A Theoretical Framework for Modeling Carbon Emission Allowance Prices: Stochastic Differential Equations in Continuous Time

Abstract

This paper establishes a theoretical framework to model carbon allowance pricing dynamics using advanced stochastic calculus. A continuous-time approach is adopted to formulate Stochastic Differential Equations (SDEs) without relying on empirical data. The price process is represented as a multifractional Brownian motion to capture irregular fluctuations. The drift term incorporates market demand, regulatory impacts, and economic variables through an abstract functional operator. The volatility term contains a Hermite chaos expansion coupled with Malliavin derivatives. Derivations utilize concepts from anticipative calculus, fractional calculus, and functional analysis to attain mathematical sophistication. Theoretical insights into market dynamics and price determinants are provided by synthesizing economic theories and environmental considerations. Overall, the paper offers a mathematically rigorous foundation for understanding carbon market interactions, with implications for further theoretical advancements in environmental economics modeling.