高頻的金融市場中，所得到交易資料不只是離散時間取樣，且為隨機時間取樣的觀測值，
本文藉由回顧Aït-Sahalia和Mykland於(2003)年的文獻，想利用高頻資料來分析，估計連續時
間擴散模型參數之最大概似估計量(MLE)時，離散取樣與忽略隨機時間間隔對於估計量效率
性的影響。該文中，針對取樣時間間隔做出三種不同之假設與限制，分別得到相對應的概似函
數，與參數估計量之漸近常態性質，以及相對應之共變異數矩陣，認為離散取樣效應小於忽略
隨機取樣之影響。本文特別針對該文之結果做理論、模擬與實證之檢驗，得到與該文概似函數
一般結果不同的表達式；然而，在特例O-U過程中，卻得到與該文一致的結果；同時也針對實
際資料做分析，提出模擬非齊次交易資料的方法。

High-frequency financial data are not only discretely sampled in time but the time
separating successive observations is often random. We review the paper of Aït-Sahalia
and Mykland (2003), that measure the effects of discreteness sampling and ignoring the
randomness of the sampling for estimating the m.l.e of a continuous-time diffusion model.
In that article, three different assumptions and restrict in one made on the sampling intervals,
and the corresponding likelihood function, asymptotic normality, and covariance
matrix are obtained. It is concluded that the effects due to discretely sampling are smaller
than the effect of simply ignoring the sampling randomness. This study focuses on rechecking
the results in the paper of A‥ıt-Sahalia and Mykland (2003) including theory, simulation
and application. We derive a different likelihood function expression from A‥ıt-Sahalia and
Mykland (2003)’s result. However, the asymptotic covariance are consistent for both approaching
in the O-U process. Furthermore, we conduct an empirical study on the high
frequency transaction time data by using non-homogeneous Poisson Processes.

1 背景簡介1
2 文獻回顧：模型與參數估計2
2.1 模型設定. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 取樣效應與估計方法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 完全資訊最大概似估計法(FIML) . . . . . . . . . . . . . . . . . . . . 5
2.2.2 部份資訊最大概似估計法(IOML) . . . . . . . . . . . . . . . . . . . . 10
2.2.3 等距化最大概似估計法(PFML) . . . . . . . . . . . . . . . . . . . . . 11
3 推導與範例12
3.1 (2.6)式推導. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 範例：O-U過程(Ornstein-Uhlenbeck process) . . . . . . . . . . . . . . . . . 13
4 模擬分析15
5 實證分析：交易時間間隔18
5.1 高頻交易筆數分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2 高頻交易時間間隔分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6 結論26
A 附錄27
A.1 (3.5)式推導. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.2 (3.6)式推導. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
A.3 (3.7)式推導. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
參考文獻32

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