Optimal Stochastic Control Problem for a Carbon Emission Reduction Process

In this paper, we study a stochastic optimization control problem for carbon emission reduction with two control variables, one of which is the stochastic emission reduction control strategy, and the other is the auction amount of carbon allowances. We optimize the total emission reduction cost by a two- step process. Fixing the auction amount, we optimize the emission reduction control strategy, and then relate the value function to a Hamilton–Jacobi–Bellman (HJB) equation. The existence and uniqueness of the classical solution of the equation are proved so as to describe the optimal control strategy. After that, we find the optimal auction amount by determining the zero point of the derivative. We then show that the auction amount minimizing the total cost is unique in a special case. Finally, the relationships between the optimal auction amount and the total cost and various parameters are discussed through numerical results.