指數追蹤在現今的交易市場中是相當重要的一門技術，而其中以技術指標的應用最為廣泛。技術指標即是以過去市場資料來預測未來股價趨勢的一門技術。然而在此篇文章中，我們透過控制問題的假設來處理指數追蹤：找出最佳化策略並提供證明。

Tracing index in the market is now a crucial and popular topic in finance. In practice, the technical analysis for forecasting the direction index based no the past market data is widely used. In this paper, we construct a model to trace an index based on the technique of the portfolio optimization problem through the linear quadratic regulator. We solve the optimal strategy using the dynamic programming and the corresponding HJB equation. The verification theorem is also provided. Furthermore, the sensitive analysis is illustrated through the numerical study. Finally, we examine the proposed strategy is by real data included S&P 500 and several individual stocks in the U.S.

[1] E. Ponsi, Technical Analysis and Chart Interpretations: A Comprehensive Guide to
Understanding Established Trading Tactics for Ultimate Profit. John Wiley & Sons,
2016.
[2] W.-K. Wong, M. Manzur, and B.-K. Chew, “How rewarding is technical analysis?
evidence from singapore stock market,” Applied Financial Economics, vol. 13, no. 7,
pp. 543–551, 2003.
[3] Friedman and M. Milton, Essays in positive economics. University of Chicago Press,
1953.
[4] B. G. Malkiel and E. F. Fama, “Efficient capital markets: A review of theory and
empirical work,” The journal of Finance, vol. 25, no. 2, pp. 383–417, 1970.
[5] I. Karatzas and S. E. Shreve, “Brownian motion,” in Brownian Motion and Stochastic
Calculus, pp. 47–127, Springer, 1998.
[6] T. Björk, Arbitrage theory in continuous time. Oxford university press, 2009.
[7] L.-H. Sun, “Systemic risk and interbank lending,” Journal of Optimization Theory
and Applications, vol. 179, no. 2, pp. 400–424, 2018.
[8] D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic
differential equations,” SIAM review, vol. 43, no. 3, pp. 525–546, 2001.
[9] G. Casella and R. L. Berger, Statistical inference, vol. 2. Duxbury Pacific Grove, CA,
2002.
41