gerber, An extension of the renewal equation and its application in the collective theory of risk, Skandinavisk Aktuarietidskrift, (1970), 205-210.   Google Scholar

[14]

H. U. Gerber and B. Landry, On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance: Mathematics and Economics, 22 (1998), 263-276.  doi: 10.1016/S0167-6687(98)00014-6.  Google Scholar

[15]

W. Y. Jiang and Z. J. Yang, Dividend payments in a risk model perturbed by diffusion with multiple thresholds, Stochastic Analysis and Applications, 31 (2013), 1097-1113.  doi: 10.1080/07362994.2013.819784.  Google Scholar

[16]

W. Y. Jiang and Z. J. Yang, The maximum surplus before ruin for dependent risk models through Farlie-Gumbel-Morgenstern copula, Scandinavian Actuarial Journal, 2016 (2016), 385-397.  doi: 10.1080/03461238.2014.936972.  Google Scholar

[17]

S. Li, The distribution of the dividend payments in the compound Poisson risk models perturbed by diffusion, Scandinavian Actuarial Journal, 2006 (2006), 73-85.  doi: 10.1080/03461230600589237.  Google Scholar

[18]

S. Li, The time of recovery and the maximum severity of ruin in a Sparre Andersen model, North American Actuarial Journal, 12 (2008), 413-427.  doi: 10.1080/10920277.2008.10597533.  Google Scholar

[19]

S. Li and D. C. M. Dickson, The maximum surplus before ruin in an Erlang$(n)$ risk process and related problems, Insurance: Mathematics and Economics, 38 (2006), 529-539.  doi: 10.1016/j.insmatheco.2005.11.005.  Google Scholar

[20]

S. Li and J. Garrido, On ruin for Erlang(n) risk process, Insurance: Mathematics and Economics, 34 (2004), 391-408.  doi: 10.1016/j.insmatheco.2004.01.002.  Google Scholar

[21]

S. Li and Y. Lu, On the maximum severity of ruin in the compound Poisson model with a threshold dividend strategy, Scandinavian Actuarial Journal, 2010 (2010), 136-147.  doi: 10.1080/03461230902850162.  Google Scholar

[22]

E. O. Mihalyko and C. Mihalyko, Mathematical investigation of the Gerber-Shiu function in the case of dependent inter-claim time and claim size, Insurance: Mathematics and Economics, 48 (2011), 378-383.  doi: 10.1016/j.insmatheco.2011.01.005.  Google Scholar

[23]

C. C. L. Tsai and G. E. Willmot, A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 30 (2002), 51-66.  doi: 10.1016/S0167-6687(01)00096-8.  Google Scholar

[24]

Z. M. Zhang and H. Yang, Gerber-Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times, Journal of Computational and Applied Mathematics, 235 (2011), 1189-1204.  doi: 10.1016/j.cam.2010.08.003.  Google Scholar

Figure 1.  $\mathcal{G}(u, 10)$ for different dependent parameters when $0\leq u< 10$
[1]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[2]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[3]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[4]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133

[5]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[6]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[7]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[8]

Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326

[9]

Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391

[10]

Kalikinkar Mandal, Guang Gong. On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020125

[11]

Andrea Braides, Antonio Tribuzio. Perturbed minimizing movements of families of functionals. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 373-393. doi: 10.3934/dcdss.2020324

[12]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065

[13]

Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040

[14]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[15]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[16]

Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109

[17]

Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100

[18]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117

[19]

Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083

[20]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (156)
  • HTML views (514)
  • Cited by (0)

Other articles
by authors

[Back to Top]