[Paper Review] Optimal Abatement Schedules for Excess Carbon Emissions Towards a Net-Zero Target

Hansjoerg Albrecher, Nora Muler|arXiv (Cornell University)|Jan 16, 2026

Risk and Portfolio Optimization0 citations

TL;DR

The paper analyzes how to optimally reduce excess carbon emissions under a down-ratcheting constraint, modeling the available budget as a diffusion process and solving via stochastic control and HJB methods.

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.

Motivation & Objective

• Motivate net-zero targets by understanding how to optimally exhaust excess carbon budgets with gradual abatement.
• Model the available excess emission budget as a stochastic process and quantify the value of leaving budget unused (Lambda).
• Derive the optimal down-ratcheting emission strategy under a cap on emission rates.

Proposed method

• Formulate the problem as maximizing expected discounted emissions plus a Lambda reward until budget depletion.
• Model X_t as x + μ t + σ W_t and constrained emission rate C_t ∈ S with C_t non-increasing.
• Derive the Hamilton-Jacobi-Bellman (HJB) equation for the continuous set S = [0, c̄] and prove V is the unique viscosity solution.
• Study discrete sets S = {c0, c1, ..., cn} and show convergence of the discrete value functions to the continuous case.
• Prove threshold-type structure in discrete settings and extend to continuum via convergence results.

Experimental results

Research questions

• RQ1 What is the optimal down-ratcheting emission strategy to maximize discounted emissions plus Lambda reward until carbon budget depletion?
• RQ2 How does restricting emission rates to be non-increasing affect the optimal schedule compared to unconstrained or linearly decreasing strategies?
• RQ3 How does the value function behave with respect to initial surplus x and initial emission rate c under the down-ratcheting constraint?
• RQ4 What is the impact of discretizing the emission-rate set on the optimal solution and its convergence to the continuous case?

Key findings

• The optimal strategy is of threshold type and arises from a discretization approach that converges to a continuous solution.
• The value function is the unique viscosity solution of the HJB equation with appropriate boundary conditions.
• A Lambda reward term governs sustainability considerations and can lead to zero threshold (stop emitting) for certain parameter regimes.
• Convergence from discrete to continuous emission-rate sets yields a Lipschitz continuous limit equal to the continuous-time optimal value.
• Quantitative illustrations show how non-increasing-abatement constraints alter the optimal schedule compared to unconstrained benchmarks.