指數追蹤在金融市場中是一種很流行的被動投資策略，追蹤問題是 藉由選取目標指數內含的股票種類所建立的資產組合來複製目標指數的 動向。此篇文章主要透過優化控制問題的方法建構模型來處理指數追蹤， 找出最佳化策略並提供證明。然而，追蹤指數時存在追蹤不穩定的問題， 當追蹤不穩定的情形發生時會造成過大的追蹤誤差。此研究中特地加入 對風險性資產的二次逞罰項及探討追蹤不穩定的情形，來減弱控制追蹤 不穩定的情形發生時造成過多的追蹤誤差。在實證研究中，使用 S&P 500 和美國股票顯示所提出的模型控制了追蹤的不穩定性，並且與無控 制風險的策略比較追蹤表現。

Index tracking is a popular passive investment strategy in finance. It refers to the problem of reproducing the performance of a stock market index by considering a portfolio of assets comprised on the index. This paper mainly attempts to construct a model based on the technique of the portfolio optimization problem through the linear quadratic regulator to trace closely an index. We obtain the optimal strategy using the dynamic programming and the corresponding HJB equation. However, we consider the problem of tracking instability when tracking the index through portfolio optimization. In this case would cause the excessive tracking error. Therefore, this research specifically joins the penalty quadratic term in risky assets and attempts to capture the tracking of unstable situations to weaken the tracking error. We show that the proposed model controls the tracking instability and compare the performance with the model that without joining the penalty quadratic term in risky assets using an empirical study of the S&P 500 and several individual stocks in the U.S.

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