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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

Wuyuan Jiang ,

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.

Keywords: Dependence, distribution of the maximum surplus, perturbed risk process, integro-differential equation, Laplace transform.
Mathematics Subject Classification: Primary: 62P05; Secondary: 91B30.
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. Albrecher, C. 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. Boudreault, H. Cossette, D. 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. Cheung, D. Landriault, G. 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. Cossette, E. 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. Cossette, E. 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. Albrecher, C. 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. Boudreault, H. Cossette, D. 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. Cheung, D. Landriault, G. 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. Cossette, E. 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. Cossette, E. 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 $0\leq u< 10$
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