# American Institute of Mathematical Sciences

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

## An optimal control model of carbon reduction and trading

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

Received  October 2015 Revised  January 2016 Published  October 2016

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.
Citation: Huaying Guo, Jin Liang. An optimal control model of carbon reduction and trading. Mathematical Control & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [5] E. Commission, The EU emissions trading system (EU ETS),, 2013, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [9] T. Dietz and E. A. Rosa, Rethinking the environmental impacts of population, affluence and technology, Human Ecology Review, 1 (1994), 277-300. Google Scholar [10] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2 edition, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] R. C. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Sciences, 4 (1973), 141-183. doi: 10.2307/3003143.  Google Scholar [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.  Google Scholar [15] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [19] X. Yang and J. Liang, Minimization of the nation cost due to carbon emission, Systems Engineering - Theory and Practice, 34 (2014), 640-647. Google Scholar [20] X. Yang, Optimal control problems associated with carbon emission abatement and leveraged credit derivatives, Ph. D Thesis, Tongji University, 2015. Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [5] E. Commission, The EU emissions trading system (EU ETS),, 2013, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [9] T. Dietz and E. A. Rosa, Rethinking the environmental impacts of population, affluence and technology, Human Ecology Review, 1 (1994), 277-300. Google Scholar [10] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2 edition, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] R. C. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Sciences, 4 (1973), 141-183. doi: 10.2307/3003143.  Google Scholar [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.  Google Scholar [15] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [19] X. Yang and J. Liang, Minimization of the nation cost due to carbon emission, Systems Engineering - Theory and Practice, 34 (2014), 640-647. Google Scholar [20] X. Yang, Optimal control problems associated with carbon emission abatement and leveraged credit derivatives, Ph. D Thesis, Tongji University, 2015. Google Scholar [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.  Google Scholar
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