Carbon transaction game prediction method based on Stackelberg game and stochastic differential equation

Abstract

The invention provides a carbon transaction game prediction method based on a Stackelberg game and a stochastic differential equation. The method comprises the following steps: step 1, determining marginal revenue functions of both carbon transaction parties; step 2, constructing utility functions of different risk preference carbon traders; 3, calculating the balanced price and quantity of carbon transactions; 4, predicting the future change trend of the market based on the historical carbon quota transaction price; and 5, determining the optimal transaction opportunity based on the future change trend of the market and the balanced transaction price. The invention aims to deeply analyze the optimal decision of the two types of participants, effectively improve the transaction income of the seller enterprise, reduce the transaction cost of the buyer enterprise and remarkably enhance the market mobility by accurately matching the market supply and demand, thereby promoting the efficient operation and sustainable development of the carbon emission permit transaction market.

Description

Technical Field

[0001] This paper belongs to the field of carbon trading technology and aims to find the optimal transaction price, transaction volume and transaction time for carbon emission rights traders. It proposes a solution method that integrates Stackelberg game, GBM, MLE and LSTM to optimize and predict the decision-making behavior of carbon traders, providing theoretical support and practical guidance for the efficient operation of the carbon emission rights market. Background Art

[0002] Climate change, driven by greenhouse gas emissions, has become a global challenge with significant impacts on human society. Observational data show that the global average surface temperature has risen by 1.1°C compared to pre-industrial levels, leading to accelerated sea level rise and more frequent extreme weather events, posing a serious threat to human safety. The carbon emissions trading system, as a market-based emission reduction mechanism, has proven to be an important policy tool for promoting carbon reduction and sustainable development. In a typical carbon emissions trading system, the government first sets an overall emission reduction target and allocates annual carbon emission quotas to key emitting enterprises based on this target. By implementing quota trading and penalties for excess emissions, the government establishes a strong incentive and constraint mechanism for emission reduction in the carbon market. Enterprises that achieve excess emission reductions can profit by selling their remaining quotas, while enterprises that exceed their emission limits must purchase additional quotas to meet their compliance requirements. This mechanism encourages enterprises to adopt more efficient production technologies to reduce costs, fostering a societal awareness of low-carbon development based on the principle that "emissions have costs, and emission reductions have rewards," thereby promoting energy structural adjustments.

[0003] In China's existing carbon emissions trading market, sellers dominate the market, strategically initiating trades by listing the price and quantity of their carbon allowances. Buyers then observe the price and quantity offered by the sellers, strategically evaluate the transaction based on the current average market price, make a purchase decision, and request the trading volume from the sellers. This listing agreement model reflects a leader-follower relationship and a dynamic game process between the two parties. Accurately predicting the optimal trading time, price, and volume within the dynamic interaction between buyers and sellers remains a key challenge that has yet to be effectively addressed. The Stackelberg game model can simulate the hierarchical decision-making process of different types of participants in the carbon emissions trading market, providing an effective approach for studying market participant behavior and revealing market dynamics. While relevant research has utilized various game models to deeply analyze the interactions among different participants in the carbon emissions trading market and their impact on emissions reduction outcomes, most focus on analyzing participants' decisions regarding whether to trade carbon allowances, neglecting specific issues such as when to trade, the quantity to trade, and the price to trade, which are crucial for market participants.

[0004] Therefore, the present invention adopts the Stackelberg game model to determine the optimal transaction price and quantity of carbon quotas when the participants maximize their utility. Secondly, the geometric Brownian motion (GBM) or long short-term memory neural network (LSTM) is used to predict the future fluctuations of carbon prices, and the maximum likelihood estimation (MLE) is applied to calibrate the model parameters, so as to derive the optimal transaction time for both parties based on the dynamic evolution of prices. In addition, considering that the carbon emission trading market is affected by uncertain factors such as policy changes, the present invention analyzes how participants can achieve the optimal transaction volume, price and time to maximize utility under different risk preferences. This provides participants with a more comprehensive decision-making perspective and is also the basis for improving market efficiency. Figure 1 This paper demonstrates the solution to the optimal trading method using the Stackelberg game model, GBM, MLE, and LSTM. The main contributions of this invention are as follows:

[0005] 1. In terms of research questions, the focus is on the micro-decision-making behavior of carbon emission trading market participants in a dynamic trading environment. The optimal carbon quota trading price, trading volume, and trading time for buyers and sellers will be studied to provide recommendations for market participants' trading decisions.

[0006] 2. Based on the current carbon emissions trading market mechanism and empirical data, a multi-objective Stackelberg game was implemented, and the dynamic evolution of carbon prices in the market was predicted by combining GBM, MLE, and LSTM, thereby obtaining the optimal trading decisions of participants.

[0007] 3. Taking into account the multiple uncertainties of real situations and the different risk preferences of participants, the present invention incorporates the subjective risk preferences of participants into the model framework, so that the results are more consistent with actual market behavior. Summary of the Invention

[0008] Technical problems solved by the present invention:

[0009] In light of the unique characteristics of carbon emissions trading, this paper proposes a carbon trading game prediction method based on Stackelberg game theory and stochastic differential equations. This method aims to deeply analyze the optimal decisions of both types of participants. By precisely matching market supply and demand, it effectively increases trading revenue for sellers, reduces transaction costs for buyers, and significantly enhances market liquidity, thereby promoting the efficient operation and sustainable development of the carbon emissions trading market.

[0010] The solution of the present invention:

[0011] Step 1: Determine the marginal revenue function of both parties in the carbon trading. Assume that the trading cycle of the carbon emission trading market is [0, T], and divide it into N equal sub-intervals, where N is a positive integer. Therefore, market participants have N+1 possible trading time points, that is, the set {0, t1,…, t n ,…,t N-1 ,T}. The buyer plans to purchase q units of carbon quota and waits for the seller's bid to fall to its expected level or below. At this time, the following two situations may occur:

[0012] 1) At t i At time (i=1,2,…,N-1), the seller's bid price drops to the buyer's expected level and the transaction is completed.

[0013] 2) Until t N-1 At time t, the seller's bid still does not meet the buyer's expectations, and the buyer waits for time T to make a decision. At time T, there are two possible outcomes: the seller's bid meets the buyer's expectations and the buyer completes the transaction; the seller's bid is still higher than the buyer's expectations and the buyer chooses to give up the transaction.

[0014] The seller plans to sell q units of carbon quota and determines its bid price based on the market price and net profit. At this point, two situations may occur:

[0015] 1) At t i At time (i=1,2,…,N-1), the seller's bid is accepted by the buyer and the transaction is completed.

[0016] 2) Until t N-1 At time t, the seller's bid is still not accepted by the buyer, so at time T, the seller adjusts its bid. There are two possible outcomes at time T: the seller's bid is accepted by the buyer and the transaction is completed; the seller's bid is still not accepted by the buyer, and the seller abandons the transaction and carries over q units of carbon quota to the next trading cycle. Therefore, the marginal revenue functions of the buyer and seller are:

[0017]

[0018] Among them, alnq represents the increase in net product revenue due to the increase in unit carbon emissions. is an indicative function, which is 1 when t=T and 0 otherwise. t and q represent the transaction price and transaction quantity of both parties respectively, F is the penalty for the buyer’s excess emission of carbon quota per unit during the settlement period, G represents the government subsidy given to the seller for selling carbon quota per unit, and B sis the potential profit that the seller can obtain by carrying over the unsold carbon quotas at time T to the next cycle for sale, θ1 and θ2 represent the probability of the buyer purchasing carbon quotas and the probability of the seller selling carbon quotas at time T, respectively, a represents the incremental net product revenue brought about by the increase in unit carbon emissions, n represents the depreciation coefficient of emission reduction equipment, and m represents the fuel conversion cost of reducing unit carbon emissions.

[0019] Step 2: Construct the utility function of carbon traders with different risk preferences. Carbon market participants usually have different risk attitudes, and the utility functions of participants with different risk attitudes show different concavity and convexity. Specifically, when the participant is risk-averse, the utility function is concave; when the participant is risk-seeking, the utility function is convex; when the participant is risk-neutral, the utility function is linear. b and U s The utility functions of the buyer and seller companies are represented by logarithmic utility function ln(x), linear utility function kx+b and quadratic utility function respectively. To characterize the risk aversion, risk neutrality and risk love of participants.

[0020] When the buyer is risk-averse, his utility function is: U t (q,p t )=ln(V b (q,p t )), where U′ b >0,U″ b <0;

[0021] When the buyer is risk-neutral, his utility function is: U b (q,p t )=k×V b (q,p t )+b, where U′ b =0;

[0022] When the buyer is risk-loving, his utility function is: Among them, U′ b >0,U″ b > 0. The seller’s utility function U s (q,p t ) are:

[0023] When the seller is risk-averse, U s (q,p t )=ln(V s (q,p t )), where U′ s >0,U″ s <0;

[0024] When the seller is risk-neutral, U s (q,p t )=k×V s (q,p t )+b, where U′ s =0;

[0025] When the seller is risk-loving, Among them, U′ s >0,U″ s >0.

[0026] Step three: Calculate the equilibrium price and quantity of carbon trading. Given that the two parties in a carbon trading transaction have a master-slave relationship, and participants with different risk profiles have different marginal utility functions, this paper uses the Stackelberg game model and solves it using backward induction to obtain the equilibrium transaction price and quantity for different combinations of risk profiles between buyers and sellers. The solution process is described in Part 2 of the Detailed Implementation: Dynamic Game between Participants with Different Risk Preferences.

[0027] Step 4: Predict future market trends based on historical carbon allowance transaction prices. In the carbon emissions trading market, carbon price fluctuations are influenced by a variety of complex factors, necessitating the selection of an appropriate model for simulation and prediction. Geometric Brownian motion (GBM) is a commonly used simulation method that effectively captures the random fluctuations of carbon prices. However, the use of GBM presupposes that historical data satisfy the normal distribution assumption, thus requiring a normality test. Common normality tests include the Shapiro-Wilk test and the Kolmogorov-Smirnov test (KS test). The Shapiro-Wilk test is well-suited to small sample data and highly sensitive to distributional variations, enabling a more accurate determination of whether data conform to a normal distribution in small sample settings. In contrast, the Kolmogorov-Smirnov test is more advantageous when working with large sample data, providing stable and reliable results. If the historical data fails the normality test, this indicates that carbon price fluctuations may be affected by nonlinear factors or long-term memory effects. In this case, an LSTM can be used to fit the carbon price trend. As a deep learning model, LSTM can effectively handle long-term dependencies in time series data and is suitable for predicting complex and fluctuating carbon prices. The specific prediction steps are as follows:

[0028] 4.1. Input historical data sequence of carbon quota transaction price: Collect historical price data of carbon quota trading market to ensure the integrity and accuracy of data. If the historical data sequence S = {S(0), S(1), ..., S(n)} is known, n+1 represents the total number of historical data collected. Calculate the logarithmic rate of return of the historical data sequence {E(1), E(2), ..., E(n)},

[0029] 4.2. Determine the scale of historical data and select an appropriate normality test method:

[0030] If the data sample size n is small (usually less than 30), use the Shapiro-Wilk test, which has good adaptability to small sample data and high distribution sensitivity.

[0031] If the data sample size n is large, the Kolmogorov-Smirnov test (KS test) is used. This method has more advantages when processing large sample data.

[0032] 4.3. Perform normality test:

[0033] The Shapiro-Wilk test or KS test was used to test the normality of the historical data of carbon quota transaction prices.

[0034] If the test results show that the data conforms to the normal distribution, the geometric Brownian motion (GBM) model can be used for simulation; otherwise, consider using the long short-term memory network (LSTM) model.

[0035] The detailed test process is described in the third section of the specific implementation method. Next, the process of performing a normality test on the sample {E(1), E(2), …, E(n)} is described. The specific test processes of the Shapiro-Wilk test and the Kolmogorov-Smirnov test are described respectively.

[0036] (1) Shapiro-Wilk test

[0037] Data sorting: Arrange {E(1), E(2),…, E(n)} in ascending order to obtain an ordered sample.

[0038] Calculate constant: Calculate constant a according to formula (3) i

[0039]

[0040] Where m=(m1,…,m n ) T is the expected value of ordered independent and identically distributed statistics sampled from standard normally distributed random variables, and V is the covariance matrix of these ordered statistics.

[0041] Calculate statistics: Calculate statistics W according to formula (4)

[0042]

[0043] in, is the sample mean.

[0044] Calculate the critical value or p-value: According to n and the significance level α (such as 0.05), look up the Shapiro-Wilk critical value table to get W α , by combining the statistic W with W α Comparing the analysis results, if W<W α If p < 0.05, the null hypothesis is rejected, and the sample data are considered to be non-normally distributed. Alternatively, use PyCharm to run scipy.stats.shapiro(E) to calculate the corresponding p-value, where E is the logarithmically processed sample data sequence {E(1), E(2), …, E(n)}. If p < 0.05, the null hypothesis is rejected; otherwise, the data are considered to be normally distributed.

[0045] (2) Kolmogorov-Smirnov test

[0046] Data sorting: Arrange the sample data {E(1), E(2),…, E(n)} in ascending order.

[0047] Calculate the empirical distribution function: For each data point E(i), calculate its empirical distribution function value in the sample Where n is the sample size and i is the serial number of the data point (from 1 to n).

[0048] Calculate the theoretical distribution function: Based on the assumed theoretical distribution (such as normal distribution), calculate the theoretical distribution function value F(E(i)) corresponding to each data point S(i).

[0049] Calculate statistics: Find the maximum absolute difference between the empirical distribution function and the theoretical distribution function, that is, D = max 1≤i≤n |F n (E(i))-F(E(i))|.

[0050] Determine the critical value or p-value: Based on the sample size n and the significance level (usually 0.05), check the critical value table of the Kolmogorov-Smirnov test and obtain the corresponding critical value D a If D>D a If p < 0.05, the null hypothesis is rejected, and the data are considered to be non-normally distributed. Alternatively, use PyCharm to run scipy.stats.kstest(E) to calculate the corresponding p-value, where E is the logarithmically processed sample data sequence {E(1), E(2), …, E(n)}. If p < 0.05, the null hypothesis is rejected; otherwise, the data are considered to be normally distributed.

[0051] 4.4. If the normality test is passed, use the Geometric Brownian Motion (GBM) model:

[0052] In a specific embodiment, the sample data passes the normality test, i.e., the geometric Brownian motion is used to predict the future price trend. Figure 3 The fitting results of predicting carbon price trends using geometric Brownian motion are presented.

[0053] Parameter estimation: Use MLE to calculate the GBM model parameters μ (drift rate) and σ (volatility). First, substitute S into equation (13) in the third section of the detailed implementation to obtain the likelihood function. Then, use equations (14)-(16) to obtain the estimated values of μ and σ.

[0054] Partial differential equation solution: Based on the partial differential equation of the GBM model, the trend of future carbon price changes is predicted by numerical methods or analytical solutions. Substitute the parameters μ and σ into Where P(0) is the current market carbon price, and P(t) represents the predicted price t units away from the current price. By changing the value of t, the trend of future carbon price changes can be obtained.

[0055] 4.5. If the normality test fails, use the long short-term memory network (LSTM) model:

[0056] Data preprocessing: normalize the historical data S = {S(0), S(1), ..., S(n)} according to Among them S min Indicates the minimum value in historical data, S max The maximum value in the historical data is used to improve the efficiency and stability of model training.

[0057] Model construction: Build an LSTM model and select an appropriate network structure, including the number of hidden layer nodes, activation function (such as tanh), optimizer (such as Adam), etc.

[0058] Model training: Use historical data to train the LSTM model and adjust the model's hyperparameters (such as learning rate, number of iterations, batch size, etc.) to optimize model performance.

[0059] Predicting future carbon prices: Use the trained LSTM model to predict future carbon prices.

[0060] 4.6. Model evaluation: Use indicators such as mean square error (MSE), mean absolute error (MAE), and root mean square error (RMSE) to evaluate the prediction performance of the model. The mean square error is calculated as follows: Where n is the number of samples, S(i) is the true value of the i-th sample, is the predicted value of the i-th sample. It is generally believed that a prediction result with a mean square error (MSE) less than 0.01 is acceptable. In the specific implementation, the present invention selects a price fitting path that minimizes the mean square error (MSE), as shown in the third part of the specific implementation.

[0061] Step 5: Determine the optimal trading time based on future market trends and equilibrium trading prices. After identifying the price trends in the carbon trading market, analyze the moments when equilibrium trading prices appear in the market under different scenarios to accurately determine the optimal trading time.

[0062] The advantages and beneficial effects of the present invention are:

[0063] 1. Improving seller profits and reducing buyer costs: This invention significantly increases sellers' trading profits while effectively reducing buyer costs through precise market participant behavior analysis and optimization algorithms. This win-win approach not only enhances the enthusiasm of market participants but also promotes the healthy development of the carbon emissions trading market.

[0064] 2. Enhanced market liquidity: The integration of the Stackelberg game model, GBM, MLE, and LSTM algorithms can optimize market supply and demand matching and improve trading efficiency. This optimization mechanism helps enhance the liquidity of the carbon emissions trading market, reduce trading friction, and improve the overall operational efficiency of the market.

[0065] 3. Adaptability to Participants with Different Risk Preferences: This invention fully considers the diverse needs and behavioral characteristics of participants with different risk preferences. Through flexible model design and algorithm optimization, it provides optimal decision-making solutions for market participants with different risk preferences. This inclusive design enables the invention to be widely applied in the complex and ever-changing carbon emissions trading market, demonstrating its strong practicality and universality. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 It is a flowchart for solving the optimal transaction quantity, price and timing in the carbon trading market.

[0067] Figure 2 This is a comparison chart of the actual carbon price and the carbon price fitted using geometric Brownian motion.

[0068] Figure 3 It is 1000 model paths of future carbon prices in Beijing's carbon emission trading market (based on geometric Brownian motion).

[0069] Figure 4(a)-Figure 4(d) It is the relationship between the optimal transaction quantity, price and timing and the changes in subsidies, emission reduction costs, depreciation coefficients and unit emission benefits. DETAILED DESCRIPTION

[0070] The simulation model data is based on market transaction data published on the official website of the Beijing Emissions Trading Center. The main simulation tool used in this paper is PyCharm Community Edition 2024.3.

[0071] ①Analysis of carbon market trading entities:

[0072] Currently, the secondary market for carbon emissions trading primarily consists of buyer and seller companies. Buyer companies are those whose carbon emissions from production and operations exceed their government-allocated quotas. To avoid financial penalties or legal sanctions for excess emissions, they purchase carbon emissions rights in the carbon emissions trading market to make up the difference. Seller companies are those that, through technological advancements or the implementation of emission reduction projects, have increased their carbon emissions quotas to exceed their own emissions. They sell these excess carbon emissions rights as commodities in the carbon emissions trading market to increase their overall profits. In this market, sellers first determine the quantity and price of carbon emissions rights they wish to sell based on the market carbon price and then list them on the market. Buyers then decide whether to purchase based on the seller's offer. If the current carbon price exceeds the buyer's expectations, the buyer may decide to wait and decide, and the seller will adjust its offer further. Conversely, if the price meets the buyer's expectations, the transaction is concluded.

[0073] For the buyer's company, in its actual production process, as the carbon quota increases, the company may face factors such as technical bottlenecks and equipment limitations, which will cause the rate of increase in output to gradually slow down. The logarithmic function can better describe this nonlinear relationship, indicating that in the early stage of the increase in carbon quotas, the output growth is faster, but as the carbon quota continues to increase, the rate of output growth will gradually decrease. Therefore, alnq is used to represent the increase in net product revenue brought about by the increase in unit carbon emissions. For the seller's company, its total emission reduction cost is composed of the depreciation cost of the clean energy equipment used by the company for production. And the premium cost mq borne by enterprises when they use expensive clean energy instead of low-cost non-clean energy for power generation. To encourage enterprises to actively sell carbon trading and increase market activity, it is assumed that the government will provide subsidies G to sellers when they sell carbon quotas in the market.

[0074] ② Dynamic game between entities with different risk preferences:

[0075] In the carbon emissions trading market, a dynamic game exists between buyers and sellers, with sellers acting as leaders and buyers acting as followers. The two parties influence each other and ultimately reach an equilibrium state that maximizes their respective profits. Therefore, this paper constructs Stackelberg game models for market participants under different risk attitude combinations to obtain equilibrium transaction prices and quantities under different scenarios. First, we analyze the case where both buyers and sellers are risk-neutral participants. In this case, the utility functions of both parties are:

[0076]

[0077] Among them, k represents the slope of the utility function, which reflects the influence of the marginal benefit function on the utility function; b is the intercept, which represents the utility value when the marginal benefit function is zero, and it reflects the basic utility level of the decision maker in the absence of marginal benefits.

[0078] Since the marginal benefits of participants are different when the transaction time is t<T and t=T, the present invention will analyze the equilibrium transaction price and quantity under the two situations of t<T and t=T respectively.

[0079] (1) When t<T; the seller first bids p t , then the buyer's utility function is formula (5). t Under known conditions, by taking the partial derivative of Equation (5) with respect to q and setting the derivative equal to 0, we can obtain the carbon quota purchase quantity that maximizes the buyer's utility, that is:

[0080]

[0081] Simplifying formula (7), we can further obtain Will Substituting the buyer's utility function (see formula (4)), we can obtain:

[0082]

[0083] For formula (8) with respect to p t The optimal carbon quota sales quantity that maximizes the seller's benefits can be obtained by taking the partial derivative:

[0084]

[0085] Solving equation (7) yields the equilibrium transaction price of the participants: The equilibrium number of transactions is

[0086] (2) When t = T, the seller first bids p T , then the buyer's utility function is formula (5). T It represents the listed price declared by the seller enterprise in the carbon trading market at time T.T Under known conditions, by taking the partial derivative of Equation (5) with respect to q and setting the derivative equal to 0, we can obtain the carbon quota purchase quantity that maximizes the buyer's utility, that is:

[0087]

[0088] By solving formula (10), we can get: Will Substitute into the seller's utility function (6), and about p T Find the partial derivative and set it equal to 0. The optimal transaction price of the market participants at time T is: The optimal transaction quantity is

[0089] Based on the above analysis, we obtain the optimal transaction price and quantity of both parties when both market participants are risk neutral. When t<T, when both the buyer and the seller are risk averse, the utility functions of both parties are U b (q,p t )=ln(V b (q,p t )) and U s (q,p t )=ln(V s (q,p t )), satisfying U′ b >0,U″ b <0,U′ s >0,U″ s <0. We also use the reverse induction method to solve the optimal transaction price and quantity in this case. First, for U b (q,p t ) Take the partial derivative with respect to q, that is

[0090]

[0091] Since Ub>0, then in formula (11) Therefore, to make equation (11) equal to 0, it is necessary to satisfy Right now The optimal transaction quantity obtained at this time is the same as when the buyer is risk-neutral. Substitute the seller's utility function and calculate U s (p t )About p t Find the partial derivative and set the derivative equal to 0;

[0092]

[0093] By U′ s >0, then Right now We found that when both parties are risk-averse, the optimal transaction price and quantity are the same as when they are risk-neutral. Similarly, the above conclusion holds true when both the buyer and seller are risk-loving, or when one party is risk-loving and the other is risk-averse or risk-neutral. In other words, the optimal transaction price and quantity are independent of the risk preferences of the parties involved.

[0094] ③Optimal trading timing prediction:

[0095] In the carbon emission trading market, due to the differences in marginal revenue functions of different participants, the corresponding equilibrium transaction prices are also different. These diverse equilibrium prices together shape the real-time average transaction price of the carbon market. By using historical data to predict the trend of the market average transaction price, the dynamic changes of the average transaction price can be captured in real time. For any specific buyer or seller, they hope to obtain a specific intended price. Buy or sell carbon quotas. Their probability of transaction increases as the average market transaction price approaches their intended price. Therefore, it is necessary to determine whether a single buyer or seller can The timing of the transaction requires their intended price By comparing it with the average transaction price in the market, we can identify the time when a transaction is most likely to occur, thus providing guidance for market participants' trading decisions.

[0096] Geometric Brownian motion (GBM) is a key model used in the financial market to predict stock prices. Although the volatility of stock prices is difficult to accurately predict, with the help of mathematical models such as GBM, possible future price trends can be simulated based on historical data. The core of the GBM model is to consider the drift and volatility of stock prices. These two parameters help to reveal the average return and price volatility characteristics of stocks. The carbon trading market is similar to the stock market in terms of trading models and other aspects. In view of this, the present invention intends to apply geometric Brownian motion to price prediction in the carbon emission trading market, in order to provide decision-making guidance for traders. However, there are differences in carbon trading market policies in different countries, and not all carbon markets are suitable for this method. In actual applications, based on the historical market price data, if the data does not pass the normality test, the currently popular time series prediction model, long short-term memory network, is used to predict the future trend of carbon price changes.

[0097] Taking the Beijing carbon emission trading market as an example, the present invention selects the daily average transaction price from January 5, 2023 to December 31, 2024 as a historical data sample. Through a rigorous data processing process, the corresponding logarithmic rate of return is calculated, and the KS test is used to test the normality of its distribution characteristics. The test results show that the obtained p-value is 0.4652220298316395, which is greater than 0.05. This result strongly supports the original hypothesis that the logarithmic rate of return of the sample data shows a normal distribution characteristic. Based on this important conclusion, the present invention further adopts the geometric Brownian motion model to accurately simulate the price evolution path, providing strong theoretical support and reliable quantitative analysis basis for in-depth exploration of the price dynamics of the Beijing carbon emission trading market.

[0098] Since the average market transaction price satisfies the stochastic differential equation dS(t)=μS(t)dt+σS(t)dB t , where μ is the drift rate, σ is the volatility, and B t is a standard Brownian motion, then the logarithmic return of the price The mean is The variance is σ 2 Therefore, the maximum likelihood function can be expressed as:

[0099]

[0100] Taking the logarithm of the likelihood function (13), we get:

[0101]

[0102] Taking partial derivatives of Equation (14) with respect to μ and σ respectively, and setting the partial derivatives equal to 0, we can obtain:

[0103]

[0104] Solving (15), we can obtain μ and σ as follows:

[0105]

[0106] Figure 2 The comparison between the carbon price data trend based on geometric Brownian motion fitting and the actual carbon price data trend is shown. After successfully obtaining the parameters of the stochastic differential equation, 1000 potential carbon price paths were further generated with the help of the geometric Brownian motion model, which are specifically presented in Figure 3 By calculating the mean absolute error, the optimal fitting path is selected, and the optimal trading time is accurately determined based on the moment when the predicted price is equal to or closest to the equilibrium price, thereby providing strong data support and scientific basis for carbon emission trading decisions. The parameter values are shown in Table 1:

[0107] Table 1: Parameter values

[0108]

[0109]

[0110] The parameters are set as follows: According to Beijing regulations, companies that exceed their emissions quotas during the period-end settlement will be fined 3 to 5 times the average market price, but not more than 250 yuan per ton. Given that the average price of Beijing's carbon emissions trading market in 2024 was 108.76 yuan per ton, the unit penalty coefficient F is set at 250 yuan per ton. Considering Midea's carbon emissions data, the company's total emissions in 2023 were 2,298,311 tons, with a net profit of 33.72 billion yuan. The calculated contribution of each unit of carbon emissions to net profit is 14,671.64 yuan. Therefore, assume a = 14,671.64 yuan per ton. Set the unit fuel conversion cost m = 118.5 yuan per ton. Given the government's significant penalties for companies with excessive emissions, it is assumed that most companies will purchase sufficient carbon allowances before the settlement period to avoid high fines. Therefore, the probability θ1 of a buyer purchasing a carbon allowance at time T is set at 1. Assuming the depreciation coefficient of emission reduction equipment, n, is 0.6, and the government subsidy to the seller for each unit of carbon quota sold, G, is 200 yuan / ton. Since there are 248 trading days in 2025, the total compliance period is set to T = 1, and the number of trading days, N, is set to 248. The time interval between each transaction is calculated as dt = T / N.

[0111] ④Numerical results analysis:

[0112] Figure 4(a)-Figure 4(d) It shows how the optimal trading method changes with the changes in the government's unit subsidy G to the seller's enterprise, the fuel conversion cost m for the seller to reduce unit carbon emissions, the depreciation coefficient n of the emission reduction equipment, and the average benefit a per unit carbon emission for the buyer, without considering the risk preferences of market participants. According to Figure 4(a), as the government's subsidy G for the seller's unit carbon quota increases, the optimal transaction quantity q * This shows that the increase in subsidies has encouraged sellers to participate more actively in emission reduction and sell more carbon quotas in the market to obtain more subsidies. It decreases as the subsidy G increases. This is because the seller companies are willing to sell carbon quotas at a lower price in order to obtain more subsidies. The reduction in the seller's bid attracts the buyer companies to purchase more carbon quotas. However, the marginal benefit of buyers purchasing carbon quotas is decreasing, that is, as the purchase volume increases, the additional benefits brought by each additional unit of carbon quota gradually decreases. When the buyer's demand reaches a certain level, the demand tends to stabilize due to the decreasing marginal benefits. This may lead to an oversupply in the market, and the buyer companies may choose to postpone the transaction in the hope that the price of carbon quotas will fall further. This postponement behavior leads to the optimal transaction time t * It shows an overall upward trend. It can be seen that increasing the government subsidy G for sellers in the carbon emission trading market to sell unit carbon quotas is conducive to lowering the market carbon price, but when the subsidy increases to a certain level, it may affect the market activity.

[0113] Figures 4(b) and 4(c) show the changes in the optimal trading method when the seller's emission reduction cost increases. Specifically, when the seller's fuel conversion cost m and the emission reduction equipment depreciation coefficient n increase, the optimal trading method is affected as follows: As the seller's emission reduction cost increases, the optimal trading quantity q * This is because sellers tend to reduce the number of carbon quotas sold in the market in order to maintain their profit level when facing higher emission reduction costs. Corresponding to the reduction in the number of carbon quota transactions, the optimal transaction price will rise accordingly. That is, as sellers' emission reduction costs increase, the number of carbon allowances they sell decreases, resulting in a supply shortage in the market. This imbalance between supply and demand drives up the price of carbon allowances. Faced with higher carbon allowance prices, buyers are willing to pay higher prices to purchase carbon allowances to avoid the high fines they face for failing to meet emission reduction requirements. This willingness to pay further drives up the market equilibrium price. When determining the optimal time to purchase carbon emission rights, market participants consider the difference between the equilibrium price and the initial carbon price. If the equilibrium price is lower than the initial carbon price, sellers are more likely to postpone sales in the hope that the price will recover; if the equilibrium price is higher than the initial carbon price, buyers may postpone purchases in the hope that the price will fall. Therefore, the larger the gap between the equilibrium price and the initial carbon price, the greater the tendency of market participants to postpone transactions; conversely, the smaller the gap, the more likely market participants are to trade promptly. In Figure 4(b), when the parameter m is set to 120, the equilibrium transaction price is approximately 110, close to the initial carbon price we set, and the optimal purchase time for market participants is at its lowest. As the equilibrium transaction price increases, the optimal buying opportunities for market participants also increase.

[0114] Figure 4(d) shows the change in the optimal transaction method when the buyer's net product revenue increment a increases due to the increase in unit carbon emissions. This shows that when the buyer's marginal revenue from purchasing unit carbon quotas increases, the optimal transaction price will be Decreases, but does not change the optimal transaction quantity q * This is because the marginal benefit of purchasing carbon allowances decreases as the purchase volume increases. Even if a increases, once the number of carbon allowances purchased exceeds a certain threshold, the marginal benefit will stabilize, so the total amount of carbon allowances purchased by the buyer remains unchanged. However, an increase in buyer benefits generally means an increase in product prices in the market, which in turn affects the seller's cost structure. As a result, sellers may increase their bids, leading to an increase in the equilibrium transaction price.

Claims

1. A carbon trading game prediction method based on Stackelberg game and stochastic differential equation, characterized in that: It includes the following steps: Step 1: Determine the marginal revenue function of both parties in the carbon trading. Assume that the trading cycle of the carbon emission trading market is [0, T], and divide it into N equal sub-intervals, where N is a positive integer. Market participants have N+1 possible trading time points, that is, the set {0, t1,…, t n ,…,t N-1 ,T}; The buyer plans to purchase q units of carbon quotas and waits for the seller's bid to drop to or below its expected level; Step 2: Construct the utility function of carbon traders with different risk preferences. When the participants are risk-averse, the utility function is a concave function; when the participants are risk-averse, the utility function is a convex function; when the participants are risk-neutral, the utility function is a linear function. b and U s The utility functions of the buyer and seller companies are represented by logarithmic utility function ln(x), linear utility function kx+b and quadratic utility function respectively. To characterize the risk aversion, risk neutrality and risk love of participants; Step 3, calculate the equilibrium price and quantity of carbon trading; use the Stackelberg game model, and solve the model by backward induction to obtain the equilibrium trading price and quantity of both buyers and sellers under different combinations of risk attitudes; Step 4, predict the future market change trend based on the historical carbon quota transaction price; use geometric Brownian motion (GBM) to capture the stochastic volatility characteristics of carbon prices; at the same time, the historical data needs to satisfy the normal distribution, and the normality test methods include the Shapiro-Wilk test and the Kolmogorov-Smirnov test (i.e., the K-S test); if the historical data fails the normality test, it indicates that the change of carbon price may be affected by non-linear factors or long-term memory effects. In this case, use LSTM to fit the change trend of carbon price; Step 5, determine the optimal trading opportunity based on the future market change trend and the equilibrium trading price; after clarifying the change trend of the carbon trading market price, analyze the emergence time of the equilibrium trading price in the market under different scenarios, and then accurately determine the optimal trading opportunity.

2. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 1 is characterized by: In Step 1, the following two situations occur: 1) At t i At time (i=1, 2, ..., N-1), the seller's bid price drops to the buyer's expected level and the transaction is completed; 2) Until t N-1 At time T, the seller's bid still does not meet the buyer's expectations, and the buyer waits for time T to make a decision. At time T, there are two possible results: the seller's bid meets the buyer's expectations and the buyer completes the transaction; the seller's bid is still higher than the buyer's expectations and the buyer chooses to give up the transaction.

3. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 1 or 2, characterized in that: In Step 1, when the seller plans to sell q units of carbon quota and determines its asking price by combining the market price and net profit; at this time, two situations will occur: 1) At t i At time (i=1, 2, ..., N-1), the seller's bid is accepted by the buyer and the transaction is completed; 2) Until t N-1 At time t, the seller's bid is still not accepted by the buyer, so at time T, the seller adjusts the bid. There are two results at time T: the seller's bid is accepted by the buyer and the transaction is completed; the seller's bid is still not accepted by the buyer, and the seller abandons the transaction and carries over q units of carbon quota to the next trading cycle. Therefore, the marginal revenue functions of the buyer and seller are: Among them, alnq represents the increase in net product revenue due to the increase in unit carbon emissions. is an indicator function, which is 1 when t=T and 0 otherwise; t and q represent the transaction price and transaction quantity of both parties respectively, F is the penalty for the buyer’s excess emission of carbon quota per unit during the settlement period, G represents the government subsidy given to the seller for selling carbon quota per unit, and B s is the potential profit that the seller can obtain by carrying over the unsold carbon quotas at time T to the next cycle for sale, θ1 and θ2 represent the probability of the buyer purchasing carbon quotas and the probability of the seller selling carbon quotas at time T, respectively, a represents the incremental net product revenue brought about by the increase in unit carbon emissions, n represents the depreciation coefficient of emission reduction equipment, and m represents the fuel conversion cost of reducing unit carbon emissions.

4. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 1 is characterized by: In step 2, when the buyer is risk-averse, his utility function is: U b (q,p t )=ln(V b (q,p t )), where U′ b >0,U″ b <0; When the buyer is risk-neutral, his utility function is: U b (q,p t )=k×V b (q,p t )+b, where U′ b =0; When the buyer is risk-loving, his utility function is: Among them, U′ b >0,U″ b >0; and the seller’s utility function U s (q,p t ) are: When the seller is risk-averse, U s (q,p t )=ln(V s (q,p t )), where U′ s >0,U″ s <0; When the seller is risk-neutral, U s (q,p t )=k×V s (q,p t )+b, where U′ s =0; When the seller is risk-loving, Among them, U′ s >0,U″ s >0.

5. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 1 is characterized by: In Step 3, construct Stackelberg game models of market participants under different combinations of risk attitudes respectively to obtain the equilibrium trading prices and quantities in different situations; first, analyze the situation where both the buyer and seller enterprises are risk-neutral participants. At this time, the utility functions of both sides are: where k represents the slope of the utility function, reflecting the influence degree of the marginal revenue function on the utility function; b is the intercept, indicating the utility value when the marginal revenue function is zero, which reflects the utility level of the decision maker without marginal revenue.

6. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 5 is characterized by: In Step 3, since the marginal revenues of the participants at trading times t < T and t = T are different, analyze the equilibrium trading prices and quantities in the two cases of t < T and t = T respectively; (1) When t < T; the seller bids first with price p t , then the buyer's utility function is given by Equation (5); under the condition that p t is known, by taking the partial derivative of Equation (5) with respect to q and setting the derivative equal to 0, the carbon quota purchase quantity that maximizes the buyer's utility is obtained, i.e.: Simplifying formula (7), we can further obtain Will Substituting into the buyer's utility function we get: For formula (8) with respect to p t Find the partial derivative to obtain the optimal carbon quota sales quantity that maximizes the seller's benefits: Solving equation (7) yields the equilibrium transaction price of the participants: The equilibrium number of transactions is (2) When t = T, the seller first bids p T , then the buyer's utility function is formula (5); p T represents the listed price declared by the seller enterprise in the carbon trading market at time T; T Under known conditions, by taking the partial derivative of Equation (5) with respect to q and setting the derivative equal to 0, we can obtain the carbon quota purchase quantity that maximizes the buyer's utility, that is: By solving formula (10), we can get Will Substituting the seller's utility function (6) and taking the partial derivative with respect to pT, setting the partial derivative equal to 0, we can obtain the optimal transaction price of the market participant at time T: The optimal transaction quantity is 7. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 1 is characterized by: In step 4, input the historical data sequence of carbon quota transaction prices: collect historical price data of carbon quota trading market to ensure the integrity and accuracy of the data; if the historical data sequence S = {S(0), S(1), ..., S(n)} is known, n+1 represents the total number of historical data collected; calculate the logarithmic rate of return of the historical data sequence {E(1), E(2), ..., E(n)}, 8. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 1 is characterized by: In Step 4, judge the scale of the historical data and select the normality test method: if the data sample size n is less than 30, use the Shapiro-Wilk test; if the data sample size n is greater than 30, use the Kolmogorov-Smirnov test.

9. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 1 or 8, characterized in that: In Step 4, use the Shapiro-Wilk test or the K-S test to conduct a normality test on the historical data of carbon quota transaction prices; If the test result shows that the data conforms to the normal distribution, use the geometric Brownian motion GBM model for simulation; otherwise, consider using the long short-term memory network LSTM model; Next, conduct a normality test on the sample {E(1), E(2), …, E(n)}, and introduce the specific test procedures of the Shapiro-Wilk test and the Kolmogorov-Smirnov test respectively; (1) Shapiro-Wilk test Data sorting: Sort {E(1), E(2), …, E(n)} in ascending order to obtain an ordered sample; Calculate constant: Calculate constant a according to formula (3) i Where m=(m1,…,m n ) T is the expected value of ordered independent and identically distributed statistics sampled from standard normal distribution random variables, and V is the covariance matrix of these ordered statistics; Calculate the statistic: Calculate the statistic W according to formula (4) in, is the mean of the sample; Calculate the critical value or p-value: According to n and significance level α, check the Shapiro-Wilk critical value table to get W α , by combining the statistic W with W α Comparing the analysis results, if W <W α Then reject the null hypothesis and think that the sample data does not obey the normal distribution; or use pycharm to run scipy.stats.shapiro(E) to calculate the corresponding p-value, where E is the logarithmically processed sample data sequence {E(1), E(2), …, E(n)}; if p < 0.05, reject the null hypothesis, otherwise it is considered that the data obeys the normal distribution; (2) Kolmogorov-Smirnov test Data sorting: Arrange the sample data {E(1), E(2),…, E(n)} in ascending order; Calculate the empirical distribution function: For each data point E(i), calculate its empirical distribution function value in the sample Where n is the sample size, i is the serial number of the data point, from 1 to n; Calculate the theoretical distribution function: According to the normal distribution, calculate the theoretical distribution function value F(E(i)) corresponding to each data point S(i); Calculate statistics: Find the maximum absolute difference between the empirical distribution function and the theoretical distribution function, that is, D = max 1≤i≤n |F n (E(i))-F(E(i))|; Determine the critical value or p-value: According to the sample size n and the significance level, check the critical value table of the Kolmogorov-Smirnov test to obtain the corresponding critical value D α ; If D>D α If p<0.05, the null hypothesis is rejected, and the data are considered to be non-normally distributed. Alternatively, the corresponding p-value can be calculated by running scipy.stats.kstest(E) with the help of pycharm, where E is the logarithmically processed sample data sequence {E(1), E(2),…, E(n)}. If p<0.05, the null hypothesis is rejected, otherwise the data are considered to be normally distributed.

10. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 10, characterized in that: In step 4, if the normality test is passed, the geometric Brownian motion GBM model is used: Parameter estimation: Use MLE to calculate the drift rate μ and volatility σ of the GBM model; Partial differential equation solution: Based on the partial differential equation of the GBM model, the trend of future carbon price changes is predicted by numerical methods or analytical solutions; the parameters μ and σ are substituted into Among them, P(0) is the current market carbon price, P(t) represents the predicted price t units away from the current price, and the trend of future carbon price changes is obtained by changing the value of t.

11. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 1, characterized in that: In step 4, if the normality test fails, the long short-term memory network LSTM model is used: Data preprocessing: normalize the historical data S = {S(0), S(1), ..., S(n)} according to Among them, S min Indicates the minimum value in historical data, S max is the maximum value in historical data; Model construction: Build an LSTM model and select an appropriate network structure, including the number of hidden layer nodes, activation function, and optimizer; Model training: Use historical data to train the LSTM model and adjust the model's hyperparameters to optimize model performance; Predicting future carbon prices: Use the trained LSTM model to predict future carbon prices.

12. The carbon trading game prediction method based on Stackelberg game and stochastic differential equation according to claim 1, characterized in that: In step 4, model evaluation: the prediction performance of the evaluation model is evaluated using the mean square error (MSE), mean absolute error (MAE), and root mean square error (RMSE). The mean square error is calculated as follows: Where n is the number of samples, S(i) is the true value of the i-th sample, is the predicted value of the i-th sample.

Images