American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension
  • MCRF Home
  • This Issue
  • Next Article
    Feedback optimal control for stochastic Volterra equations with completely monotone kernels
June  2015, 5(2): 237-258. doi: 10.3934/mcrf.2015.5.237

Lookback option pricing for regime-switching jump diffusion models

Zhuo Jin 1, and  Linyi Qian 2,

1. 

Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia
2. 

School of Finance and Statistics, East China Normal University, 500 Dongchuan Road, Shanghai, 200241

Received  June 2014 Revised  August 2014 Published  April 2015

In this paper, we will introduce a numerical method to price the European lookback floating strike put options where the underlying asset price is modeled by a generalized regime-switching jump diffusion process. In the Markov regime-switching model, the option value is a solution of a coupled system of nonlinear integro-differential partial differential equations. Due to the complexity of regime-switching model, the jump process involved, and the nonlinearity, closed-form solutions are virtually impossible to obtain. We use Markov chain approximation techniques to construct a discrete-time Markov chain to approximate the option value. Convergence of the approximation algorithms is proved. Examples are presented to demonstrate the applicability of the numerical methods.
Keywords: floating strike, Lookback option, Markov chain approximation..
Mathematics Subject Classification: 60H35, 91G20, 91G80, 65C0.
Citation: Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control & Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237
References:
[1]

L. Andersen and J. Andreasen, Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing,, Review of Derivatives Research, 4 (2000), 231.   Google Scholar

[2]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, J. Asymptotic Analysis, 4 (1991), 271.   Google Scholar

[3]

G. K. Basak, M. K. Ghosh and A. Goswami, Risk minimizing option pricing for a class of exotic options in a Markov-modulated market,, Stochastic Analysis and Applications, 29 (2011), 259.  doi: 10.1080/07362994.2011.548665.  Google Scholar

[4]

S. Boyarchenko and S. Levendorskii, Barrier options and touch-and-out options under regular Lévy processes of exponential type,, Ann. Appl. Probab., 12 (2002), 1261.  doi: 10.1214/aoap/1037125863.  Google Scholar

[5]

P. Boyle and T. Draviam, Pricing exotic options under regime switching,, Insurance: Mathematics and Economics, 40 (2007), 267.  doi: 10.1016/j.insmatheco.2006.05.001.  Google Scholar

[6]

M. Broadie, P. Glasserman and S. G. Kou, Connecting discrete and continuous path-dependent options,, Finance Stochastics, 3 (1999), 55.  doi: 10.1007/s007800050052.  Google Scholar

[7]

P. Buchen and O. Konstandatos, A new method of pricing lookback options,, Mathematical Finance, 15 (2005), 245.  doi: 10.1111/j.0960-1627.2005.00219.x.  Google Scholar

[8]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model,, Management Science, 57 (2011), 2067.  doi: 10.1287/mnsc.1110.1393.  Google Scholar

[9]

P. Carr and J. Crosby, A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options,, Quantitative Finance, 10 (2010), 1115.  doi: 10.1080/14697680903413605.  Google Scholar

[10]

A. Conze and R. Vishwanathan, Path-dependent options: The case of lookback options,, J. Finance, 46 (1991), 1893.  doi: 10.2307/2328577.  Google Scholar

[11]

M. Dai and Y. K. Kwok, Characterization of optimal stopping regions of American Asian and lookback options,, Mathematical Finance, 16 (2006), 63.  doi: 10.1111/j.1467-9965.2006.00261.x.  Google Scholar

[12]

J. DiCesare and D. Mcleish, Simulation of jump diffusions and the pricing of options,, Insurance: Mathematics and Economics, 43 (2008), 316.  doi: 10.1016/j.insmatheco.2008.06.001.  Google Scholar

[13]

H. U. Gerber and E. S. Shiu, Pricing lookback options and dynamic guarantees,, North American Actuarial Journal, 7 (2003), 48.  doi: 10.1080/10920277.2003.10596076.  Google Scholar

[14]

M. B. Goldman, H. B. Sosin and M. A. Gatto, Path-dependent options,, J. Finance, 34 (1979), 1111.   Google Scholar

[15]

J. Hamilton, A new approach to the economic analysis of non-stationary time series,, Econometrica, 57 (1989), 357.  doi: 10.2307/1912559.  Google Scholar

[16]

C. He, J. S. Kennedy, T. F. Coleman, P. A. Forsyth, Y. Li and K. R. Vetzal, Calibration and hedging under jump diffusion,, Review of Derivatives Research, 9 (2006), 1.  doi: 10.1007/s11147-006-9003-1.  Google Scholar

[17]

S. Jaimungal and T. Wang, Catastrophe options with stochastic interest rates and compound Poisson losses,, Insurance: Mathematics and Economics, 38 (2006), 469.  doi: 10.1016/j.insmatheco.2005.11.008.  Google Scholar

[18]

Z. Jin, G. Yin and F. Wu, Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods,, Insurance Mathematics and Economics, 53 (2013), 733.  doi: 10.1016/j.insmatheco.2013.09.015.  Google Scholar

[19]

Z. Jin, G. Yin and H. L. Yang, Numerical methods for dividend optimization using regime-switching jump-diffusion models,, Mathematical Control and Related Fields, 1 (2011), 21.  doi: 10.3934/mcrf.2011.1.21.  Google Scholar

[20]

Z. Jin, G. Yin and C. Zhu, Numerical solutions of optimal risk control and dividend optimization policies under a generalized singular control formulation,, Automatica, 48 (2012), 1489.  doi: 10.1016/j.automatica.2012.05.039.  Google Scholar

[21]

Z. Jin, H. Yang and G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections,, Automatica, 49 (2013), 2317.  doi: 10.1016/j.automatica.2013.04.043.  Google Scholar

[22]

K. I. Kim, H. S. Park and X. S. Qian, A mathematical modeling for the lookback option with jump-diffusion using binomial tree method,, Journal of Computational and Applied Mathematics, 235 (2011), 5140.  doi: 10.1016/j.cam.2011.05.002.  Google Scholar

[23]

S. G. Kou, A jump-diffusion model for option pricing,, Management Science, 48 (2002), 1086.  doi: 10.1287/mnsc.48.8.1086.166.  Google Scholar

[24]

S. Kou and H. Wang, Option pricing under a double exponential jump diffusion model,, Management Science, 50 (2004), 1178.  doi: 10.1287/mnsc.1030.0163.  Google Scholar

[25]

H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory,, MIT Press, (1984).   Google Scholar

[26]

H. Kushner and P. Dupuis, Numerical Methods for Stochstic Control Problems in Continuous Time,, Second edition, (2001).  doi: 10.1007/978-1-4613-0007-6.  Google Scholar

[27]

H. J. Kushner and L. F. Martins, Numerical methods for stochastic singular control problems,, SIAM J. Control Optim., 29 (1991), 1443.  doi: 10.1137/0329073.  Google Scholar

[28]

H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications,, Second edition, (2003).   Google Scholar

[29]

Y. K. Kwok, Mathematical Models of Financial Derivatives,, Second edition, (2008).   Google Scholar

[30]

R. C. Merton, Option pricing when underlying stock returns are discontinuous,, Journal of Financial Economics, 3 (1976), 125.  doi: 10.1016/0304-405X(76)90022-2.  Google Scholar

[31]

H. Pham, Optimal stopping, free boundary, and American option in a jump-diffusion model,, Applied Mathematics and Optimization, 35 (1997), 145.  doi: 10.1007/BF02683325.  Google Scholar

[32]

R. Poulsen, Barrier options and their static hedges: Simple derivations and extensions,, Quantitative Finance, 6 (2006), 327.  doi: 10.1080/14697680600690331.  Google Scholar

[33]

Q. S. Song, G. Yin and Z. Zhang, Numerical method for controlled regime-switching diffusions and regime-switching jump diffusions,, Automatica, 42 (2006), 1147.  doi: 10.1016/j.automatica.2006.03.016.  Google Scholar

[34]

Q. S. Song, G. Yin and Z. Zhang, Numerical solutions for stochastic differential games with regime swiching,, IEEE Transaction on Automatic Control, 53 (2008), 509.  doi: 10.1109/TAC.2007.915169.  Google Scholar

[35]

P. Wilmott, Paul Wilmott on Quantitative Finance,, Wiley, (2000).   Google Scholar

[36]

P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation,, Oxford Financial Press, (1994).   Google Scholar

[37]

C. Xu and Y. K. Kwok, Integral price formulas for lookback options,, Journal of Applied Mathematics, (2005), 117.  doi: 10.1155/JAM.2005.117.  Google Scholar

[38]

H. Yang and G. Yin, Ruin probability for a model under Markovian switching regime,, in Probability, (2004), 206.   Google Scholar

[39]

G. Yin, H. Jin and Z. Jin, Numerical methods for portfolio selection with bounded constraints,, J. Computational Appl. Math., 233 (2009), 564.  doi: 10.1016/j.cam.2009.08.055.  Google Scholar

[40]

G. Yin, Q. Zhang and G. Badowski, Discrete-time singularly perturbed Markov chains: Aggregation, occupation measures, and switching diffusion limit,, Adv. Appl. Probab., 35 (2003), 449.  doi: 10.1239/aap/1051201656.  Google Scholar

[41]

Z. Zhang, G. Yin and Z. Liang, A stochastic approximation algorithm for American lookback put options,, Stochastic Analysis and Applications, 29 (2011), 332.  doi: 10.1080/07362994.2010.503473.  Google Scholar

[42]

X. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: A continuous-time model,, SIAM J. Control Optim., 42 (2003), 1466.  doi: 10.1137/S0363012902405583.  Google Scholar

show all references

References:
[1]

L. Andersen and J. Andreasen, Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing,, Review of Derivatives Research, 4 (2000), 231.   Google Scholar

[2]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, J. Asymptotic Analysis, 4 (1991), 271.   Google Scholar

[3]

G. K. Basak, M. K. Ghosh and A. Goswami, Risk minimizing option pricing for a class of exotic options in a Markov-modulated market,, Stochastic Analysis and Applications, 29 (2011), 259.  doi: 10.1080/07362994.2011.548665.  Google Scholar

[4]

S. Boyarchenko and S. Levendorskii, Barrier options and touch-and-out options under regular Lévy processes of exponential type,, Ann. Appl. Probab., 12 (2002), 1261.  doi: 10.1214/aoap/1037125863.  Google Scholar

[5]

P. Boyle and T. Draviam, Pricing exotic options under regime switching,, Insurance: Mathematics and Economics, 40 (2007), 267.  doi: 10.1016/j.insmatheco.2006.05.001.  Google Scholar

[6]

M. Broadie, P. Glasserman and S. G. Kou, Connecting discrete and continuous path-dependent options,, Finance Stochastics, 3 (1999), 55.  doi: 10.1007/s007800050052.  Google Scholar

[7]

P. Buchen and O. Konstandatos, A new method of pricing lookback options,, Mathematical Finance, 15 (2005), 245.  doi: 10.1111/j.0960-1627.2005.00219.x.  Google Scholar

[8]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model,, Management Science, 57 (2011), 2067.  doi: 10.1287/mnsc.1110.1393.  Google Scholar

[9]

P. Carr and J. Crosby, A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options,, Quantitative Finance, 10 (2010), 1115.  doi: 10.1080/14697680903413605.  Google Scholar

[10]

A. Conze and R. Vishwanathan, Path-dependent options: The case of lookback options,, J. Finance, 46 (1991), 1893.  doi: 10.2307/2328577.  Google Scholar

[11]

M. Dai and Y. K. Kwok, Characterization of optimal stopping regions of American Asian and lookback options,, Mathematical Finance, 16 (2006), 63.  doi: 10.1111/j.1467-9965.2006.00261.x.  Google Scholar

[12]

J. DiCesare and D. Mcleish, Simulation of jump diffusions and the pricing of options,, Insurance: Mathematics and Economics, 43 (2008), 316.  doi: 10.1016/j.insmatheco.2008.06.001.  Google Scholar

[13]

H. U. Gerber and E. S. Shiu, Pricing lookback options and dynamic guarantees,, North American Actuarial Journal, 7 (2003), 48.  doi: 10.1080/10920277.2003.10596076.  Google Scholar

[14]

M. B. Goldman, H. B. Sosin and M. A. Gatto, Path-dependent options,, J. Finance, 34 (1979), 1111.   Google Scholar

[15]

J. Hamilton, A new approach to the economic analysis of non-stationary time series,, Econometrica, 57 (1989), 357.  doi: 10.2307/1912559.  Google Scholar

[16]

C. He, J. S. Kennedy, T. F. Coleman, P. A. Forsyth, Y. Li and K. R. Vetzal, Calibration and hedging under jump diffusion,, Review of Derivatives Research, 9 (2006), 1.  doi: 10.1007/s11147-006-9003-1.  Google Scholar

[17]

S. Jaimungal and T. Wang, Catastrophe options with stochastic interest rates and compound Poisson losses,, Insurance: Mathematics and Economics, 38 (2006), 469.  doi: 10.1016/j.insmatheco.2005.11.008.  Google Scholar

[18]

Z. Jin, G. Yin and F. Wu, Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods,, Insurance Mathematics and Economics, 53 (2013), 733.  doi: 10.1016/j.insmatheco.2013.09.015.  Google Scholar

[19]

Z. Jin, G. Yin and H. L. Yang, Numerical methods for dividend optimization using regime-switching jump-diffusion models,, Mathematical Control and Related Fields, 1 (2011), 21.  doi: 10.3934/mcrf.2011.1.21.  Google Scholar

[20]

Z. Jin, G. Yin and C. Zhu, Numerical solutions of optimal risk control and dividend optimization policies under a generalized singular control formulation,, Automatica, 48 (2012), 1489.  doi: 10.1016/j.automatica.2012.05.039.  Google Scholar

[21]

Z. Jin, H. Yang and G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections,, Automatica, 49 (2013), 2317.  doi: 10.1016/j.automatica.2013.04.043.  Google Scholar

[22]

K. I. Kim, H. S. Park and X. S. Qian, A mathematical modeling for the lookback option with jump-diffusion using binomial tree method,, Journal of Computational and Applied Mathematics, 235 (2011), 5140.  doi: 10.1016/j.cam.2011.05.002.  Google Scholar

[23]

S. G. Kou, A jump-diffusion model for option pricing,, Management Science, 48 (2002), 1086.  doi: 10.1287/mnsc.48.8.1086.166.  Google Scholar

[24]

S. Kou and H. Wang, Option pricing under a double exponential jump diffusion model,, Management Science, 50 (2004), 1178.  doi: 10.1287/mnsc.1030.0163.  Google Scholar

[25]

H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory,, MIT Press, (1984).   Google Scholar

[26]

H. Kushner and P. Dupuis, Numerical Methods for Stochstic Control Problems in Continuous Time,, Second edition, (2001).  doi: 10.1007/978-1-4613-0007-6.  Google Scholar

[27]

H. J. Kushner and L. F. Martins, Numerical methods for stochastic singular control problems,, SIAM J. Control Optim., 29 (1991), 1443.  doi: 10.1137/0329073.  Google Scholar

[28]

H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications,, Second edition, (2003).   Google Scholar

[29]

Y. K. Kwok, Mathematical Models of Financial Derivatives,, Second edition, (2008).   Google Scholar

[30]

R. C. Merton, Option pricing when underlying stock returns are discontinuous,, Journal of Financial Economics, 3 (1976), 125.  doi: 10.1016/0304-405X(76)90022-2.  Google Scholar

[31]

H. Pham, Optimal stopping, free boundary, and American option in a jump-diffusion model,, Applied Mathematics and Optimization, 35 (1997), 145.  doi: 10.1007/BF02683325.  Google Scholar

[32]

R. Poulsen, Barrier options and their static hedges: Simple derivations and extensions,, Quantitative Finance, 6 (2006), 327.  doi: 10.1080/14697680600690331.  Google Scholar

[33]

Q. S. Song, G. Yin and Z. Zhang, Numerical method for controlled regime-switching diffusions and regime-switching jump diffusions,, Automatica, 42 (2006), 1147.  doi: 10.1016/j.automatica.2006.03.016.  Google Scholar

[34]

Q. S. Song, G. Yin and Z. Zhang, Numerical solutions for stochastic differential games with regime swiching,, IEEE Transaction on Automatic Control, 53 (2008), 509.  doi: 10.1109/TAC.2007.915169.  Google Scholar

[35]

P. Wilmott, Paul Wilmott on Quantitative Finance,, Wiley, (2000).   Google Scholar

[36]

P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation,, Oxford Financial Press, (1994).   Google Scholar

[37]

C. Xu and Y. K. Kwok, Integral price formulas for lookback options,, Journal of Applied Mathematics, (2005), 117.  doi: 10.1155/JAM.2005.117.  Google Scholar

[38]

H. Yang and G. Yin, Ruin probability for a model under Markovian switching regime,, in Probability, (2004), 206.   Google Scholar

[39]

G. Yin, H. Jin and Z. Jin, Numerical methods for portfolio selection with bounded constraints,, J. Computational Appl. Math., 233 (2009), 564.  doi: 10.1016/j.cam.2009.08.055.  Google Scholar

[40]

G. Yin, Q. Zhang and G. Badowski, Discrete-time singularly perturbed Markov chains: Aggregation, occupation measures, and switching diffusion limit,, Adv. Appl. Probab., 35 (2003), 449.  doi: 10.1239/aap/1051201656.  Google Scholar

[41]

Z. Zhang, G. Yin and Z. Liang, A stochastic approximation algorithm for American lookback put options,, Stochastic Analysis and Applications, 29 (2011), 332.  doi: 10.1080/07362994.2010.503473.  Google Scholar

[42]

X. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: A continuous-time model,, SIAM J. Control Optim., 42 (2003), 1466.  doi: 10.1137/S0363012902405583.  Google Scholar

[1]

Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529
[2]

Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control & Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007
[3]

Ralf Banisch, Carsten Hartmann. Addendum to "A sparse Markov chain approximation of LQ-type stochastic control problems". Mathematical Control & Related Fields, 2017, 7 (4) : 623-623. doi: 10.3934/mcrf.2017023
[4]

Na Song, Ximin Huang, Yue Xie, Wai-Ki Ching, Tak-Kuen Siu. Impact of reorder option in supply chain coordination. Journal of Industrial & Management Optimization, 2017, 13 (1) : 449-475. doi: 10.3934/jimo.2016026
[5]

Samuel N. Cohen, Lukasz Szpruch. On Markovian solutions to Markov Chain BSDEs. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 257-269. doi: 10.3934/naco.2012.2.257
[6]

Jingzhi Tie, Qing Zhang. An optimal mean-reversion trading rule under a Markov chain model. Mathematical Control & Related Fields, 2016, 6 (3) : 467-488. doi: 10.3934/mcrf.2016012
[7]

Alan Elcrat, Ray Treinen. Numerical results for floating drops. Conference Publications, 2005, 2005 (Special) : 241-249. doi: 10.3934/proc.2005.2005.241
[8]

Lin Xu, Rongming Wang. Upper bounds for ruin probabilities in an autoregressive risk model with a Markov chain interest rate. Journal of Industrial & Management Optimization, 2006, 2 (2) : 165-175. doi: 10.3934/jimo.2006.2.165
[9]

Kazuhiko Kuraya, Hiroyuki Masuyama, Shoji Kasahara. Load distribution performance of super-node based peer-to-peer communication networks: A nonstationary Markov chain approach. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 593-610. doi: 10.3934/naco.2011.1.593
[10]

Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks & Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803
[11]

Badal Joshi. A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1077-1105. doi: 10.3934/dcdsb.2015.20.1077
[12]

Mingzheng Wang, M. Montaz Ali, Guihua Lin. Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks. Journal of Industrial & Management Optimization, 2011, 7 (2) : 317-345. doi: 10.3934/jimo.2011.7.317
[13]

Yanming Ge, Ziwen Yin, Yifan Xu. Overbooking with transference option for flights. Journal of Industrial & Management Optimization, 2011, 7 (2) : 449-465. doi: 10.3934/jimo.2011.7.449
[14]

Jingmei Zhou, Xiangmo Zhao, Xin Cheng, Zhigang Xu. Visualization analysis of traffic congestion based on floating car data. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1423-1433. doi: 10.3934/dcdss.2015.8.1423
[15]

Weiwei Wang, Ping Chen. A mean-reverting currency model with floating interest rates in uncertain environment. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1921-1936. doi: 10.3934/jimo.2018129
[16]

Ethan Akin. On chain continuity. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 111-120. doi: 10.3934/dcds.1996.2.111
[17]

Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1751-1759. doi: 10.3934/dcdsb.2012.17.1751
[18]

Na Song, Yue Xie, Wai-Ki Ching, Tak-Kuen Siu. A real option approach for investment opportunity valuation. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1213-1235. doi: 10.3934/jimo.2016069
[19]

Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206
[20]

Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial & Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences