本研究主要在探討隨機波動度模型中參數與狀態變數的聯合估計問題，並提出一種結合copula函數與粒子濾波法的多維狀態相依估計方法—狀態相依Copula粒子濾波法。本研究主要分三部分進行分析。第一部分驗證粒子濾波法結合最大期望演算法與狀態相依粒子濾波法的估計結果，顯示狀態相依粒子濾波法能穩定且精確地估計模型參數，特別在長期觀測資料下表現優異。第二部分提出狀態相依Copula粒子濾波法，建立粒子間與狀態間的相關性，並驗證其在多維金融資料下的適用性與準確性。第三部分探討五種常見的copula 函數（Gaussian、t、Clayton、Frank、Gumbel）在狀態相依Copula粒子濾波法中的應用表現，並透過均方平均誤差與平均相對誤差進行評估。結果顯示，狀態相依Copula粒子濾波法具備良好的穩定性與彈性，能有效估計參數與捕捉市場動態變化。不同copula函數對估計結果的影響亦呈現顯著差異，顯示誤差衡量標準與copula選擇應視應用場景而定。;This study investigates the estimation problem of parameters and state variables in stochastic volatility models, proposing a multi-dimensional state-dependent estimation method that combines copula functions and particle filters. This study is structured into three main parts. The first part validates the estimation results of the State-Dependent Particle Filter (SD-PF), demonstrating that the SD-PF can stably and accurately estimate model parameters, performing particularly well with long-term observational data. The second part introduces the State-Dependent Copula Particle Filter (SD-COPF), establishing correlations among particles and between states, and validates its applicability and accuracy for multi-dimensional financial data. The third part explores the performance of five copula functions (Gaussian, t, Clayton, Frank, Gumbel) within the SD-COPF framework, evaluating them using Mean Squared Average Error (MSAE) and Mean Relative Error (MRE). The results indicate that the SD-COPF exhibits good stability and flexibility, effectively estimating parameters and capturing dynamic market changes. Significant differences in estimation results were also observed among different copula functions, suggesting that error metrics and copula selection should depend on the application scenario.
