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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
Department of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China |
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 and 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
M. Denuit, J. Dhaene, M. J. Goovaerts and R. Kaas,
Actuarial Theory for Dependent Risks-Measures, Orders and Models, Wiley, New York, 2005. |
| [13] |
H. U. Gerber,
An extension of the renewal equation and its application in the collective theory of risk, Skandinavisk Aktuarietidskrift, (1970), 205-210.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
M. Denuit, J. Dhaene, M. J. Goovaerts and R. Kaas,
Actuarial Theory for Dependent Risks-Measures, Orders and Models, Wiley, New York, 2005. |
| [13] |
H. U. Gerber,
An extension of the renewal equation and its application in the collective theory of risk, Skandinavisk Aktuarietidskrift, (1970), 205-210.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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