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doi: 10.3934/jimo.2020179
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General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes

Wenyuan Wang 1, and  Ran Xu 2,,

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 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.

Keywords: De Finetti's dividend problem, Spectrally negative Lévy process, General drawdown time, Impulse dividend strategy.
Mathematics Subject Classification: Primary: 60G51, 91G50; Secondary: 60E10.
Citation: Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020179
References:
[1]

S. Asmussen, F. 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.  Google Scholar

[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.  Google Scholar

[3]

F. Avram, A. 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.  Google Scholar

[4]

F. Avram, Z. 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.  Google Scholar

[5]

F. Avram, Z. 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.  Google Scholar

[6]

F. Avram, N. 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.  Google Scholar

[7]

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

[8]

E. J. Baurdoux, Z. 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.  Google Scholar

[9]

E. Bayraktar, A. 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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

A. E. Kyprianou, R. 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.  Google Scholar

[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.  Google Scholar

[20]

D. Landriault, B. 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.  Google Scholar

[21]

D. Landriault, B. 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.  Google Scholar

[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.  Google Scholar

[23]

B. Li, N. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

X. Peng, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

Y. Zhao, P. 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.  Google Scholar

show all references

References:
[1]

S. Asmussen, F. 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.  Google Scholar

[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.  Google Scholar

[3]

F. Avram, A. 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.  Google Scholar

[4]

F. Avram, Z. 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.  Google Scholar

[5]

F. Avram, Z. 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.  Google Scholar

[6]

F. Avram, N. 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.  Google Scholar

[7]

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

[8]

E. J. Baurdoux, Z. 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.  Google Scholar

[9]

E. Bayraktar, A. 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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

A. E. Kyprianou, R. 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.  Google Scholar

[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.  Google Scholar

[20]

D. Landriault, B. 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.  Google Scholar

[21]

D. Landriault, B. 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.  Google Scholar

[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.  Google Scholar

[23]

B. Li, N. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

X. Peng, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

Y. Zhao, P. 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.  Google Scholar

Figure 1.  Illustration of various drawdown times
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Figure 2.  General drawdown times based on Eq.(43)
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Figure 3.  Cramér-Lundberg model: $ \varsigma(z) $
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Figure 4.  Cramér-Lundberg model: $ V^{z_2^*}_{z_1^*}(x) $
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Figure 5.  Brownian Motion with drift: $ \varsigma(z) $
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Figure 6.  Brownian Motion with drift: $ V^{z_2^*}_{z_1^*}(x) $
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Figure 7.  Jump-diffusion process: $ \varsigma(z) $
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Figure 8.  Jump-diffusion process: $ V^{z_2^*}_{z_1^*}(x) $
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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)
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$ (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)
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$ (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 $(2.8442, 11.3162)(2.8439, 11.3162)(2.8441, 11.3166)(2.2989, 10.8580)
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$ (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)
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$ (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)
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$ \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)
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$ (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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