**5. Conclusions**

In summary, we introduced a multivariate Hawkes process to model the limit order generation processes of individual HFTs participating in the USD/JPY foreign exchange market for 5 days and analyzed their limit order generation mechanisms. First, we confirmed that an eight-variable Hawkes process, which consisted of each HFTs' own buy–sell limit orders and the six types of orders in the order book, could adequately model the limit order generation processes of 104 of the 134 HFTs. Then, we categorized the 104 properly modeled HFTs into three categories based on the similarity of the excitation mechanisms measured by the parameter values of the Hawkes process. As a result, we confirmed that the majority of the HFTs in our dataset reacted to the execution of trades, while 12 of the 134 HFTs only reacted to limit orders and 15 of the 134 HFTs reacted to their orders. By evaluating the time constants of the estimated excitations of individual HFTs, we found that many HFTs responded to the most recent change in the order book in a very short time, by placing or canceling new orders. Since HFTs currently account for the majority of limit orders shown in the order book, the results of this analysis provide more microscopic insight into the dynamics of the order book than previous studies.

The following issues will be studied in the future as a generalization of the present work. The first goal is to clarify the limit order generation processes of the remaining 30 HFTs who could not be adequately modeled by the present analysis. The Hawkes process adopted in this study only included the impact of the occurrence of a recent order event and ignored important financial market influences such as the volume of orders, market price fluctuations and trends, and the positions of the HFTs. We believe that the information ignored in this study could contain variables that would explain their order generation processes. Second, although this study only focused on the generation of limit

orders by HFTs, it is also important to clarify the cancellation process for limit orders by HFTs and the generation of market orders. Third, we did not pay attention to profit and loss; however, practically, a key factor in an HFT strategy is the ability to make stable profits.

As the period of our data is very short, we did not observe any abnormal behavior in the market; however, we cannot deny the possibility that HFTs may overreact and result in serious synchronization during other periods or in other markets. Further studies of the relations among Hawkes parameters and the case of crashes are needed to prevent the excessive synchronization of biased orders of buy or sell. Our results are important since the model we derived in this paper provides a foundation for performing such studies through simulations. HFTs play a central role in providing liquidity to the market, and further detailed analyses of HFT strategies will contribute to the development of modern financial markets in general.

**Author Contributions:** Conceptualization, M.T.; formal analysis, H.W.; data curation, H.W.; investigation, H.W., H.T. and M.T.; methodology, H.W., H.T.; project administration, M.T.; resources, M.T.; software, H.W.; supervision, M.T.; writing—original draft, H.W.; writing—review and editing, H.T., M.T. All authors have read and agreed to submit this version of the manuscript.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Restrictions apply to the availability of these data. Data were obtained from the EBS. We obtained permission for publication.

**Acknowledgments:** We appreciate EBS, NEX Group plc. for their provision of the EBS data.

**Conflicts of Interest:** The authors declare no conflict of interest.
