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April  2019, 15(2): 517-535. doi: 10.3934/jimo.2018055

Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model

Manman Li 1,, and  George Yin 2,

1. 

College of Economics and Business Administration, Chongqing University, Chongqing 400030, China
2. 

Department of Mathematics, Wayne State University, MI, USA, 48202

* Corresponding author: Manman Li

Received  August 2017 Revised  October 2017 Published  April 2018

Fund Project: The research of M. Li was supported in part by MOE Project of Humanities and Social Sciences on the west and the border area (No.14XJC910001) and the Fundamental Research Funds for the Central Universities (No.106112016CDJXY100002). The research of G. Yin was supported in part by the National Science Foundation under DMS-1207667.

This paper focuses on optimal threshold strategies for a spectrally negative Lévy (SNL) risk process with capital injections and proportional transaction costs. Restricted to solvency constraint, our model requires the shareholders of dividends prevent ruin by injecting capitals. Value function of the firm is assumed to be an expected discounted total [dividends less discounted capital injection]. Under such a setup, we derive certain key identities in connection with value function of the firm of a maximum dividend rate. Under restricted dividend rates and capital injection, we give analytical description of the maximum value function of the firm and the optimal threshold strategy explicitly.

Keywords: Spectrally negative Lévy process, threshold strategy, capital injection, scale function, fluctuation theory.
Mathematics Subject Classification: Primary: 93E20; Secondary: 91B70, 60H30.
Citation: Manman Li, George Yin. Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model. Journal of Industrial & Management Optimization, 2019, 15 (2) : 517-535. doi: 10.3934/jimo.2018055
References:
[1]

H. Albrecher, J. Hartinger and S. Thonhauser, On exact solutions for dividend strategies of threshold and linear barrier type in a Sparre Andersen model, ASTIN Bull., 37 (2007), 203-233.  doi: 10.1017/S0515036100014847.  Google Scholar

[2]

S. Asmussen, Applied Probability and Queues, World Scientific, 2003,  Google Scholar

[3]

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

[4]

B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bull., 41 (2011), 611-644.   Google Scholar

[5]

F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a SNLP, Ann. Appl. Prob., 17 (2007), 156-180.  doi: 10.1214/105051606000000709.  Google Scholar

[6]

J. Bertoin, Lévy Processes, Cambridge University Press, 1996.  Google Scholar

[7]

T. Chan, A. E. Kyprianou and M. Savov, Smoothness of scale functions for spectrally negative Lévy processes, Probab. Theory Relat. Fields, 150 (2011), 691-708.  doi: 10.1007/s00440-010-0289-4.  Google Scholar

[8]

H. S. Dai, Z. M. Liu and N. Luan, Optimal dividend strategies in a dual model with capital injections, Math. Meth. Oper. Res., 72 (2010), 129-143.  doi: 10.1007/s00186-010-0312-7.  Google Scholar

[9]

H. Gerber and E. Shiu, On optimal dividend strategies in the compound poisson model, North American Actuarial J., 10 (2006), 76-93.  doi: 10.1080/10920277.2006.10596249.  Google Scholar

[10]

H. Gerber and E. Shiu, On optimal dividends: from reflection to refraction, J.Comput. Appl. Math., 186 (2006), 4-22.  doi: 10.1016/j.cam.2005.03.062.  Google Scholar

[11]

J. M. Harrison and A. J. Taylor, Optimal control of a Brownian storage system, Stoch. Process. Appl., 6 (1978), 179-194.   Google Scholar

[12]

M. Jeanblanc-Picqué and A. N. Shiryaev, Optimization of the flow of dividends, Russian Math. Surveys, 50 (1995), 257-277.   Google Scholar

[13]

N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cramer-Lundberg model with capital injections, Insurance Math. Econom., 43 (2008), 270-278.  doi: 10.1016/j.insmatheco.2008.05.013.  Google Scholar

[14]

A. E. Kyprianou and R. L. Loeffen, Refracted Lévy processes, Annales de l'Instut Henri Poincaré, 46 (2010), 24-44.  doi: 10.1214/08-AIHP307.  Google Scholar

[15]

A. E. Kyprianou, V. Rivero and R. Song, Convexity and smoothness of scale functions and de Finetti's control problem, J. Th. Probab., 23 (2010), 547-564.  doi: 10.1007/s10959-009-0220-z.  Google Scholar

[16]

A. E. Kyprianou and F. Hubalek, Old, new examples of scale functions for spectrally negative Lévy processes, Seminar on Stochastic Analysis, Random Fields and Applications VI Progress in Probability, 63 (2011), 119-145.   Google Scholar

[17]

A. E. Kyprianou, R. Loeffen and J. Perez, Optimal control with absolutely continuous strategies for spectrally negative Lévy prcoesses, J. Appl. Probab., 49 (2012), 150-166.  doi: 10.1239/jap/1331216839.  Google Scholar

[18]

A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd ed. Springer, 2014.  Google Scholar

[19]

A. Lambert, Completely asymmetric Lévy processes confined in a finite interval, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 251-274.  doi: 10.1016/S0246-0203(00)00126-6.  Google Scholar

[20]

M. Li and G. Yin, Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model, preprint, 2014. Google Scholar

[21]

X. S. Lin and K. P. Pavlova, The compound Poisson risk midel with a threhsold dividend strategy, Insurance Math. Econom., 38 (2006), 57-80.  doi: 10.1016/j.insmatheco.2005.08.001.  Google Scholar

[22]

R. L. Loeffen, On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes, Ann. Appl. Probab., 18 (2008), 1669-1680.  doi: 10.1214/07-AAP504.  Google Scholar

[23]

R. L. Loeffen, An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density, J. Appl. Probab., 46 (2009), 85-98.  doi: 10.1239/jap/1238592118.  Google Scholar

[24]

R. L. Loeffen, J. F. Renaud and X. W. Zhou, Occupation times of intervals until first passage times for spectrally negative Lévy processes, Stoch. Proc. Appl., 124 (2014), 1408-1435.  doi: 10.1016/j.spa.2013.11.005.  Google Scholar

[25]

A. Løkka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance Math. Econom., 42 (2008), 954-961.  doi: 10.1016/j.insmatheco.2007.10.013.  Google Scholar

[26]

A. C. Y. Ng, On a dual model with a dividend threshold, Insurance Math. Econom., 44 (2009), 315-324.  doi: 10.1016/j.insmatheco.2008.11.011.  Google Scholar

[27]

M. R. Pistorius, On doubly reflected completely asymmetric Lévy processes, Stoch. Proc. Appl., 107 (2003), 131-143.  doi: 10.1016/S0304-4149(03)00049-8.  Google Scholar

[28]

M. R. Pistorius, On exit and ergodicity of the completely asymmetric Lévy process reflected at its infimum, J. Th. Probab., 17 (2004), 183-220.  doi: 10.1023/B:JOTP.0000020481.14371.37.  Google Scholar

[29]

J. L. Pérez and K. Yamazakic, On the refracted-reflected spectrally negative Lévy processes, Stochastic Process. Appl., 128 (2018), 306-331.  doi: 10.1016/j.spa.2017.03.024.  Google Scholar

[30]

K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, 1999.  Google Scholar

[31]

S. E. Shreve, J. P. Lehoczky and D. P. Gaver, Optimal consumption for general diffusions with absorbing and refelecting barriers, SIAM J. Control Optim., 22 (1984), 55-75.  doi: 10.1137/0322005.  Google Scholar

[32]

D. J. Yao, H. L. Yang and R. M. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64.  doi: 10.1016/j.econmod.2013.10.026.  Google Scholar

[33]

C.C. Yin and K.C. Yuen, Optimality of the threshold dividend strategy for the compound Poisson model, Statistics and Probability Letters, 81 (2011), 1841-1846.  doi: 10.1016/j.spl.2011.07.022.  Google Scholar

[34]

M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207.  doi: 10.1016/j.econmod.2011.09.007.  Google Scholar

[35]

J. X. Zhu and H. L. Yang, Optimal financing and dividend distribution in a general diffusion model with regime switching, Advances in Applied Probability, 48 (2016), 406-422.  doi: 10.1017/apr.2016.7.  Google Scholar

show all references

References:
[1]

H. Albrecher, J. Hartinger and S. Thonhauser, On exact solutions for dividend strategies of threshold and linear barrier type in a Sparre Andersen model, ASTIN Bull., 37 (2007), 203-233.  doi: 10.1017/S0515036100014847.  Google Scholar

[2]

S. Asmussen, Applied Probability and Queues, World Scientific, 2003,  Google Scholar

[3]

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

[4]

B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bull., 41 (2011), 611-644.   Google Scholar

[5]

F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a SNLP, Ann. Appl. Prob., 17 (2007), 156-180.  doi: 10.1214/105051606000000709.  Google Scholar

[6]

J. Bertoin, Lévy Processes, Cambridge University Press, 1996.  Google Scholar

[7]

T. Chan, A. E. Kyprianou and M. Savov, Smoothness of scale functions for spectrally negative Lévy processes, Probab. Theory Relat. Fields, 150 (2011), 691-708.  doi: 10.1007/s00440-010-0289-4.  Google Scholar

[8]

H. S. Dai, Z. M. Liu and N. Luan, Optimal dividend strategies in a dual model with capital injections, Math. Meth. Oper. Res., 72 (2010), 129-143.  doi: 10.1007/s00186-010-0312-7.  Google Scholar

[9]

H. Gerber and E. Shiu, On optimal dividend strategies in the compound poisson model, North American Actuarial J., 10 (2006), 76-93.  doi: 10.1080/10920277.2006.10596249.  Google Scholar

[10]

H. Gerber and E. Shiu, On optimal dividends: from reflection to refraction, J.Comput. Appl. Math., 186 (2006), 4-22.  doi: 10.1016/j.cam.2005.03.062.  Google Scholar

[11]

J. M. Harrison and A. J. Taylor, Optimal control of a Brownian storage system, Stoch. Process. Appl., 6 (1978), 179-194.   Google Scholar

[12]

M. Jeanblanc-Picqué and A. N. Shiryaev, Optimization of the flow of dividends, Russian Math. Surveys, 50 (1995), 257-277.   Google Scholar

[13]

N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cramer-Lundberg model with capital injections, Insurance Math. Econom., 43 (2008), 270-278.  doi: 10.1016/j.insmatheco.2008.05.013.  Google Scholar

[14]

A. E. Kyprianou and R. L. Loeffen, Refracted Lévy processes, Annales de l'Instut Henri Poincaré, 46 (2010), 24-44.  doi: 10.1214/08-AIHP307.  Google Scholar

[15]

A. E. Kyprianou, V. Rivero and R. Song, Convexity and smoothness of scale functions and de Finetti's control problem, J. Th. Probab., 23 (2010), 547-564.  doi: 10.1007/s10959-009-0220-z.  Google Scholar

[16]

A. E. Kyprianou and F. Hubalek, Old, new examples of scale functions for spectrally negative Lévy processes, Seminar on Stochastic Analysis, Random Fields and Applications VI Progress in Probability, 63 (2011), 119-145.   Google Scholar

[17]

A. E. Kyprianou, R. Loeffen and J. Perez, Optimal control with absolutely continuous strategies for spectrally negative Lévy prcoesses, J. Appl. Probab., 49 (2012), 150-166.  doi: 10.1239/jap/1331216839.  Google Scholar

[18]

A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd ed. Springer, 2014.  Google Scholar

[19]

A. Lambert, Completely asymmetric Lévy processes confined in a finite interval, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 251-274.  doi: 10.1016/S0246-0203(00)00126-6.  Google Scholar

[20]

M. Li and G. Yin, Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model, preprint, 2014. Google Scholar

[21]

X. S. Lin and K. P. Pavlova, The compound Poisson risk midel with a threhsold dividend strategy, Insurance Math. Econom., 38 (2006), 57-80.  doi: 10.1016/j.insmatheco.2005.08.001.  Google Scholar

[22]

R. L. Loeffen, On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes, Ann. Appl. Probab., 18 (2008), 1669-1680.  doi: 10.1214/07-AAP504.  Google Scholar

[23]

R. L. Loeffen, An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density, J. Appl. Probab., 46 (2009), 85-98.  doi: 10.1239/jap/1238592118.  Google Scholar

[24]

R. L. Loeffen, J. F. Renaud and X. W. Zhou, Occupation times of intervals until first passage times for spectrally negative Lévy processes, Stoch. Proc. Appl., 124 (2014), 1408-1435.  doi: 10.1016/j.spa.2013.11.005.  Google Scholar

[25]

A. Løkka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance Math. Econom., 42 (2008), 954-961.  doi: 10.1016/j.insmatheco.2007.10.013.  Google Scholar

[26]

A. C. Y. Ng, On a dual model with a dividend threshold, Insurance Math. Econom., 44 (2009), 315-324.  doi: 10.1016/j.insmatheco.2008.11.011.  Google Scholar

[27]

M. R. Pistorius, On doubly reflected completely asymmetric Lévy processes, Stoch. Proc. Appl., 107 (2003), 131-143.  doi: 10.1016/S0304-4149(03)00049-8.  Google Scholar

[28]

M. R. Pistorius, On exit and ergodicity of the completely asymmetric Lévy process reflected at its infimum, J. Th. Probab., 17 (2004), 183-220.  doi: 10.1023/B:JOTP.0000020481.14371.37.  Google Scholar

[29]

J. L. Pérez and K. Yamazakic, On the refracted-reflected spectrally negative Lévy processes, Stochastic Process. Appl., 128 (2018), 306-331.  doi: 10.1016/j.spa.2017.03.024.  Google Scholar

[30]

K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, 1999.  Google Scholar

[31]

S. E. Shreve, J. P. Lehoczky and D. P. Gaver, Optimal consumption for general diffusions with absorbing and refelecting barriers, SIAM J. Control Optim., 22 (1984), 55-75.  doi: 10.1137/0322005.  Google Scholar

[32]

D. J. Yao, H. L. Yang and R. M. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64.  doi: 10.1016/j.econmod.2013.10.026.  Google Scholar

[33]

C.C. Yin and K.C. Yuen, Optimality of the threshold dividend strategy for the compound Poisson model, Statistics and Probability Letters, 81 (2011), 1841-1846.  doi: 10.1016/j.spl.2011.07.022.  Google Scholar

[34]

M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207.  doi: 10.1016/j.econmod.2011.09.007.  Google Scholar

[35]

J. X. Zhu and H. L. Yang, Optimal financing and dividend distribution in a general diffusion model with regime switching, Advances in Applied Probability, 48 (2016), 406-422.  doi: 10.1017/apr.2016.7.  Google Scholar

Figure 1.  The modified Lévy risk process
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