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Numerical methods for two person games arising from transboundary pollution with emission permit trading

Author

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  • Lai, Junyu
  • Wan, Justin W.L.
  • Zhang, Shuhua

Abstract

We propose finite difference methods for solving two dimensional Hamilton–Jacobi–Bellman (HJB) equations and systems arising from the modelling of transboundary pollution with emission permits trading. We prove that our numerical scheme for the HJB equations from cooperative game is consistent, stable, and monotone, therefore guarantees convergence to viscosity solutions. To ensure our scheme is fully implicit and unconditionally monotone, we combine wide stencil with narrow stencil in the discretization. We address coupling between unknown variables and emission controls using policy iteration. We solve the coupled systems of HJB equations from non-cooperative game efficiently through policy-like iteration. We give a theoretical analysis of our scheme for solving the coupled system and find that it converges under certain conditions. Finally, we show that our numerical scheme achieves higher order of convergence than the finite volume method proposed originally.

Suggested Citation

  • Lai, Junyu & Wan, Justin W.L. & Zhang, Shuhua, 2019. "Numerical methods for two person games arising from transboundary pollution with emission permit trading," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 11-31.
    • Handle: RePEc:eee:apmaco:v:350:y:2019:i:c:p:11-31
    DOI: 10.1016/j.amc.2018.12.053
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    References listed on IDEAS

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    1. Swathi Amarala & Justin W. L. Wan, 2016. "Numerical methods for dynamic Bertrand oligopoly and American options under regime switching," Quantitative Finance, Taylor & Francis Journals, vol. 16(11), pages 1741-1762, November.
    2. Shuhua Chang & Xinyu Wang & Zheng Wang, 2015. "Modeling and Computation of Transboundary Industrial Pollution with Emission Permits Trading by Stochastic Differential Game," PLOS ONE, Public Library of Science, vol. 10(9), pages 1-29, September.
    3. Seifert, Jan & Uhrig-Homburg, Marliese & Wagner, Michael, 2008. "Dynamic behavior of CO2 spot prices," Journal of Environmental Economics and Management, Elsevier, vol. 56(2), pages 180-194, September.
    4. Shoude Li, 2014. "A Differential Game of Transboundary Industrial Pollution with Emission Permits Trading," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 642-659, November.
    5. Daskalakis, George & Psychoyios, Dimitris & Markellos, Raphael N., 2009. "Modeling CO2 emission allowance prices and derivatives: Evidence from the European trading scheme," Journal of Banking & Finance, Elsevier, vol. 33(7), pages 1230-1241, July.
    6. Copeland, Brian R & Taylor, M Scott, 1995. "Trade and Transboundary Pollution," American Economic Review, American Economic Association, vol. 85(4), pages 716-737, September.
    7. D. W. K. Yeung, 2007. "Dynamically Consistent Cooperative Solution in a Differential Game of Transboundary Industrial Pollution," Journal of Optimization Theory and Applications, Springer, vol. 134(1), pages 143-160, July.
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