Carbon Price Prediction and Risk Assessment Considering Energy Prices Based on Uncertain Differential Equations

Against the backdrop of escalating atmospheric carbon dioxide concentrations, carbon emission trading systems (ETS) have emerged as pivotal policy instruments, with China’s ETS playing a prominent role globally. The carbon price, central to ETS functionality, guides resource allocation and corporate strategies. Due to unexpected events, political conflicts, limited access to data information, and insufficient cognitive levels of market participants, there are epistemic uncertainties in the fluctuations of carbon and energy prices. Existing studies often lack effective handling of these epistemic uncertainties in energy prices and carbon prices. Therefore, the core objective of this study is to reveal the dynamic linkage patterns between energy prices and carbon prices, and to quantify the impact mechanism of epistemic uncertainties on their relationship with the help of uncertain differential equations. Methodologically, a dynamic model of carbon and energy prices was constructed, and analytical solutions were derived and their mathematical properties were analyzed to characterize the linkage between carbon and energy prices. Furthermore, based on the observation data of coal prices in Qinhuangdao Port and national carbon prices, the unknown parameters of the proposed model were estimated, and uncertain hypothesis tests were conducted to verify the rationality of the proposed model. Results showed that the mean squared error of the established model for fitting the linkage relationship between carbon and energy prices was 0.76, with the fitting error controlled within 3.72 % . Moreover, the prediction error was 1. 88 % . Meanwhile, the 5 % value at risk (VaR) of the logarithmic return rate of carbon prices was predicted to be − 0.0369 . The research indicates that this methodology provides a feasible framework for capturing the uncertain interactions in the carbon-energy market. The price linkage mechanism revealed by it helps market participants optimize their risk management strategies and provides more accurate decision-making references for policymakers.