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March  2022, 18(2): 795-823. doi: 10.3934/jimo.2020179

General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes

1. 

School of Mathematical Sciences, Xiamen University, Fujian 361005, China

2. 

Department of Statistics and Actuarial Science, Xi'an Jiaotong-Liverpool University, Suzhou, 215123, China

* Corresponding author: Ran Xu

Received  May 2020 Revised  August 2020 Published  March 2022 Early access  December 2020

For spectrally negative Lévy risk processes we consider a generalized version of the De Finetti's optimal dividend problem with fixed transaction costs, where the ruin time is replaced by a general drawdown time in the framework. We identify a condition under which a band–type impulse dividend strategy is optimal among all admissible impulse strategies. As a consequence, we are able to extend the previous results on ruin time based impulse dividend optimization problem to those on drawdown time based impulse dividend optimization problems. A new type of drawdown function is proposed at end, and various numerical examples are presented to illustrate the existence of those optimal impulse dividend strategies under different assumptions.

Citation: Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial and Management Optimization, 2022, 18 (2) : 795-823. doi: 10.3934/jimo.2020179
References:
[1]

S. AsmussenF. Avram and M. R. Pistorius, Russian and american put options under exponential phase-type lévy models, Stochastic Processes and their Applications, 109 (2004), 79-111.  doi: 10.1016/j.spa.2003.07.005.

[2]

S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0.

[3]

F. AvramA. Kyprianouy and M. Pistoriusz, Exit problems for spectrally negative l evy processes and applications to russian, american and canadized options, Ann. Appl. Probab, 14 (2004), 215-238.  doi: 10.1214/aoap/1075828052.

[4]

F. AvramZ. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative lévy process, The Annals of Applied Probability, 17 (2007), 156-180.  doi: 10.1214/105051606000000709.

[5]

F. AvramZ. Palmowski and M. R. Pistorius, On gerber–shiu functions and optimal dividend distribution for a lévy risk process in the presence of a penalty function, The Annals of Applied Probability, 25 (2015), 1868-1935.  doi: 10.1214/14-AAP1038.

[6]

F. AvramN. L. Vu and X. Zhou, On taxed spectrally negative lévy processes with draw-down stopping, Insurance: Mathematics and Economics, 76 (2017), 69-74.  doi: 10.1016/j.insmatheco.2017.06.005.

[7]

J. Azéma and M. Yor, Une solution simple au probleme de skorokhod, in Séminaire de Probabilités XIII, Springer, 721 (1979), 90–115.

[8]

E. J. BaurdouxZ. Palmowski and M. R. Pistorius, On future drawdowns of lévy processes, Stochastic Processes and their Applications, 127 (2017), 2679-2698.  doi: 10.1016/j.spa.2016.12.008.

[9]

E. BayraktarA. E. Kyprianou and K. Yamazaki, On optimal dividends in the dual model, ASTIN Bulletin: The Journal of the IAA, 43 (2013), 359-372.  doi: 10.1017/asb.2013.17.

[10]

J. Bertoin, Lévy Processes, vol. 121, Cambridge university press Cambridge, 1996.

[11]

G. Burghardt and R. Duncan, Deciphering drawdown, Risk management for investors, September, S16–S20.

[12]

P. Carr, First-order calculus and option pricing, Journal of Financial Engineering, 1 (2014), 1450009. doi: 10.1142/s2345768614500093.

[13]

B. De Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, in Transactions of the XVth international congress of Actuaries, vol. 2, New York, 1957,433–443.

[14]

D. C. Dickson and H. R. Waters, Some optimal dividends problems, ASTIN Bulletin: The Journal of the IAA, 34 (2004), 49-74.  doi: 10.1017/S0515036100013878.

[15]

B. HøJgaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.  doi: 10.1111/1467-9965.00066.

[16]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Elsevier, 2014.

[17]

J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-662-02514-7.

[18]

A. E. KyprianouR. Loeffen and J.-L. Pérez, Optimal control with absolutely continuous strategies for spectrally negative lévy processes, Journal of Applied Probability, 49 (2012), 150-166.  doi: 10.1239/jap/1331216839.

[19]

A. E. Kyprianou and Z. Palmowski, Distributional study of de finetti's dividend problem for a general lévy insurance risk process, Journal of Applied Probability, 44 (2007), 428-443.  doi: 10.1239/jap/1183667412.

[20]

D. LandriaultB. Li and S. Li, Analysis of a drawdown-based regime-switching lévy insurance model, Insurance: Mathematics and Economics, 60 (2015), 98-107.  doi: 10.1016/j.insmatheco.2014.11.005.

[21]

D. LandriaultB. Li and S. Li, Drawdown analysis for the renewal insurance risk process, Scandinavian Actuarial Journal, 2017 (2017), 267-285.  doi: 10.1080/03461238.2015.1123174.

[22]

J. P. Lehoczky, Formulas for stopped diffusion processes with stopping times based on the maximum, The Annals of Probability, 5 (1977), 601-607.  doi: 10.1214/aop/1176995770.

[23]

B. LiN. L. Vu and X. Zhou, Exit problems for general draw-down times of spectrally negative lévy processes, Journal of Applied Probability, 56 (2019), 441-457.  doi: 10.1017/jpr.2019.31.

[24]

R. L. Loeffen, On optimality of the barrier strategy in de finetti's dividend problem for spectrally negative lévy processes, The Annals of Applied Probability, 18 (2008), 1669-1680.  doi: 10.1214/07-AAP504.

[25]

R. L. Loeffen, An optimal dividends problem with a terminal value for spectrally negative lévy processes with a completely monotone jump density, Journal of Applied Probability, 46 (2009), 85-98.  doi: 10.1017/S0021900200005246.

[26]

R. L. Loeffen, An optimal dividends problem with transaction costs for spectrally negative lévy processes, Insurance: Mathematics and Economics, 45 (2009), 41-48.  doi: 10.1016/j.insmatheco.2009.03.002.

[27]

R. L. Loeffen and J.-F. Renaud, De finetti's optimal dividends problem with an affine penalty function at ruin, Insurance: Mathematics and Economics, 46 (2010), 98-108.  doi: 10.1016/j.insmatheco.2009.09.006.

[28]

D. B. Madan and M. Yor, Making markov martingales meet marginals: With explicit constructions, Bernoulli, 8 (2002), 509-536. 

[29]

X. PengM. Chen and J. Guo, Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.  doi: 10.1016/j.insmatheco.2012.08.004.

[30]

G. Peskir, Designing options given the risk: The optimal skorokhod-embedding problem, Stochastic Processes and Their Applications, 81 (1999), 25-38.  doi: 10.1016/S0304-4149(98)00097-0.

[31]

J.-F. Renaud and X. Zhou, Distribution of the present value of dividend payments in a lévy risk model, Journal of Applied Probability, 44 (2007), 420-427.  doi: 10.1239/jap/1183667411.

[32]

N. Scheer and H. Schmidli, Optimal dividend strategies in a cramer–lundberg model with capital injections and administration costs, European Actuarial Journal, 1 (2011), 57-92.  doi: 10.1007/s13385-011-0007-3.

[33]

F. Schuhmacher and M. Eling, Sufficient conditions for expected utility to imply drawdown-based performance rankings, Journal of Banking & Finance, 35 (2011), 2311-2318. 

[34]

L. Shepp and A. N. Shiryaev, The russian option: Reduced regret, The Annals of Applied Probability, 3 (1993), 631-640.  doi: 10.1214/aoap/1177005355.

[35]

H. M. Taylor, A stopped brownian motion formula, The Annals of Probability, 3 (1975), 234-246.  doi: 10.1214/aop/1176996395.

[36]

S. Thonhauser and H. Albrecher, Dividend maximization under consideration of the time value of ruin, Insurance: Mathematics and Economics, 41 (2007), 163-184.  doi: 10.1016/j.insmatheco.2006.10.013.

[37]

M. Vierkötter and H. Schmidli, On optimal dividends with exponential and linear penalty payments, Insurance: Mathematics and Economics, 72 (2017), 265-270.  doi: 10.1016/j.insmatheco.2016.12.001.

[38]

W. Wang, Y. Wang and X. Wu, Dividend and capital injection optimization with transaction cost for spectrally negative lévy risk processes, arXiv preprint, arXiv: 1807.11171.

[39]

W. Wang and Z. Zhang, Optimal loss-carry-forward taxation for lévy risk processes stopped at general draw-down time, Advances in Applied Probability, 51 (2019), 865-897.  doi: 10.1017/apr.2019.33.

[40]

W. Wang and X. Zhou, General drawdown-based de finetti optimization for spectrally negative lévy risk processes, Journal of Applied Probability, 55 (2018), 513-542.  doi: 10.1017/jpr.2018.33.

[41]

R. Xu and J.-K. Woo, Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments, Insurance: Mathematics and Economics, 92 (2020), 1-16.  doi: 10.1016/j.insmatheco.2020.02.008.

[42]

C. Yin and Y. Wen, Optimal dividend problem with a terminal value for spectrally positive levy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773.  doi: 10.1016/j.insmatheco.2013.09.019.

[43]

Y. ZhaoP. Chen and H. Yang, Optimal periodic dividend and capital injection problem for spectrally positive lévy processes, Insurance: Mathematics and Economics, 74 (2017), 135-146.  doi: 10.1016/j.insmatheco.2017.03.006.

show all references

References:
[1]

S. AsmussenF. Avram and M. R. Pistorius, Russian and american put options under exponential phase-type lévy models, Stochastic Processes and their Applications, 109 (2004), 79-111.  doi: 10.1016/j.spa.2003.07.005.

[2]

S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0.

[3]

F. AvramA. Kyprianouy and M. Pistoriusz, Exit problems for spectrally negative l evy processes and applications to russian, american and canadized options, Ann. Appl. Probab, 14 (2004), 215-238.  doi: 10.1214/aoap/1075828052.

[4]

F. AvramZ. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative lévy process, The Annals of Applied Probability, 17 (2007), 156-180.  doi: 10.1214/105051606000000709.

[5]

F. AvramZ. Palmowski and M. R. Pistorius, On gerber–shiu functions and optimal dividend distribution for a lévy risk process in the presence of a penalty function, The Annals of Applied Probability, 25 (2015), 1868-1935.  doi: 10.1214/14-AAP1038.

[6]

F. AvramN. L. Vu and X. Zhou, On taxed spectrally negative lévy processes with draw-down stopping, Insurance: Mathematics and Economics, 76 (2017), 69-74.  doi: 10.1016/j.insmatheco.2017.06.005.

[7]

J. Azéma and M. Yor, Une solution simple au probleme de skorokhod, in Séminaire de Probabilités XIII, Springer, 721 (1979), 90–115.

[8]

E. J. BaurdouxZ. Palmowski and M. R. Pistorius, On future drawdowns of lévy processes, Stochastic Processes and their Applications, 127 (2017), 2679-2698.  doi: 10.1016/j.spa.2016.12.008.

[9]

E. BayraktarA. E. Kyprianou and K. Yamazaki, On optimal dividends in the dual model, ASTIN Bulletin: The Journal of the IAA, 43 (2013), 359-372.  doi: 10.1017/asb.2013.17.

[10]

J. Bertoin, Lévy Processes, vol. 121, Cambridge university press Cambridge, 1996.

[11]

G. Burghardt and R. Duncan, Deciphering drawdown, Risk management for investors, September, S16–S20.

[12]

P. Carr, First-order calculus and option pricing, Journal of Financial Engineering, 1 (2014), 1450009. doi: 10.1142/s2345768614500093.

[13]

B. De Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, in Transactions of the XVth international congress of Actuaries, vol. 2, New York, 1957,433–443.

[14]

D. C. Dickson and H. R. Waters, Some optimal dividends problems, ASTIN Bulletin: The Journal of the IAA, 34 (2004), 49-74.  doi: 10.1017/S0515036100013878.

[15]

B. HøJgaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.  doi: 10.1111/1467-9965.00066.

[16]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Elsevier, 2014.

[17]

J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-662-02514-7.

[18]

A. E. KyprianouR. Loeffen and J.-L. Pérez, Optimal control with absolutely continuous strategies for spectrally negative lévy processes, Journal of Applied Probability, 49 (2012), 150-166.  doi: 10.1239/jap/1331216839.

[19]

A. E. Kyprianou and Z. Palmowski, Distributional study of de finetti's dividend problem for a general lévy insurance risk process, Journal of Applied Probability, 44 (2007), 428-443.  doi: 10.1239/jap/1183667412.

[20]

D. LandriaultB. Li and S. Li, Analysis of a drawdown-based regime-switching lévy insurance model, Insurance: Mathematics and Economics, 60 (2015), 98-107.  doi: 10.1016/j.insmatheco.2014.11.005.

[21]

D. LandriaultB. Li and S. Li, Drawdown analysis for the renewal insurance risk process, Scandinavian Actuarial Journal, 2017 (2017), 267-285.  doi: 10.1080/03461238.2015.1123174.

[22]

J. P. Lehoczky, Formulas for stopped diffusion processes with stopping times based on the maximum, The Annals of Probability, 5 (1977), 601-607.  doi: 10.1214/aop/1176995770.

[23]

B. LiN. L. Vu and X. Zhou, Exit problems for general draw-down times of spectrally negative lévy processes, Journal of Applied Probability, 56 (2019), 441-457.  doi: 10.1017/jpr.2019.31.

[24]

R. L. Loeffen, On optimality of the barrier strategy in de finetti's dividend problem for spectrally negative lévy processes, The Annals of Applied Probability, 18 (2008), 1669-1680.  doi: 10.1214/07-AAP504.

[25]

R. L. Loeffen, An optimal dividends problem with a terminal value for spectrally negative lévy processes with a completely monotone jump density, Journal of Applied Probability, 46 (2009), 85-98.  doi: 10.1017/S0021900200005246.

[26]

R. L. Loeffen, An optimal dividends problem with transaction costs for spectrally negative lévy processes, Insurance: Mathematics and Economics, 45 (2009), 41-48.  doi: 10.1016/j.insmatheco.2009.03.002.

[27]

R. L. Loeffen and J.-F. Renaud, De finetti's optimal dividends problem with an affine penalty function at ruin, Insurance: Mathematics and Economics, 46 (2010), 98-108.  doi: 10.1016/j.insmatheco.2009.09.006.

[28]

D. B. Madan and M. Yor, Making markov martingales meet marginals: With explicit constructions, Bernoulli, 8 (2002), 509-536. 

[29]

X. PengM. Chen and J. Guo, Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.  doi: 10.1016/j.insmatheco.2012.08.004.

[30]

G. Peskir, Designing options given the risk: The optimal skorokhod-embedding problem, Stochastic Processes and Their Applications, 81 (1999), 25-38.  doi: 10.1016/S0304-4149(98)00097-0.

[31]

J.-F. Renaud and X. Zhou, Distribution of the present value of dividend payments in a lévy risk model, Journal of Applied Probability, 44 (2007), 420-427.  doi: 10.1239/jap/1183667411.

[32]

N. Scheer and H. Schmidli, Optimal dividend strategies in a cramer–lundberg model with capital injections and administration costs, European Actuarial Journal, 1 (2011), 57-92.  doi: 10.1007/s13385-011-0007-3.

[33]

F. Schuhmacher and M. Eling, Sufficient conditions for expected utility to imply drawdown-based performance rankings, Journal of Banking & Finance, 35 (2011), 2311-2318. 

[34]

L. Shepp and A. N. Shiryaev, The russian option: Reduced regret, The Annals of Applied Probability, 3 (1993), 631-640.  doi: 10.1214/aoap/1177005355.

[35]

H. M. Taylor, A stopped brownian motion formula, The Annals of Probability, 3 (1975), 234-246.  doi: 10.1214/aop/1176996395.

[36]

S. Thonhauser and H. Albrecher, Dividend maximization under consideration of the time value of ruin, Insurance: Mathematics and Economics, 41 (2007), 163-184.  doi: 10.1016/j.insmatheco.2006.10.013.

[37]

M. Vierkötter and H. Schmidli, On optimal dividends with exponential and linear penalty payments, Insurance: Mathematics and Economics, 72 (2017), 265-270.  doi: 10.1016/j.insmatheco.2016.12.001.

[38]

W. Wang, Y. Wang and X. Wu, Dividend and capital injection optimization with transaction cost for spectrally negative lévy risk processes, arXiv preprint, arXiv: 1807.11171.

[39]

W. Wang and Z. Zhang, Optimal loss-carry-forward taxation for lévy risk processes stopped at general draw-down time, Advances in Applied Probability, 51 (2019), 865-897.  doi: 10.1017/apr.2019.33.

[40]

W. Wang and X. Zhou, General drawdown-based de finetti optimization for spectrally negative lévy risk processes, Journal of Applied Probability, 55 (2018), 513-542.  doi: 10.1017/jpr.2018.33.

[41]

R. Xu and J.-K. Woo, Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments, Insurance: Mathematics and Economics, 92 (2020), 1-16.  doi: 10.1016/j.insmatheco.2020.02.008.

[42]

C. Yin and Y. Wen, Optimal dividend problem with a terminal value for spectrally positive levy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773.  doi: 10.1016/j.insmatheco.2013.09.019.

[43]

Y. ZhaoP. Chen and H. Yang, Optimal periodic dividend and capital injection problem for spectrally positive lévy processes, Insurance: Mathematics and Economics, 74 (2017), 135-146.  doi: 10.1016/j.insmatheco.2017.03.006.

Figure 1.  Illustration of various drawdown times
Figure 2.  General drawdown times based on Eq.(43)
Figure 3.  Cramér-Lundberg model: $ \varsigma(z) $
Figure 4.  Cramér-Lundberg model: $ V^{z_2^*}_{z_1^*}(x) $
Figure 5.  Brownian Motion with drift: $ \varsigma(z) $
Figure 6.  Brownian Motion with drift: $ V^{z_2^*}_{z_1^*}(x) $
Figure 7.  Jump-diffusion process: $ \varsigma(z) $
Figure 8.  Jump-diffusion process: $ V^{z_2^*}_{z_1^*}(x) $
Table 1.  Cramér-Lundberg model
$ (z_1^*, z_2^*) $ $ k= \infty $ $ k=0.6 $ $ k=0.4 $ $ k=0.2 $
$ \alpha=2.0 $ (3.9689, 11.6898) (3.9689, 11.6902) (2.0000, 11.2002) (2.0000, 10.1503)
$ \alpha=1.5 $ (3.4688, 11.1898) (3.4688, 11.1893) (1.5000, 10.0581) (1.5330, 10.3051)
$ \alpha=1.2 $ (3.1688, 10.8898) (3.1680, 10.8917) (3.1688, 10.8898) (1.6154, 10.3557)
$ \alpha=1.0 $ (2.9688, 10.6898) (2.9687, 10.6901) (2.9688, 10.6894) (1.7244, 10.3693)
$ (z_1^*, z_2^*) $ $ k= \infty $ $ k=0.6 $ $ k=0.4 $ $ k=0.2 $
$ \alpha=2.0 $ (3.9689, 11.6898) (3.9689, 11.6902) (2.0000, 11.2002) (2.0000, 10.1503)
$ \alpha=1.5 $ (3.4688, 11.1898) (3.4688, 11.1893) (1.5000, 10.0581) (1.5330, 10.3051)
$ \alpha=1.2 $ (3.1688, 10.8898) (3.1680, 10.8917) (3.1688, 10.8898) (1.6154, 10.3557)
$ \alpha=1.0 $ (2.9688, 10.6898) (2.9687, 10.6901) (2.9688, 10.6894) (1.7244, 10.3693)
Table 2.  Cramér-Lundberg model: $ (z_1^*, z_2^*) $ vs. $ \beta $
$ (z_1^*, z_2^*) $ $ \alpha=2.0, k=\infty $ $ \alpha=1.5, k=0.6 $ $ \alpha=1.0, k=0.2 $ Ruin Time
$ \beta=0.5 $ (4.5855, 10.5305) (4.0855, 10.0305) (2.6070, 9.3218) (2.5855, 8.5305)
$ \beta=1.0 $ (3.9689, 11.6898) (3.4688, 11.1893) (1.7244, 10.3693) (1.9688, 9.6898)
$ \beta=1.5 $ (3.5369, 12.6025) (3.0369, 12.1025) (1.0605, 11.1901) (1.5369, 10.6025)
$ \beta=2.0 $ (3.1911, 13.3968) (1.5000, 12.8618) (1.0000, 11.9132) (1.1912, 11.3968)
$ \beta= 2.5 $ (2.8968, 14.1194) (1.5000, 13.5114) (1.0000, 12.6039) (0.8968, 12.1194)
$ (z_1^*, z_2^*) $ $ \alpha=2.0, k=\infty $ $ \alpha=1.5, k=0.6 $ $ \alpha=1.0, k=0.2 $ Ruin Time
$ \beta=0.5 $ (4.5855, 10.5305) (4.0855, 10.0305) (2.6070, 9.3218) (2.5855, 8.5305)
$ \beta=1.0 $ (3.9689, 11.6898) (3.4688, 11.1893) (1.7244, 10.3693) (1.9688, 9.6898)
$ \beta=1.5 $ (3.5369, 12.6025) (3.0369, 12.1025) (1.0605, 11.1901) (1.5369, 10.6025)
$ \beta=2.0 $ (3.1911, 13.3968) (1.5000, 12.8618) (1.0000, 11.9132) (1.1912, 11.3968)
$ \beta= 2.5 $ (2.8968, 14.1194) (1.5000, 13.5114) (1.0000, 12.6039) (0.8968, 12.1194)
Table 3.  Brownian Motion with drift
$ (z_1^*, z_2^*) $$ k = \infty $$ k = 0.6 $$ k = 0.4 $$ k = 0.2 $
$ \alpha = 2.0 $(3.8442, 12.3162)(3.8441, 12.3176)(3.0519, 11.7522)(2.2989, 10.8580)
$ \alpha = 1.5 $(3.3442, 11.8162)(3.3441, 11.8162)(3.0575, 11.6979)(2.2989, 10.8580)
$ \alpha = 1.2 $(3.0442, 11.5162)(3.0441, 11.5162)(3.0397, 11.5094)(2.2989, 10.8580)
$ \alpha = 1.0 $(2.8442, 11.3162)(2.8439, 11.3162)(2.8441, 11.3166)(2.2989, 10.8580)
$ (z_1^*, z_2^*) $$ k = \infty $$ k = 0.6 $$ k = 0.4 $$ k = 0.2 $
$ \alpha = 2.0 $(3.8442, 12.3162)(3.8441, 12.3176)(3.0519, 11.7522)(2.2989, 10.8580)
$ \alpha = 1.5 $(3.3442, 11.8162)(3.3441, 11.8162)(3.0575, 11.6979)(2.2989, 10.8580)
$ \alpha = 1.2 $(3.0442, 11.5162)(3.0441, 11.5162)(3.0397, 11.5094)(2.2989, 10.8580)
$ \alpha = 1.0 $(2.8442, 11.3162)(2.8439, 11.3162)(2.8441, 11.3166)(2.2989, 10.8580)
Table 4.  Brownian motion with drift: $ (z_1^*, z_2^*) $ vs. $ \beta $
$ (z_1^*, z_2^*) $ $ \alpha=2.0, k=\infty $ $ \alpha=1.5, k=0.6 $ $ \alpha=1.0, k=0.2 $ Ruin Time
$ \beta=0.5 $ (3.9695, 9.9869) (3.4695, 9.4869) (2.4539, 8.5596) (1.9695, 7.9869)
$ \beta=1.0 $ (3.8442, 12.3162) (3.3441, 11.8162) (2.2989, 10.8580) (1.8442, 10.3162)
$ \beta=1.5 $ (3.7664, 14.1668) (3.2664, 13.6668) (2.2024, 12.6891) (1.7664, 12.1668)
$ \beta=2.0 $ (3.7087, 15.7657) (3.2087, 15.2657) (2.1307, 14.2736) (1.7087, 13.7657)
$ \beta= 2.5 $ (3.6623, 17.2027) (3.1623, 16.7027) (2.0730, 15.6991) (1.6623, 15.2027)
$ (z_1^*, z_2^*) $ $ \alpha=2.0, k=\infty $ $ \alpha=1.5, k=0.6 $ $ \alpha=1.0, k=0.2 $ Ruin Time
$ \beta=0.5 $ (3.9695, 9.9869) (3.4695, 9.4869) (2.4539, 8.5596) (1.9695, 7.9869)
$ \beta=1.0 $ (3.8442, 12.3162) (3.3441, 11.8162) (2.2989, 10.8580) (1.8442, 10.3162)
$ \beta=1.5 $ (3.7664, 14.1668) (3.2664, 13.6668) (2.2024, 12.6891) (1.7664, 12.1668)
$ \beta=2.0 $ (3.7087, 15.7657) (3.2087, 15.2657) (2.1307, 14.2736) (1.7087, 13.7657)
$ \beta= 2.5 $ (3.6623, 17.2027) (3.1623, 16.7027) (2.0730, 15.6991) (1.6623, 15.2027)
Table 5.  Jump-diffusion process
$ \alpha=2.0 $ (5.4394, 14.7877) (5.4394, 14.7879) (5.4393, 14.7878) (2.0000, 13.4730)
$ \alpha=1.5 $ (4.9394, 14.2876) (4.9394, 14.2876) (4.9394, 14.2875) (2.1303, 13.7094)
$ \alpha=1.2 $ (4.6394, 13.9876) (4.6394, 13.9876) (4.6386, 13.9915) (2.7529, 13.7678)
$ \alpha=1.0 $ (4.4394, 13.7876) (4.4394, 13.7865) (4.4394, 13.7877) (3.5068, 13.7285)
$ \alpha=2.0 $ (5.4394, 14.7877) (5.4394, 14.7879) (5.4393, 14.7878) (2.0000, 13.4730)
$ \alpha=1.5 $ (4.9394, 14.2876) (4.9394, 14.2876) (4.9394, 14.2875) (2.1303, 13.7094)
$ \alpha=1.2 $ (4.6394, 13.9876) (4.6394, 13.9876) (4.6386, 13.9915) (2.7529, 13.7678)
$ \alpha=1.0 $ (4.4394, 13.7876) (4.4394, 13.7865) (4.4394, 13.7877) (3.5068, 13.7285)
Table 6.  Jump-diffusion process: $ (z_1^*, z_2^*) $ vs. $ \beta $
$ (z_1^*, z_2^*) $ $ \alpha=2.0, k=\infty $ $ \alpha=1.5, k=0.6 $ $ \alpha=1.2, k=0.4 $ Ruin Time
$ \beta=0.5 $ (6.2530, 13.5120) (5.7530, 13.0120) (5.2514, 12.5135) (4.2530, 11.5120)
$ \beta=1.0 $ (5.4394, 14.7876) (4.9394, 14.2876) (3.5068, 13.7285) (3.4394, 12.7876)
$ \beta=1.5 $ (4.8685, 15.7668) (4.3685, 15.2668) (2.0611, 14.5761) (2.8685, 13.7667)
$ \beta=2.0 $ (4.4172, 16.6037) (3.9172, 16.1037) (1.2175, 15.3164) (2.4172, 14.6037)
$ \beta= 2.5 $ (4.0462, 17.3555) (3.5462, 16.8555) (1.0000, 15.9652) (2.0462, 15.3555)
$ (z_1^*, z_2^*) $ $ \alpha=2.0, k=\infty $ $ \alpha=1.5, k=0.6 $ $ \alpha=1.2, k=0.4 $ Ruin Time
$ \beta=0.5 $ (6.2530, 13.5120) (5.7530, 13.0120) (5.2514, 12.5135) (4.2530, 11.5120)
$ \beta=1.0 $ (5.4394, 14.7876) (4.9394, 14.2876) (3.5068, 13.7285) (3.4394, 12.7876)
$ \beta=1.5 $ (4.8685, 15.7668) (4.3685, 15.2668) (2.0611, 14.5761) (2.8685, 13.7667)
$ \beta=2.0 $ (4.4172, 16.6037) (3.9172, 16.1037) (1.2175, 15.3164) (2.4172, 14.6037)
$ \beta= 2.5 $ (4.0462, 17.3555) (3.5462, 16.8555) (1.0000, 15.9652) (2.0462, 15.3555)
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