An optimal control model of carbon reduction and trading

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December 2016, 6(4): 535-550. doi: 10.3934/mcrf.2016015

An optimal control model of carbon reduction and trading

Huaying Guo ^1, and Jin Liang ^1,

1.	Department of Mathematics, Tongji University, Shanghai 200092, China

Received October 2015 Revised January 2016 Published October 2016

Abstract
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In this study, a stochastic control model is established for a country to formulate a carbon abatement policy to minimize the total carbon reduction costs. Under Merton's consumption framework, by considering carbon trading, carbon abatement and penalties in a synthetic manner, the model is converted into a two-dimensional Hamilton--Jacobi--Bellman equation. We rigorously prove the existence and uniqueness of its viscosity solution. We also present the numerical results and discuss the properties of the optimal carbon reduction policy and the minimum total costs.

Keywords: viscosity solution., Hamilton--Jacobi--Bellman equation, Carbon reduction, optimal control, carbon trading.

Mathematics Subject Classification: 93E20, 49L25, 35K5.

Citation: Huaying Guo, Jin Liang. An optimal control model of carbon reduction and trading. Mathematical Control and Related Fields, 2016, 6 (4) : 535-550. doi: 10.3934/mcrf.2016015

References:

[1]	F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062.
[2]	R. Carmona, M. Fehr and J. Hinz, Optimal stochastic control and carbon price formation, SIAM Journal on Control and Optimization, 48 (2009), 2168-2190. doi: 10.1137/080736910.
[3]	R. Carmona, M. Fehr, J. Hinz and A. Porchet, Market design for emission trading schemes, SIAM Review, 52 (2010), 403-452. doi: 10.1137/080722813.
[4]	B. Commoner, The environmental cost of economic growth, In R.G. Ridker (Ed.), Population, Resources and the Environment, Washington, DC, U.S. Government Printing Office, (1972), 339-363.
[5]	E. Commission, The EU emissions trading system (EU ETS),, 2013, ().
[6]	M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.
[7]	M. G. Crandall and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[8]	G. Daskalakis, D. Psychoyios and P. N. Markellos, Modeling CO$_2$ emission allowance prices and derivatives: Evidence from the European trading scheme, Journal of Banking and Finance, 33 (2009), 1230-1241.
[9]	T. Dietz and E. A. Rosa, Rethinking the environmental impacts of population, affluence and technology, Human Ecology Review, 1 (1994), 277-300.
[10]	W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2 edition, 2006.
[11]	H. Guo and J. Liang, An optimal control model for reducing and trading of carbon emissions, Physica A: Statistical Mechanics and its Applications, 446 (2016), 11-21, Available from: http://dx.doi.org/10.1016/j.physa.2015.10.076. doi: 10.1016/j.physa.2015.10.076.
[12]	C. Hepburn, Carbon trading: A review of the Kyoto mechanisms, The Annual Review of Environment and Resources, 32 (2007), 375-393. doi: 10.1146/annurev.energy.32.053006.141203.
[13]	R. C. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Sciences, 4 (1973), 141-183. doi: 10.2307/3003143.
[14]	R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X.
[15]	R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257.
[16]	J. Seifert, M. Uhrig-Homburg and M. Wagner, Dynamic behavior of $CO_2$ spot prices, Journal of Environmental Economics and Management, 56 (2008), 180-194.
[17]	A. Tsoularis and J. Wallace, Analysis of logistic growth models, Mathematical Biosciences, 179 (2002), 21-55. doi: 10.1016/S0025-5564(02)00096-2.
[18]	M. Wang, M. Wang and S. Wang, Optimal investment and uncertainty on China's carbon emission abatement, Energy Policy, 41 (2012), 871-877. doi: 10.1016/j.enpol.2011.11.077.
[19]	X. Yang and J. Liang, Minimization of the nation cost due to carbon emission, Systems Engineering - Theory and Practice, 34 (2014), 640-647.
[20]	X. Yang, Optimal control problems associated with carbon emission abatement and leveraged credit derivatives, Ph. D Thesis, Tongji University, 2015.
[21]	E. Zagheni and F. C. Billari, A cost valuation model based on a stochastic representation of the IPAT equation, Population and Environment, 29 (2007), 68-82. doi: 10.1007/s11111-008-0061-1.

show all references

References:

[1]	F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062.
[2]	R. Carmona, M. Fehr and J. Hinz, Optimal stochastic control and carbon price formation, SIAM Journal on Control and Optimization, 48 (2009), 2168-2190. doi: 10.1137/080736910.
[3]	R. Carmona, M. Fehr, J. Hinz and A. Porchet, Market design for emission trading schemes, SIAM Review, 52 (2010), 403-452. doi: 10.1137/080722813.
[4]	B. Commoner, The environmental cost of economic growth, In R.G. Ridker (Ed.), Population, Resources and the Environment, Washington, DC, U.S. Government Printing Office, (1972), 339-363.
[5]	E. Commission, The EU emissions trading system (EU ETS),, 2013, ().
[6]	M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.
[7]	M. G. Crandall and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[8]	G. Daskalakis, D. Psychoyios and P. N. Markellos, Modeling CO$_2$ emission allowance prices and derivatives: Evidence from the European trading scheme, Journal of Banking and Finance, 33 (2009), 1230-1241.
[9]	T. Dietz and E. A. Rosa, Rethinking the environmental impacts of population, affluence and technology, Human Ecology Review, 1 (1994), 277-300.
[10]	W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2 edition, 2006.
[11]	H. Guo and J. Liang, An optimal control model for reducing and trading of carbon emissions, Physica A: Statistical Mechanics and its Applications, 446 (2016), 11-21, Available from: http://dx.doi.org/10.1016/j.physa.2015.10.076. doi: 10.1016/j.physa.2015.10.076.
[12]	C. Hepburn, Carbon trading: A review of the Kyoto mechanisms, The Annual Review of Environment and Resources, 32 (2007), 375-393. doi: 10.1146/annurev.energy.32.053006.141203.
[13]	R. C. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Sciences, 4 (1973), 141-183. doi: 10.2307/3003143.
[14]	R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X.
[15]	R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257.
[16]	J. Seifert, M. Uhrig-Homburg and M. Wagner, Dynamic behavior of $CO_2$ spot prices, Journal of Environmental Economics and Management, 56 (2008), 180-194.
[17]	A. Tsoularis and J. Wallace, Analysis of logistic growth models, Mathematical Biosciences, 179 (2002), 21-55. doi: 10.1016/S0025-5564(02)00096-2.
[18]	M. Wang, M. Wang and S. Wang, Optimal investment and uncertainty on China's carbon emission abatement, Energy Policy, 41 (2012), 871-877. doi: 10.1016/j.enpol.2011.11.077.
[19]	X. Yang and J. Liang, Minimization of the nation cost due to carbon emission, Systems Engineering - Theory and Practice, 34 (2014), 640-647.
[20]	X. Yang, Optimal control problems associated with carbon emission abatement and leveraged credit derivatives, Ph. D Thesis, Tongji University, 2015.
[21]	E. Zagheni and F. C. Billari, A cost valuation model based on a stochastic representation of the IPAT equation, Population and Environment, 29 (2007), 68-82. doi: 10.1007/s11111-008-0061-1.

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