Optimal Abatement Schedules for Excess Carbon Emissions Towards a Net-Zero Target

TL;DR

This work models the available excess carbon budget as a diffusion process $X_t = x + \mu t + \sigma W_t$ and seeks an optimal, non-increasing emission policy $C_t$ to maximize the discounted cumulative emissions plus a sustainability reward $\Lambda$, up to the depletion time $\tau = \inf\{t: X_t^C<0\}$. It develops a rigorous stochastic-control framework where the value function $V^S(x,c)$ solves a Hamilton-Jacobi-Bellman equation and is characterized as the unique viscosity solution, with a detailed treatment of both continuous and discrete emission-rate sets. A key contribution is showing that, starting from a discrete set of admissible rates, the optimal strategies are of threshold type and that the discrete problems converge to the continuous problem, enabling efficient numerical construction of the optimal abatement schedule. The results reveal that gradual, downward ratcheting of emissions yields a small loss relative to unconstrained optimality, with the sustainability parameter $\Lambda$ significantly influencing the value and the shape of the abatement curve, offering practical guidance for net-zero transition policies.

Abstract

Achieving net-zero carbon emissions requires a transformation of energy systems, industrial processes, and consumption patterns. In particular, a transition towards that goal involves a gradual reduction of excess carbon emissions that are not essential for the well-functioning of society. In this paper we study this problem from a stochastic control perspective to identify the optimal gradual reduction of the emission rate, when an allocated excess carbon budget is used up over time. Assuming that updates of the available carbon budget follow a diffusion process, we identify the emission strategy that maximizes expected discounted emissions under the constraint of a non-increasing emission rate, with an additional term rewarding the amount of time for which the budget is not yet depleted. We establish a link of this topic to optimal dividend problems in insurance risk theory under ratcheting constraints and show that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. We provide numerical illustrations of the resulting optimal abatement schedule of emissions and a quantitative evaluation of the effect of the non-increasing rate constraint on the value function.

Figures (4)

• Figure 7.1: Optimal value function $V(x,4)$ (solid line) and unconstrained value function $V_{\text{class}}(x)$ (dashed line) as well as optimal threshold $z^*(c)$ (right) for $\mu=3$, $\sigma=2$, $q=0.1$, $\Lambda=4$ and $S=[0,4]$.
• Figure 7.2: Optimal value function $V(x,2)$ and optimal threshold $z^*(c)$ for $\sigma=1$, $q=0.1$, $\Lambda=1.5$ and $S=[0,2]$ for $\mu=1$ (solid line), $\mu=0.5$ (dashed line), $\mu=0$ (dotted line) and $\mu=-0.5$ (dash-dotted line).
• Figure 7.3: Sample path $X_t^C$ and resulting emission patterns for the parameters of Figure \ref{['Ex2_Str']} with $\mu=0$ for the optimal strategy according to $z^*(c)$ and a linear decreasing emission rate $c(t)=2-0.4t$.
• Figure 7.4: Optimal value function $V(x,2)$ and optimal threshold $z^*$ as a function of $c$ for $\sigma=1$, $q=0.1$, $\mu=1$ and $S=[0,2]$ for $\Lambda=1.5$ (solid), $\Lambda=1$ (dashed), $\Lambda=0.5$ (dotted) and $\Lambda=0$ (dash-dotted).

Theorems & Definitions (28)

• Remark 2.1
• Remark 2.2
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• Remark 3.1
• Definition 3.1
• Remark 3.2
• Proposition 3.1
• Lemma 3.2