Paper: arXiv 2407.08477

Authors: Xinfu Chen, Yuchao Dong, Wenlin Huang, Jin Liang

Abstract

In this paper, we consider a company can simultaneously reduce its emissions and buy carbon allowances at any time. We establish an optimal control model involving two stochastic processes with two control variables, which is a singular control problem. This model can then be converted into a Hamilton-Jacobi-Bellman (HJB) equation, which is a two-dimensional variational equality with gradient barrier, so that the free boundary is a surface. We prove the existence and uniqueness of the solution. Finally, some numerical results are shown.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework for optimal carbon emission control using a singular control approach, establishing the HJB variational inequality and proving existence and uniqueness. However, it is almost purely theoretical with minimal empirical validation, relying only on basic numerical simulations without real market data backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Optimal Carbon Emission Control & Allowance Purchasing"] --> B["Inputs: Stochastic Processes for Carbon Emissions & Allowance Prices"]
    B --> C["Formulate Model: Singular Control Problem with Two Control Variables"]
    C --> D["Transform to HJB Equation: 2D Variational Inequality with Gradient Barrier"]
    D --> E["Mathematical Proof: Prove Existence & Uniqueness of Solution"]
    E --> F["Numerical Analysis: Solve Free Boundary Surface Computationally"]
    F --> G["Key Outcomes: Optimal Carbon Abatement & Purchasing Policy"]

Cited References (26)

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  6. A Note on Market Power in an Emission Permits Market with Banking (2005)
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and 18 more
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  2. Minimization of Carbon Abatement Cost: Modeling, Analysis and Simulation (2017)
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  5. Controlled Diffusion Processes (1980)
  6. An optimal control model for reducing and trading of carbon emissions (2016)
  7. An optimal control model of carbon reduction and trading (2016)
  8. Optimal Control Strategy of Companies: Inheriting Period and Carbon Emission Reduction (2020) — link
  9. Optimal Investment Timing For Carbon Emission Reduction Technology With A Jump-Diffusion Process (2021)
  10. Optimal control model of an enterprise for single and inheriting periods of carbon emission reduction (2022)
  11. Optimal stochastic control problem for a carbon emission reduction process (2023)
  12. Variational Principles and Free-boundary Problems (1982)
  13. Partial Differential Equations of Parabolic Type (1964)
  14. Linear and Quasi-Linear Equations of Parabolic Type (1968)
  15. Free boundary problem concerning pricing convertible bond (2011)
  16. Second-Order Parabolic Partial Differential Equations (2003)
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  18. Nonconcave utility maximization with portfolio bounds (2022)

0 of 26 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.