統計数理（2021）
第 69 巻 第 2 号 123–143©2021 統計数理研究所

特集「Hawkes 過程の新展開と応用」［総合報告］

Hawkes 過程の個人史∗
Alan Hawkes1 ・Jing Chen2（受付 2021 年 1 月 2 日；改訂 2 月 9 日；採択 2 月 17 日）

要旨

この文章は Hawkes 過程として知られることになった，自己励起性点過程と相互励起性点過程について，1971 年から 1974 年の間に発表された一連の 5 本の論文についての，Alan Hawkes
へのインタビューに基づくものです．加えて，これらの論文の発表前の Alan Hawkes の経歴，
Hawkes 過程の研究に戻ってくるまで，なぜ 40 年もかかったのか，またその間に彼が行ったことについてもいくつか語ってもらっています．
キーワード：キュー，ETAS モデル，イオンチャネル，信頼性，金融，確率過程．
編集者：私は，Hawkes 過程の考案者である Alan Hawkes 氏をお招きし，Hawkes 過程の初期の歴史について，こちらからのいくつかの質問に回答していただくというのは素晴らしい思いつきだと思いました．非常に嬉しいことには，Alan が私が依頼した以上のことをしたいと申し出てくれました．Alan を Hawkes 過程の研究に引き戻したキーパーソンである Jing Chen（陳静）教授も近年の出来事を読者に語ってくれるそうです．
この文章の前段は Alan が書いたもので 4.1 章の主執筆者は陳静となりますが，各章の内容に二人の著者が関わっています．

1.

はじめに

このような著名な研究機関が発行するジャーナルの特集号に，Hawkes 過程の歴史について寄稿の依頼をいただき大変名誉に感じました．最初はいくつかの質問へ私が回答する形でという提案をいただきましたが，もう少しお話しさせていただきたいと思いました．その結果がこの文章です．この文章は質問に対する私の回答を含みますが，より多くのテーマについても書いています．このテーマについての私の研究は初期の研究と 2012 年以降この分野に戻ってからの研究の 2 つに明確に分けられます．後者については，私の同僚である Jing Chen 教授なしでは実現することはなかったでしょう．彼女の貢献は非常に大きいものがあります．そのため，
私は彼女に共同執筆者になってくれるようにお願いをしました．この文章の前段は私（Alan）が書いたもので，Jing が後段の主執筆者となりますが，各章の内容に二人の著者が関わっています．
1

School of Management, Swansea University, Bay Campus, Fabian Way, Swansea SA1 8EN, UK;

2

School of Mathematics, Cardiﬀ University, Senghennydd Road, Cardiﬀ CF24 4AG, UK;∗

本論文のもとになっている英文原稿を付録に掲載している．

a.g.hawkes@swansea.ac.uk

Chenj60@cardiﬀ.ac.uk

統計数理

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第 69 巻 第 2 号 2021

1.1 経歴私は 1960 年にキングス・カレッジ・ロンドンの数学部を卒業し，さらに学位保持者向けの統計学の 1 年課程を修めるためにユニバーシティ・カレッジ・ロンドンの統計学部に移りました．
その年の終わりには，道路交通問題に応用されている待ち行列理論を研究している Maurice
Bartlett 教授の指導の下で研究生をはじめました．1 年後私はその学部で講師の職を得，教育とその他の業務をこなしながら研究を続けました．私の学位論文は 1965 年に完成しましたが，その時の外部試験官は David Cox 教授でした．David Cox 教授には独創的で重要な研究実績が数多くありますが，Cox 教授は Cox 過程と呼ばれるよく知られた点過程モデルを最初に広めました．
学位論文終了後，私は待ち行列理論について更に 2 つの論文を発表しました．発表後は自分の研究分野をもっと広げるべき時期だと判断しました．その後は様々なテーマについて更に 4 つの論文を発表しました．ユニバーシティ・カレッジ・ロンドンは研究には最高の場所を提供してくれました．ロンドン大学が，ロンドンの他の大学の学者（Cox や David Brillingerのような学者）
や外国からの著名な訪問者と交流することができるセミナーを開催してくれま
した．外国からの著名な訪問者には，数名を挙げるならば点過程理論に貢献した研究者 Jerzy
は，いくつかの理論モデルと一緒
Neyman や Peter Lewis が含まれます．Cox and Lewis（1966）に興味深いモノグラフを発表しましたが，ほぼデータ分析に集中していました．私はとりわけ私の博士課程の指導教官である Bartlett（1963）に刺激を受けました．Bartlett はその論文の中で，その後ほどなくして点過程における Bartlett スペクトルとして知られることになるものを初めて紹介しました．私はしばらくの間は点過程が集中すべき興味深い分野を提供してくれるであろうという判断に至りました．
2.

いくつかの励起性点過程

この章では Hawkes 過程について初期の 5 本の論文（Hawkes, 1971a, 1971b, 1972; Hawkesが書かれた 1971 年から 1974 年の頃についand Adamopoulos, 1973; Hawkes and Oakes, 1974）て書きます．その当時はもちろん Hawkes 過程とは呼ばれておらず，自己励起及び相互励起点過程と呼ばれていました．私がここで語らなければならないことには，私が先に言及した私への 5 つの質問の内の 3 つへの回答が含まれています．
編集者からの質問 1：Hawkes 過程が提唱された時代，どのような点過程モデルに人気がありましたか．Hawkes 過程モデルや他モデルを応用するにあたっての困難な点で主だったものはどのようなものでしたか．
1969 年にダラム大学に移った頃，私は既に点過程について真剣に考え始めていました．
当時，点過程はある程度の関心を集めてはいましたが，Poisson 過程，リニューアル過程，
semi-Markov 過程（Hawkes, 1970 は最後のテーマに関する論文），それに加えて Neyman-Scott
過程や Bartlett-Lewis 過程，Cox 過程を除けば，多くのモデルはそれ程の関心を集めているよ
うに私には見えませんでした．私は特に Bartlett に影響を受けていたため，自分が新しいモデルを導入し，その Bartlett スペクトルを見つけることができるかもしれないと思っていました．
今日私は点過程についてもっと広い定義をし，待ち行列理論や連続時間 Markov 連鎖（即ち，，ニューロンの発火，感染症流行過程，システムの信連続時間で離散状態をもつ Markov 過程）
頼性，等のようなイベントの流れをモデル化するものであればどのようなものであっても定義に含めています．
その当時難しかった主な点は，計算と正確で十分な量のデータを利用可能な形で確保するこHawkes 過程の個人史125とでした．コンピュータは物理的に巨大で遠隔地にありましたが，容量的に小さく速度も遅いものでした．ソフトウェアは使用範囲の点で限定的でした．現在では個人のデスク上で信じられないくらいパワフルなコンピュータと膨大な種類のソフトウェアが利用できますし，ソフトウェアの多くは世界中の研究者の努力のお陰で無料で利用することができます．コンピュータのハードウェアとソフトウェアの発展は，信頼性の高い大量のデータ，例えば生理や金融市場での数ミリ秒ごとの取引記録のデータ，を集めることも容易にしました．
編集者からの質問 2：Hawkes 過程はどうやって思いついたのですか．Hawkes 過程は私が生まれる以前に世に出ていました．私より若い研究者の多くはこの質問の答えに興味があるのではないかと思います．
正確にどうやってそうなったか，またなぜそのようになったかについては，はっきりとは覚えていません．いくつかの応用の可能性，特に感染症流行過程への応用可能性については言われていましたが，動機は純粋に数学的なものでした．多くの人が Hawkes 過程は元々は地震の連鎖を記述するために発明されたといって Hawkes 過程の論文を書き始めますが，これは正し
くないことは明らかです．ただ，Vere-Jones（1970）の興味深い論文が Hawkes（1971a）で引用されてはいますが．このとき博士課程の大学院生を一人採用しましたが，私は彼がなにか実用的なことをすべきと考えていたため，彼に地震の分析をやらせました．Hawkes（1972）ではマー
ク付き励起過程を紹介し，「マーク付き過程のモデルの地震データへの適合はダラム大学の L.
Adamopoulos による」という注釈を含めていました．
このことは Hawkes and Adamopoulos（1973）で実を結びました．実際 Hawkes 過程の実用的な対象への初めての適用でした．ただ，これらの過程が最初に導入された時，私の頭の中にはもちろん地震のことはありませんでした．我々には地震の知識がほとんどありませんでしたので，この論文は特別良い論文ではありませんでしたが，地震学者は面白い可能性があると感じていました．地震学者は地震についてよく知っていますから，地震学者，特に著名な日本人の地震専門家は，道理にかなった励起カーネルを導入しずっと上手くやることができていました
し，重要な時空間版モデルを含む有名な ETAS モデルを開発していました（Ogata, 1998）．これらのことは犯罪活動を予測する研究者も取り上げていました．犯罪活動予測の例についてはMohler et al.（2011）を参照してください．
編集者からの質問 3：統計的推論について，私の印象ではスペクトル，生成汎関数及びモーメントに基づく方法が，点過程データ解析で主に用いられている技術だったのですが，尤度に基づく方法（最尤推定）が最近紹介されました．これは本当ですか．
本当だと思います．最も使用されている方法は Cox and Lewis（1966）が書いていますが，
Lewis et al.（1969）
はそれらの方法を適用するために役立つコンピュータプログラム（SASE IV）
を提供しました．Metropolis-Hastings 法のようなコンピュータを多用した MCMC Bayes 法は
揺籃期にあり（私の記憶の限りでは）
点過程解析の一般的なツールにはまだなっていなかったのは，サンプルのスペクトル（ピリオドグラムと呼ばれです．Hawkes and Adamopoulos（1973）
ることもある）と慎重に選ばれた様々な周波数で計算した理論的な Bartlett スペクトルの比は，およそ独立同分布指数確率変数のような分布となるという考えに基づく「スペクトル尤度法」
を紹介しました．
私は 1974 年にダラム大学からスウォンジー大学に異動し，1978 年頃に David Vere-Jones がスウォンジーに講演に来てくれました．講演で彼は，一般的な点過程についての Rubin（1972）が発表した Hawkes 過程の今日ではよく知られた尤度公の結果に基づいて尾崎（Ozaki, 1979）
式について話してくれました．Vere-Jones はその問題に取り組むように尾崎を勇気付けること

統計数理

126

第 69 巻 第 2 号 2021に一役かったと思います．最大尤度の EM アルゴリズムは Dempster et al.（1977）の注目すべき研究以前には一般的には知られていませんでした．今では Hawkes 過程（Hawkes and Oakes,の分枝過程を表す際によく使われています．
1974）
3.

中期

編集者からの質問 4：多くの人が認識していますが，貴方は長年の間，自己励起型過程の研究を継続せずやめていました．その大きな理由は何ですか．その期間中，何が貴方の主な研究対象だったのですか．
私の研究者としてのキャリアを通じて，常に私は自分のことを本来は応用確率論者だと思っていました．私は数学理論のために理論を開発することには特に関心はありませんでした．私はある程度実用的見地から重要な具体的問題を研究することを好みます．それらの自己励起型
過程はそれほどの興奮を（研究者の間に）もたらしたようには見えませんでした．グーグル・ス
カラーによれば，最初の Hawkes の論文 Hawkes（1971a）が受理されてから 30 年の間，引用は年平均約 1.5 回です．今日では年間約 200 回の引用があります．私は自分が例えば地震や感染症について実際に役立つ貢献をするに足る知識を持ち合わせているとは思えませんでした．私はもっと興味が持てることを見つけていました．
私はユニバーシティ・カレッジ・ロンドンで過ごした頃から薬理学者の David Colquhoun のことを知っていました．彼が書いていた本でちょっとお手伝いをしたのです．生物統計学の本の確率過程に関する章で彼のお手伝いをしました．神経系で一連のスパイ（Colquhoun, 1971）
クを観察する際に，どのくらいの頻度で異常に大きなスパイクを観察することが期待されるかという質問を彼は私へ投げ掛けましたが，Hawkes（1970）はその質問に対する回答として書かれたものでもあります．論文はそのことを一般化し原型をとどめないほど抽象化していましたが，それが彼が回答を求めていた質問でした．
その後彼はイオンチャンネルを通る微小電流について私に質問をしてきました．イオンチャ
ンネルは体中に電気信号を伝える複雑な伝達系の不可欠な部分で，それなしでは我々は感じることも考えることも動くことも全く何もできなくなります．イオンチャンネルは薬物作用においても重要なサイトになります．例えば，局所麻酔はチャンネルを閉鎖し信号が通過できない
ようにしますが，そのことによって患者は痛みを感じないようになります．私はイオンチャン
ネルについてあまり知識がありませんでしたが，David には知識があり，何が必要であるかを私に伝えることができ，私の数学を理解することができました．
そのようにして我々の間で約 40 年もの間続いた素晴らしい共同研究が始まりました．Davidはもの凄く才能のある人間です．彼はユニバーシティ・カレッジ・ロンドンで薬理学の教授および王立学会フェローとなり，3 人のノーベル賞受賞者と研究しました．彼は本当に優しい人
間で良き友人です．彼と働くのは大きな喜びでした．
我々は共にシングル・チャンネルの動力学の確率モデルの先駆者となりました．そのモデル
は，薬理学者や生理学者，生物物理学者が日常的に測定する実験結果について詳細な理解を引き出すことを可能にし，彼らの世界中のコミュニティで欠くことができないものとなりました．
私はこれを，人間や動物の身体，更には植物について理解を深める実用的見地から，これまで行った最も重要な仕事と考えています．それに加えその数学も素晴らしいものでした．我々は21 の論文を発表しましたが，その最も重要な 10 の論文を参考文献（Colquhoun and Hawkes,
1977, 1982, 1987, 1995a, 1995b; Colquhoun et al., 1996, 1997, 2003; Hawkes et al., 1990, 1992）に掲載しました．私の同僚である Assad Jalali 博士も，チャンネル開閉時間が約 25 マイクロ秒以下という非常に短時間しか続かないために観察できない，いくつかのチャンネル開閉時間

Hawkes 過程の個人史

127についての time interval ommision の重要な問題の解決に大きな貢献をされたこと（Hawkes etも述べておかねばなりません．このテーマでは私の唯al., 1990, 1992; Jalali and Hawkes, 1992）
一の日本語論文があることも特に言及しておきます（蛯名 他, 1989）．
私はシステムの信頼性についても 14 の論文を発表しました．このテーマについては，その数学の一部がイオンチャンネルを説明する際に用いたものと似てはいましたが，私にそれほど興味があったわけではありません．このテーマに関心があった生徒を指導したことが主な経緯です．参考文献にはこのテーマに関する 8 つの論文（Cui and Hawkes, 1994, 2008; Cui et al.,
1995, 2012; Fawzi and Hawkes, 1990; Hawkes et al., 2011; Jalali et al., 2005; Zheng et al., 2006）
を含めてあります．ほとんどは北京理工大学の Lirong Cui 教授と何名かの彼の研究生との共著です．
その他様々なテーマについて永続的な重要性はあまり高くない 15 本の論文もありました．
4.

Hawkes 過程への回帰

2012 年に Jing Chen と知り合う機会がありました．彼女は Hawkes 過程の研究テーマに戻る
であること
ように私を説得しました．彼女は，Hawkes 過程が「金融分野ではホットなテーマ」
を私に納得させ，彼女との共同研究に私を誘ってくれました．私は金融については全く知識がありませんでしたが，私を導くのに十分によく数学を理解したそれぞれの分野の才能ある専門
家と，再び一緒に仕事することになりました．同時に私はあらゆる新しい数学と Hawkes 過程の非常に広い応用範囲について追いつくために努力しようと思いました．ただ，これは不可能な仕事であることが明白になりました．最初の論文（Hawkes, 1971a）は今では年間 200 以上引用されています．
我々が金融分野で Hawkes 過程についてやったほぼ全てのこと（ワークショップ，ジャーナル
特集号，等）
は Jing が意義を見出し準備をしてくれましたので，その話は彼女の方から続けていただきます．

編集者：4.1 章は陳静が語り手
4.1 Hawkes 過程と金融の最近の歴史
2010 年に私はスウォンジー大学の経営管理学部に金融学の講師として着任しました．2011
年には金融の研究，特に高頻度取引と市場のマーケットマイクロストラクチャーの研究で，Hawkes 過程が重要なツールとなりつつあることに気がつきました．私はまた，所属する学部のAlan Hawkes 名誉教授が実は Hawkes 過程の著者であることにも気がつきました．高頻度取引（例えば，取引とクォート）が非常に高速で（ミリ秒で）
生じるの重要な特徴の 1 つは，イベント
ことと，それがしばしばクラスターとなって発生することです．このことにより Hawkes 過程を適用する自然な環境ができあがります．実際，まさに当てはまる事例であることが証明されてきており，金融における初期の Hawkes 過程の適用は，取引値を動かすことになる買値と売値に指値注文ブック行動がどのように関わっているか理解するためのものです（Bowsher, 2007と Large, 2007 を参照）．保険では，Hawkes 過程を使ってクラスターとなって発生する保険請
求をモデル化する研究がいくつか出てきました（Stabile and Torrisi, 2010 と Dassios and Zhao,
2011 を参照）．保険請求過程を，自己励起的または相互励起的特徴を示す「動的感染過程」とみなすという考え方です．そのような応用は，最終的には破産確率の古典的問題に関わってくるので，与信リスクモデル化において多くの場合で非常に有益ということがわかっています．
これらの初期における Hawkes 過程の応用は非常に興味深いものです．Hawkes 過程が金融のより多くの分野で極めて重要であり役に立つということを Alan に確信させるのに，私にも

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第 69 巻 第 2 号 2021

う迷いはありませんでした．2012 年に我々は共同研究を始めました．Alan は私に確率過程（特と統計学を教え，同時に私が Alan に金融を教えました．Alan にとっての最初に Hawkes 過程）の金融の訓練の一部は，私が金融系ジャーナルに提出していたいくつかの論文で数学と統計学をチェックする作業でした．
我々が取り掛かった最初の研究テーマは，Hawkes ジャンプ過程を GARCH 金融モデルに組み込むことでした．理論モデルは開発しましたが，我々には一連の金融リターンにおいてジャンプを検出するための方法が必要でした．我々は最も普及している方法の 1 つを使い始めましたが，ジャンプを検出する能力が，特にジャンプが感染特性を示した際に，非常に劣っている
ことが時々あることがわかりました．そのため，我々はそのような状況下でもよい反応を示すと思われる，新しい方法を考案しました．我々は極めて重要な結果と考えることを出版したいのですが，残念ながらこれまでのところ一流ジャーナルを説得することに難しさを感じています．この件は挑戦し続けています．ただ，我々のジャンプ検出方法は，ワシントン DC において，米国政府金融規制機関である米国商品先物取引委員会（CFTC）講演会で発表した際，高く評価されました．最終的に我々の新しい検出方法を用いることで自分たちが計画したジャンプ GARCH モデルに戻りました．モデルは完成間近です．我々は自分たちのモデルを通じて，ジャンプの発生のような日中の市場の振る舞いが，どのようにボラティリティに関係しているのかを理解したいと思います．そうすれば，トレーダーは 1 日の終わりにボラティリティのより正確な予測を得られるようになる可能性があります．このことはトレーダーの日常的な意思決定において実務的に達成，実現が可能です．
同時に，私は Hawkes 過程の特集号を発行してもらえるように Quantitative Finance ジャー
ナルを説得しました．ジャーナルからは私がその特集号の主編集者になることを依頼されました．特集号は 2018 年の初めに出版されました．Chen and Hawkes と様々な共同執筆者によって書かれた 3 つの論文（Chen et al., 2018; Khashanah et al., 2018; Yang et al., 2018）と Hawkes
による総括（Hawkes, 2018）が掲載されています．
の出生死滅移民 Hawkes モデルを通じて金融市場での注文
最初に，Khashanah et al.（2018）の流れをモデル化します．これは正の励起過程を用いる大半の文献とは逆のものです．我々は一連のイベントが他のタイプのイベントとの相互作用により減少するレートで発生するようにした，相互励起型二変量設定で取引とクォートをモデル化しています．
は，存在しうる励起カーネルの集合から選択するため，赤池情報量
次に，Chen et al.（2018）
及び Hannan-Quinn 情報量規準（HQ）の 3 つの典型的な情基準（AIC），Bayes 情報量規準（BIC）
報量基準を比較しています．
同じ頃，行動ファイナンス研究が広がりつつありました．とりわけ，ニュースによって生じる感情が存在し市場価格を動かしているのは明白です．Yang et al.（2018）は，どのようにして楽観的／悲観的な価格と感情の動きが相互作用するのかを理解するために，相互励起 Hawkes
過程を適用しました．我々は，特に 2008 年の金融危機以降において，市場に対するトレーダーの信念を反映するニュース感情が，マーケットの変動を誘導する上でより支配的になってきていることを発見しています．
この特集号にはさらに別の 6 本の素晴らしい論文も掲載されています．テーマは理論的なものと実用的なもの様々で，例を挙げると，指値注文ブック用の高次非線形 Hawkes モデル（Achab et al., 2018; Lu and Abergel, 2018），ダーク・プール・トレーディング（Gao et al.,2018），金融におけるジャンプと共ジャンプ（Calcagnile et al., 2018），流動性／非流動性波及及び定率ポートフォリオ保険（Buccioli and Kokholm, 2018）です．
効果（Schneider et al., 2018）
金融分野での Hawkes 過程の応用をさらに進めるため，我々は現在もう 1 冊，The European
Journal of Finance の特集号の編集を進めています．ボラティリティ・クラスタリング，金融Hawkes 過程の個人史

129

ジャンプ，金融ネットワーク，ポートフォリオ最適化，等のような金融における多くのより重要な課題について考察する予定です．この特集号には金融における Hawkes ジャンプ拡散についての Hawkes による評論（Hawkes, 2021）も掲載されます．
我々は，カーディフ（2017 年），スウォンジー（2018 年）及びニュージャージー州ホーボー
ケンのスティーブンス工科大学（2019 年）での一連の 3 つのワークショップで Hawkes 過程の啓発活動もしてきました．スウォンジーでの会議は Alan の 80 回目の誕生日を祝う 2 日間にわたるものでした．David Colquhoun がイオンチャンネルに関する Alan の業績について話をしました．他の話の大部分は金融での Hawkes 過程に関するものでした．メインスピーカー，Valérie
は Mathieu Rosenbaum，Frédéric Abergel. Steve Hardiman と Judith Rousseau（Paris）（Switzerland），Khaldoun Khashanah と Steve Yang（NewChavez-Demoulin と Matthias Kirchner，Fabrizio Lillo（Italy），Mark Tippett（Australia），Lirong Cui（China），Anton Merlushkin
Jersey）（Credit Suisse, London），Enrico Scalas（Sussex, UK）そしてもちろん Alan Hawkes．素晴らしい誕生日でした．
Hawkes 過程とその応用の発展のもう 1 つの道筋は，複素カーネルをもっと使用することを検は，古典的な ETAS モデルのべき乗則カーネルを置き換え討することです．Chen et al.（2021）
るために，Mittag-Leﬄer 型カーネルを導入しています．これはよく研究された Mittag-Leﬄer
関数のラプラス変換を利用することが可能で，それによって Hawkes モデルのいくつかの特性の計算の簡略化を可能にします．Cui et al.（2020）はいくつかの Hawkes 過程の特性のモーメン
トを見つける方法を開発しています．
我々はニュース感情と価格行動の研究も継続しています．Liu et al. の 2 つの論文（2020a,
は Hawkes 過程を全く使わずエントロピーに基づいています．Liu et al.（2020a）は，価2020b）
格またはニュースによって動かされる情報の流れを定量化するエントロピーに基づく方法を導入し，それによって様々なタイプの取引行動や市場のレジームを確認しています．11 年間のニュースと市場データに基づき，金融市場がダブル・クライシス（2008 年の流動性とユーロ圏
負債危機）の時代の前後で様々な情報の流れに支配されていたことを発見しています．情報の流れへの反応は，グレンジャー因果性の研究に関わる金融関係の文献でしっかりと研究されては再びエントロピーを用いて，ニュースに対するトレーダーの反応
います．Liu et al.（2020b）
が危機の期間中はずっとより顕著となり，したがってより効果的な価格発見過程には，価格とニュースの両方によって動かされる情報を取り入れる必要があるということを発見しています．
5.

Hawkes 過程の現在と将来

編集者：再び Alan の話に戻りましょう．では最後の質問です．
編集者からの質問 5：Hawkes モデルの最近の発展と応用についてどうお考えですか．Hawkes
モデルの将来の発展はどのようなものになると期待していますか．
Hawkes 過程が約 30 年もの間実質無視されていたことを考えると，この 5 年間の Hawkes 過
程の応用範囲と量を見て驚いています．昔からのユーザーである地震学者たちは今もなお活発です．最近のユーザーの中では金融，ソーシャルメディアや数学理論の分野で活発な動きが多く見られますし，神経科学の分野，犯罪や暴力行動の分野でも一定の研究活動が見られます．
Hawkes 過程を用いた出版物の発行頻度はいずれ落ち着いていくと予想していますが，Poisson
過程がそうであるように，消え去るということはないでしょう．消え去るには便利過ぎます．
Hawkes 過程は，ベース過程，励起カーネル（必要ならばマークと空間効果付き），外生過程との相互作用の選択においてフレキシブルです．特別な状況に合うように様々なやり方で変更す

130

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第 69 巻 第 2 号 2021

ることも可能です．我々は様々な分野でこの特性の独創的な利用の研究が継続的に増えていくと予想しています．
興味深いことには，私の論文（Hawkes, 1971a）での示唆にも関わらず，Hawkes 過程は病気をモデル化するために役立つかもしれません．ETAS というタイトルが示唆する対象そのものですが，疫学者はまだあまり Hawkes 過程を活用していません．新型コロナウイルス感染症の出現は急激な関心をもたらしました．参考文献の中ではこのタイプの出版物については特別触れてはいません．どれが重要なものなのか見極めるには早すぎるためです．興味深いことには，
しばしば R と呼ばれる分岐比は，パンデミックを伝える日々のニュースではいつも出てくる部
分です．
Hawkes 過程の一般的な利用のために，我々はしっかりとテストされた無料で入手可能なソフトウェアとモデル適合法の整合的なセットの開発をみたいと思っています．時変特性とモデ
ル学習特性を組み合わせたベイズ法は特に有望です．

参

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Proceedings of the Institute of Statistical Mathematics Vol. 69, No. 2, 123–143 (2021)

133

A Personal History of Hawkes Processes
Alan Hawkes1 and Jing Chen2
1 School of Management, Swansea University
2 School of Mathematics, Cardiﬀ University

This paper is based on an interview with Alan Hawkes about the series of five papers
published 1971–1974 on self-exciting and mutually-exciting point processes that came tobe known as Hawkes processes. This is supplemented by additional material describing
the background before the papers were published, why it was 40 years before he returnedto the subject of Hawkes processes and some of the things that have since been achieved.

Key words: Queue, ETAS model, ion-channel, reliability, ﬁnance, stochastic process.

134

Appendix

A Personal History of Hawkes Processes
Alan Hawkes1 and Jing Chen2
1 School of Management, Swansea University, Bay Campus, Fabian Way, Swansea SA1 8EN, UK;a.g.hawkes@swansea.ac.uk
2 School of Mathematics, Cardiﬀ University, Senghennydd Road, Cardiﬀ CF24 4AG, UK;

Chenj60@cardiﬀ.ac.uk

The editor: I thought that it was a great idea to invite Alan Hawkes, the inventor of the Hawkes
process, to answer some questions about the early history related to it. I was so delighted that
Alan told me that he would like to do more than that. Prof. Jing Chen, the key person who hadbrought Alan back to the direction of Hawkes processes, would also tell the readers the facts inrecent years.
The ﬁrst part of this article has been written by Alan, and Jing is the main author of Section
4.1. Both authors contributed to each section.
Abstract
This paper is based on an interview with Alan Hawkes about the series of ﬁve papers published 1971–1974 on self-exciting and mutually-exciting point processes that came to be known as
Hawkes processes. This is supplemented by additional material describing the background before
the papers were published, why it was 40 years before he returned to the subject of Hawkes
processes and some of the things that have since been achieved.
1.

Introduction

I was very honoured to be asked to contribute something on the history of Hawkes processes
to a special issue of a journal produced by such an illustrious Institute. At ﬁrst it was suggested
that I might answer a few questions, but I thought that I would like to say a bit more: theresult is this article which covers more topics as well as including answers to those questions. My
work on this topic falls distinctly into two parts: the early work and a late return starting in
2012. The latter would never have happened without my colleague, Professor Jing Chen, who has
contributed so much. I therefore asked her to be co-author. The ﬁrst part of this article has been
written by me (Alan) and Jing is the main author of the second part, but we both contributed
to each section.
1.1 BackgroundI graduated in Mathematics from King’s College London in 1960 and moved to the Statistics
Department in University College London (UCL) to study a one-year postgraduate diploma in
Statistics. At the end of that year, I started as a research student under the direction of Professor
Maurice Bartlett, working on queueing theory applied to problems in road traﬃc. One year later
I was appointed to a lectureship in the department and continued my research alongside teachingand other duties. My thesis was completed in 1965: the external examiner was Professor DavidHawkes 過程の個人史

135

Cox who, amongst many of his important original works, introduced a well-known point processmodel called the Cox process.
On completion of my thesis, I published two more queuing theory papers then decided that it
was time to broaden my research area. I published another four papers on a variety of topics.
UCL provided excellent facilities for research. London University had seminars that enabled one
to meet scholars from other colleges in London (such as Cox and David Brillinger) and distinguished visitors from abroad, including Jerzy Neyman and Peter Lewis, to name just a few who
made contributions to point process theory. Cox and Lewis [1] published an interesting monograph with some theoretical models, but mostly concentrating on data analysis. Most of all, I
was stimulated by a paper by my PhD supervisor, Bartlett [2], in which he introduced what soon
became known as the Bartlett spectrum for a point process. I decided that point processes mightprovide an interesting area to concentrate on for a while.
2.

Some exciting point processes

In this section I discuss the time, 1971–1974, when those ﬁve early papers on Hawkes processes, [3]–[7], were written. Of course, they were not called Hawkes processes at that time but
self-exciting and mutually-exciting point processes. What I have to say includes answers to threeof the ﬁve questions that I mentioned were posed to me.
Editor’s Question 1 : At the time when the Hawkes process was proposed, what point
process models were popular? What were the main diﬃculties of applying Hawkes
models and other models?
When I moved to Durham University 1n 1969 I had already begun to think seriously about
point processes. There was some interest in point processes at that time but, it seemed to me
then, not many models apart from the obvious Poisson, renewal, semi-Markov processes (Hawkes,
1970, [8] was on the latter topic) together with Neyman-Scott, Bartlett-Lewis and Cox processes.
I was inﬂuenced particularly by Bartlett so that I thought maybe I could introduce a new modeland ﬁnd its Bartlett spectrum.
Nowadays I have a much broader deﬁnition of point processes to include anything that models
streams of events, such as queueing theory, continuous-time Markov chains (i.e. Markov processes
with discrete states in continuous time), neuron ﬁring, epidemic processes, systems reliability etc.
The main diﬃculties in those days were computing and availability of ample accurate data.
Computers were physically large and remote, though small in capacity and slow in speed. Software was limited in scope. Today you can have an incredibly powerful machine on your desk anda huge range of software, much of it freely available as the result of work of scholars from around
the World. The developments in computer hardware and software have also made it much easier
today to get large amounts of reliable data, for example in physiology or recording transactions
on ﬁnancial markets every few milliseconds.
Editor’s Question 2 : How did the process come out in your idea? This process appeared before I was born. I wonder many researchers who are younger than I might beinterested in the answer to this question.
I do not have a clear memory of exactly how or why things turned out the way they did.

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The motivation was purely mathematical, although some possible applications were mentioned,
particularly epidemic processes. Many people begin articles on Hawkes processes saying thatthey were originally invented to describe earthquake sequences. This is clearly not true, although
the interesting paper by Vere-Jones [9] was cited in Hawkes [3]. At this time I acquired a PhD
student and I thought that he should do something practical, so I set him to work on analysing
earthquake data. Hawkes [5] introduced marked exciting processes and included the remark “Amarked process model is being ﬁtted to earthquake data by L. Adamopoulos at Durham”.
This came to fruition in Hawkes and Adamopoulos [6], which was indeed the ﬁrst applicationof Hawkes processes to any practical subject: but earthquakes were certainly not on my mind
when those processes were ﬁrst introduced. It was not a particularly good paper, because we
knew little about earthquakes, but Seismologists felt that it had interesting possibilities. Since
they do know about earthquakes they, particularly the famous Japanese experts, have been able
to do much better with sensible exciting kernels and developing the well-known ETAS models,
including important spatio-temporal versions (Ogata, 1998, [10]). These have also been taken up
by people predicting criminal activity, see for example Mohler et al., 2011, [11].
Editor’s Question 3 : On statistical inference, in my impression, spectrum, generating
functional and moment-based methods were the main techniques used in point processdata analysis. Likelihood based method (MLE) was just introduced. Is this true?
I think that is true. The methods most in use were described in Cox and Lewis [1] while
Lewis et al. (1969, [12]) provided useful computer programs (SASE IV) to apply those methods. Computer-intensive MCMC Bayes methods, such as Metropolis-Hastings, were in theirinfancy but (as far as I can recall) had not yet become common tools for analysing point processes. Hawkes and Adamopoulos [6] introduced “a spectral likelihood” method that relied on theidea that the ratio of the sample spectrum (sometimes called periodogram) and the theoretical
Bartlett spectrum computed at various well-chosen frequencies were approximately distributed
as i.i.d exponential random variables.
I moved from Durham to Swansea university in 1974 and about 1978 David Vere-Jones visited
Swansea to give a talk in which he described the now well-known likelihood formula for Hawkes
processes published by Ozaki (1979, [13]) following a result of Rubin (1972, [14]) for general point
processes. I believe Vere-Jones had some part in encouraging Ozaki to work on that problem.
The EM algorithm for maximum likelihood was not generally known before the notable work of
Dempster et al. (1977, [15]): now it is often used on the branching process representation of the
Hawkes process [7].
3.

The middle years
Editor’s Question 4 : As many people have noticed, you gave up to continue yourwork in self-exciting process for many years, what are the main reasons? During theseyears what were your main research focuses?

Throughout my academic career I have always thought of myself primarily as an applied probabilist. I have not been particularly interested in developing mathematical theory for its own sake.
I like to work on a concrete problem that has some practical importance. Those self-exciting
processes did not seem to have generated all that much excitement: according to Google Scholar,

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over the next 30 years the ﬁrst Hawkes paper [3] received on average about 1.5 citations per year:
nowadays it gets about 200 citations per year. I did not feel that I had suﬃcient knowledge tomake a practical contribution to, say, earthquakes or epidemics. But I found something much
more interesting to do.
I had known pharmacologist, David Colquhoun, from my time at UCL. I had made a small
contribution to a book he was writing, on biostatistics [16] by helping him with the chapter
on stochastic processes. Also, Hawkes [8] was written in response to a question he asked me
about how often you might expect to see extra-large spikes when observing a series of spikes ina nerve system. The paper generalised it and abstracted it out of all recognition, but that was
the question he wanted answered.
Then he started asking me questions about minute electric currents passing through ionchannels, an essential part of the complex communication system that carries electrical signals
around the body, without which we would be unable to feel, think, move or do anything at all.
They are also important sites of drug action. For example, local anaesthetics close channels sothat no signal can pass and therefore the patient feels no pain. I did not know much about ion
channels, but David did and was able to communicate what he needed to know and to understandthe mathematics that I produced.
So began a wonderful collaboration between us that lasted for about 40 years. David is immensely talented: he became professor of Pharmacology at UCL, a Fellow of the Royal Society(FRS) who worked with three Nobel prize-winners. Such a nice man and good friend, it was agreat pleasure to work with him.
Together we pioneered the stochastic modelling of single-channel dynamics which came to beindispensable to the worldwide community of pharmacologists, physiologists and biophysicists
who were thereby enabled to extract detailed understanding of the experimental results thatthey routinely measure. I regard this as the most important work that I ever did, having practical importance for our understanding of the bodies of people, animals and even plants. And
the mathematics was nice too! We published 21 papers, of which 10 of the most important areincluded in the bibliography [17]–[26]. I should also mention that my colleague, Dr Assad Jalali,also made a signiﬁcant contribution to solving the important problem of time interval omission([20], [21], [27]), when some open or shut channel periods may not be observed because they
last for such a short time; less than about 25 µs. Under this topic I also note my only previous
publication in Japanese [28].
I also published 14 papers on systems reliability. I was not so motivated by this topic, althoughsome of the mathematics was similar to that used for describing ion-channels. I was mainly drawn
into this by students or former students of mine who were interested in that subject and wanted
my help from time to time. The bibliography includes 8 of these papers, [29]–[36], mostly incollaboration with Professor Lirong Cui, from Beijing Institute of Technology, and some of his
research students.
There were also 15 papers of little lasting importance on a miscellaneous collection of topics.
4.

Return to Hawkes processes

In 2012 I got to know Jing Chen, who persuaded me to return to the subject of Hawkes processes, which she assured me was a “hot topic in ﬁnance” and invited me to work with her on

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that subject. I knew nothing about ﬁnance but once again I found myself working with a talented
specialist in their ﬁeld who understood the mathematics suﬃciently well to guide my eﬀorts. At
the same time, I thought I might try to catch up with all the new mathematics and the very broad
range of applications of Hawkes processes. This proved to be an impossible task: the original
paper [3] now gets over 200 citations per year.
As Jing motivated and organised almost everything we did on Hawkes processes in Finance(workshops, journal special issues . . . ), she will continue the story.
The Editor: Section 4.1 was narrated by Jing Chen
4.1 Recent history of Hawkes processes and FinanceIn 2010 I joined Swansea School of Management as a lecturer in Finance. In 2011 I realizedthat Hawkes processes were becoming important tools in Finance research, particularly in highfrequency trading (HFT) and market microstructure. I also realized that Emeritus Professor
Alan Hawkes, in my department, was actually the author of Hawkes processes. One of the important features of HFT is that the events (e.g. trades and quotes) occur at extremely high speed(in milliseconds) and, often, in clusters. This provides a natural environment to apply Hawkes
processes. This, indeed, has proved to be the case and the early Hawkes process applications in
ﬁnance are to understand how limit order book activities related to bids and asks that would
move trading prices (See Bowsher, 2007 [37] and Large, 2007 [38]). In insurance, a few researches
also appeared using Hawkes processes to model the clusters in arrivals of insurance claims (seeStabile & Torrisi, 2010 [39] and Dassios & Zhao, 2011, [40]). The idea is to consider the claim
process as a “dynamic contagion process” that exhibits the self- or mutual-exciting characteristics. Such applications are often found highly useful in credit risk modelling as they ultimately
relate to the classic problem of the probability of ruin
These early applications of Hawkes processes are fascinating, and I hesitated no more to convince Alan that Hawkes processes will be hugely important and helpful in many more areas in
Finance. In 2012, we started to work together, Alan teaching me Stochastic processes — especially Hawkes processes — and Statistics, while I taught him Finance. Part of his initial training
in Finance was to check the Mathematics and Statistics in several papers that I was submitting
to Finance journals.
Our ﬁrst research idea that we worked on was to incorporate a Hawkes jump process into a
GARCH ﬁnancial model. Having developed a theoretical model, we needed a method of identifying jumps in a series of ﬁnancial returns. We started using one of the most popular methods: wefound that it was sometimes very poor at identifying jumps, especially when the jumps showeda contagious property. Therefore, we devised a new method that would behave well in thosecircumstances. Unfortunately, we have so far found it diﬃcult to persuade quality journals to
publish what we think are quite important results — but we keep trying. Our method of jump detection was, however, well received when presented in a talk to the Commodities Futures Trading
Commission (CFTC), a US government ﬁnancial regulatory body in Washington DC. Eventually,
we returned to our planned jump-GARCH model, using our new methods of detection, and areclose to completion. We hope, through our model, to understand how intra-day behaviour, suchas occurrence of jumps, is related to volatility. Subsequently, traders may be able to get moreaccurate forecasts of volatility at the end of each day, which are practically achievable and feasible

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for their routine decision making.
Meanwhile, I persuaded the journal Quantitative Finance to publish a special issue on Hawkes
processes, and they asked me to be its chief editor. The issue was published in early 2018. It
included three papers written by Chen and Hawkes with various other co-authors, [41], [42] and[43], and a review paper by Hawkes [44].
First, we model order ﬂows in a ﬁnancial market through a birth-death-immigration Hawkes
model in Khashanah, Chen and Hawkes [41]. This is opposite to most literature using positively
exciting processes: we model trades and quotes in a mutual-exciting bi-variate setting so thatone stream of events occurs at a decreasing rate due to interactions with the other type of events.
Second, Chen, Hawkes, Scalas and Trinh [42] compare three typical information criteria forchoosing from a collection of possible exciting kernels: Akaike’s information criterion (AIC),
Bayesian information criterion (BIC) and the Hannan-Quinn criterion (HQ).
In the meanwhile, behaviour ﬁnance research has been prevailing. Especially, it is clear thatnews sentiment exists and drives market prices to move. Yang, Liu, Chen and Hawkes [43]applied a mutually-exciting Hawkes process to understand how positive/negative price and sentiment movements interact. We ﬁnd that, in particular post the 2008 ﬁnancial crisis, the news
sentiment that reﬂects traders’ belief about the market has become more dominant in leading
the market to ﬂuctuate.
Another half dozen excellent papers are also published in this special issue, addressing various
theoretical and practical issues such as high-dimensional, non-linear Hawkes model for limit orderbook (Achab et al., [45]; Lu & Abergel, [46]), dark pool trading (Gao et al., [47]), ﬁnancial jumps
and co-jumps (Calcagnile et al., [48]), liquidity/illiquidity spillover (Schneider et al., [49]) andconstant proportion portfolio insurance (Buccioli & Kokholm, [50]).
To further advance the applications of Hawkes processes in Finance, we are currently editing
another special issue of The European Journal of Finance. Many more important issues in ﬁnancesuch as volatility clustering, ﬁnancial jumps, ﬁnancial networks, portfolio optimisation etc. will
be discussed. This will also include a review by Hawkes (2021, [51]) on Hawkes jump-diﬀusions
in Finance.
We have also promoted Hawkes processes by a series of three Workshops in Cardiﬀ (2017),
Swansea (2018) and Stevens Institute in Hoboken, New Jersey (2019). The Swansea meeting
was a 2-day aﬀair to celebrate Alan’s 80th birthday. David Colquhoun talked about Alan’s workon ion channels. Most of the other talks were about Hawkes processes in Finance. Main speakers were Mathieu Rosenbaum, Frédéric Abergel. Steve Hardiman and Judith Rousseau (Paris);
Valérie Chavez-Demoulin and Matthias Kirchner (Switzerland); Khaldoun Khashanah and Steve
Yang (New Jersey); Fabrizio Lillo (Italy); Mark Tippett (Australia), Lirong Cui (China), Anton
Merlushkin (Credit Suisse, London), Enrico Scalas (Sussex, UK) and, of course, Alan Hawkes. It
was a great birthday!
Another strand of development in Hawkes processes and their applications is to consider using
more complex kernels.: Chen, Hawkes and Scalas (2020, [52]) introduce a Mittag-Leﬄer typekernel to replace the classic ETAS models’ power law kernel. This can take advantage of thewell-studied Laplace transform of the Mittag-Leﬄer function, thus providing a simpler calculation of some properties of such a Hawkes model. Cui, Hawkes and Yi (2020, [53]) develop amethod for ﬁnding moments of properties of some Hawkes processes.
We also continue the research in news sentiment and price behaviour. Two papers by Liu, Yang,

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Chen and Hawkes (2020) do not involve Hawkes processes at all, but are based on entropy. [54]introduces entropy-based measures to quantify information ﬂows driven by prices or news, andthus identify diﬀerent types of trading behaviours and market regimes. Based on 11 years of news
and market data, we ﬁnd that the ﬁnancial market has been dominated by diﬀerent information
ﬂows before and after the double crises period (the 2008 liquidity and euro-zone debt crises).
The responses to information ﬂows are well studied in ﬁnance literature of Granger causality
studies. [55] again uses entropy to further ﬁnd traders’ responses to news have become much
more pre-dominant during the crisis and, therefore, the more eﬀective price discovery process
will need to adopt both price- and news-driven information.
5.

The present and the future of Hawkes processes

Editor: Let us back to Alan again. And now the ﬁnal question.
Editor’s Question 5 : How do you think about recent developments and applicationsof Hawkes models? How do you expect future development of Hawkes models will be?
It has been amazing to see the volume and scope of applications of Hawkes processes over thelast ﬁve years, considering that they were virtually ignored for about 30 years. The traditional
users, the seismologists, are still active. Among the more recent users we ﬁnd plenty of activity
in finance, social media and mathematical theory, with some regular work also in Neuroscience;
crime and acts of violence. We suppose that the frequency of publications using Hawkes processes
will die down eventually but, like the Poisson process, it will never entirely fade away: it is just
too useful. The processes are ﬂexible in choice of base process, and exciting kernels, with marks
and spatial eﬀects if needed, and interaction with exogenous processes. They can also be modiﬁed
in various ways to suit particular circumstances. We expect an increasing number of researches
in various ﬁelds of study will continue to make ingenious use of these properties.
It is interesting to note that, despite the suggestion in [3], that these processes might be usefulfor modelling disease, and the obvious implications of the title ETAS, epidemiologists have notmade a great deal of use of Hawkes processes. The arrival of COVID-19 has brought a sudden
spurt of interest. We do not refer to any particular publication of this type in the bibliography as
it is too early to decide which are the important ones. It is interesting to note that the branchingratio, often called R, is a regular part of the daily news reporting of the pandemic.
For the general use of Hawkes processes we would like to see development of a coherent setof model-ﬁtting techniques with well-tested, freely available, software. Bayesian methods show
particular promise, combining time-varying properties and model-learning features.
References[1] Cox, D.R. and Lewis, P.A.W. 1966. The statistical analysis of series of events. Methuen, London.
[2] Bartlett, M.S. 1963. The spectral analysis of point processes. J. Royal Statistical Society B 25, 264296.
[3] Hawkes, A.G. 1971a. Spectra of some self-exciting and mutually-exciting point processes. Biometrika,
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[4] Hawkes, A.G. 1971b. Point spectra of some mutually-exciting point processes. J. Royal StatisticalSociety, B 33, 438-443.
[5] Hawkes, A.G. 1972. Spectra of some mutually exciting point processes with associated variables.

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Chapter in Stochastic Point Processes: Statistical Analysis, Theory, and Applications, P.A.W. Lewis(ed.), Wiley, New York, 261-271.
[6] Hawkes, A.G. and Adamopoulos, L. 1973. Cluster models for earthquakes - regional comparisons. Invited paper at the ISI conference, Vienna. Bulletin International Statistical Institute 45(3), 454-461.
[7] Hawkes, A.G. and Oakes, D. 1974. A cluster process representation of a self-exciting process. J.
Applied Probability 11(3), 493-503.
[8] Hawkes, A.G. 1970. Bunching in a semi-Markov process. J. Applied Probability 7, 175-182.
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[10] Ogata, Y. (1998), Space–Time Point Process Models for Earthquake Occurrences, Annals of theInstitute of Statistical Mathematics, 50(2), 379–402.
[11] Mohler, G.O., Short, M.B., Brantingham, P.J., Schoenberg, F.P. & Tita, G.E. (2011) Self-Exciting
Point Process Modeling of Crime, Journal of the American Statistical Association, 106:493, 100-108.
[12] Lewis, P.A.W., Katcher, A.M. and Weis, A.H. 1969. An improved program for the statistical analysisof series of events. I.B.M. New York.
[13] Ozaki, T. 1979. Maximum likelihood estimation of Hawkes’ self-exciting point processes. Ann. Institute of Statistical Mathematics 31(B), 145-155.
[14] Rubin, I. 1972. Regular point processes and their detection. IEEE Trans. Information Theory, IT-18,
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[15] Dempster, A.P., Laird, N.M. and Rubin, D.B. 1977. Maximum Likelihood from Incomplete Data via
the EM Algorithm. Journal of the Royal Statistical Society, B 39 (1), 1–38.
[16] Colquhoun, D. 1971. Lectures on Biostatistics. Clarendon, Oxford, 425+xviii pp.
[17] Colquhoun, D. and Hawkes, A.G. 1977. Relaxation and ﬂuctuations of currents that ﬂow throughdrug operated ion channels. Proceedings of the Royal Society of London B199, 231-262.
[18] Colquhoun, D. and Hawkes, A.G. 1982. On the stochastic properties of bursts of single ion channel
openings and of clusters of bursts. Philosophical Transactions of the Royal Society of London B300,
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[19] Colquhoun, D. and Hawkes, A.G. 1987. A note on correlations in single ion channel records. Proceedings of the Royal Society of London B230, 15-52.
[20] Hawkes, A.G., Jalali, A. and Colquhoun, D. 1990. The distributions of the apparent open times andshut times in a single ion channel when brief events cannot be detected. Philosophical Transactionsof the Royal Society of London A332, 511-538.
[21] Hawkes, A.G., Jalali, A. and Colquhoun, D. 1992. Asymptotic distributions of apparent open times
and shut times in a single channel record allowing for the omission of brief events. PhilosophicalTransactions of the Royal Society of London B337, 383-404.
[22] Colquhoun, D. and Hawkes, A.G. 1995. The principles of the stochastic interpretation of ion-channel
mechanism. In Single-Channel Recording 2nd. ed., B. Sakmann and E. Neher (eds.), 397-482. New
York: Plenum Press.
[23] Colquhoun, D. and Hawkes, A.G. 1995. Desensitisation of N-methyl-D-aspartate receptors: a problem
of interpretation. Proceedings of the National Academy of Sciences, USA, 92, 10327-10329.
[24] Colquhoun, D., Hawkes, A.G. and Srodzinski, K. 1996. Joint distributions of apparent open times andshut times of single ion channels and the maximum likelihood ﬁtting of mechanisms. PhilosophicalTransactions of the Royal Society of London A354, 2555-2590.
[25] Colquhoun, D., Hawkes, A.G., Merlushkin, A. and Edmonds, B. 1997. Properties of single ion channel currents elicited by a pulse of agonist concentration or voltage. Philosophical Transactions of theRoyal Society of London A355, 1743-1786.
[26] Colquhoun, D., Hatton, C.J. and Hawkes, A.G. 2003. The quality of maximum likelihood estimationof ion channel rate constants. J. Physiology 547(3), 699-728.
[27] Jalali, A. and Hawkes, A.G.1992. Generalised eigenproblems arising in aggregated Markov processes
allowing for time interval omission. Advances in Applied Probability 24(2), 302-321.

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[28] Ebina, Y., Mukuno, M., Shingai, R., Nakajima, K. and Hawkes, A.G. 1989. Power spectrum density
equation of ﬂuctuating membrane current based on discrete time Markov chain model - Analysis ofion channels with 2, 3 states. Transactions of the Institute of Electronics, Information and Communication Engineers J72-D-II(11), 1926-1934. (in Japanese)[29] Fawzi, B.B. and Hawkes, A.G. 1990. Availability of a series system with replacement and repair. J.
Applied Probability 27(4), 873-887.
[30] Cui, L. and Hawkes, A.G. 1994. Availability of a series system with warm spares.
Microelectronics Reliability 34(6), 1057-1069.
[31] Cui, L., Hawkes, A.G. and Jalali, A. 1995. The increasing failure rate property of consecutive-k-outof-n systems. Probability in Engineering and Informational Sciences 9(2), 217-225[32] Jalali, A., Hawkes, A.G., Cui, L. and Hwang, F.K. 2005. The optimal consecutive-k-out-of-n : G line
for n ≤ 2k. Journal of Statistical Planning and Inference 128(1), 281-287.
[33] Zheng, Z., Cui, L. and Hawkes, A.G. 2006. A study on a single-unit Markov repairable system withrepair time omission. IEEE Transactions on Reliability 55(2), 182-188.
[34] Cui, L. and Hawkes, A.G. 2008. A note on the proof of the optimal consecutive-k-out-of-n : Glineforn ≤ 2k. Journal of Statistical Planning and Inference 138(5), 1516-1520.
[35] Hawkes, A.G., Cui, L. and Zheng, Z. 2011. Modeling the evolution of system performance underalternative environments. IIE Transactions 43(11), 761-772.
[36] Cui, L., Du, S. and Hawkes, A.G. 2012. A study on a single-unit repairable system with state aggregations, IIE Transactions, 44(11), 1022-1032.
[37] Bowsher, C.G. 2007. Modelling security market events in continuous time: Intensity based, multivariate point process models. Journal of Econometrics, 141(2), 876–912.
[38] Large, J. 2007. Measuring the resiliency of an electronic limit order book. Journal of Financial
Markets, 10(1), 1–25.
[39] Stabile, G. and Torrisi, G.L. 2010. Risk processes with non-stationary Hawkes claims arrivals. Methodology and Computing in Applied Probability, 12, 415-429.
[40] Dassios, A. and Zhao, H. 2011. A dynamic contagion process. Advances in Applied Probability, 43(3),814–846.
[41] Khashanah, K., Chen, J. and Hawkes, A. 2018. A slightly depressing jump model: intraday volatility
pattern simulation. Quantitative Finance 18(2), 213-224.
[42] Chen, J., Hawkes, A.G., Scalas, E. and Trinh, M. 2018. Performance of information criteria for
selection of Hawkes process models of ﬁnancial data. Quantitative Finance 18(2), 225-236.
[43] Yang, S.Y., Liu, A., Chen, J. and Hawkes, A. 2018. Applications of a multivariate Hawkes processto
joint modeling of sentiment and market return events. Quantitative Finance 18(2), 295-310.
[44] Hawkes, A.G. 2018. Hawkes processes and their applications to ﬁnance: a review. Quantitative Finance 18(2), 193-198.
[45] Achab, M., Bacry, E., Muzy, J.F. and Rambaldi, M. 2018. Analysis of order book ﬂows using a
non-parametric estimation of the branching ratio matrix. Quantitative Finance 18(2), 199-212.
[46] Lu, X. and Abergel, F. 2018. High-dimensional Hawkes processes for limit order books: modelling,
empirical analysis and numerical calibration. Quantitative Finance 18(2), 249-264.
[47] Gao, X., Zhou, X. and Zhu, L. Transform analysis for Hawkes processes with applications to darkpool trading. Quantitative Finance 18(2), 265-282.
[48] Calcagnile, L.M., Bormetti, G., Treccani, M., Marmi, S. and Lillo, F. 2018. Collective synchronizationand high frequency systemic instabilities in ﬁnancial markets. Quantitative Finance 18(2), 237-248.
[49] Schneider, M., Lillo, F. and Pelizzon, L. 2018. Modelling illiquidity spillovers with Hawkes processes:
an application to the sovereign bond market. Quantitative Finance 18(2), 283-294.
[50] Buccioli, A. and Kokholm, T. 2018. Constant proportion portfolio insurance strategies in contagious
markets. Quantitative Finance 18(2), 311-331.
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Mainardi, R. Garrappa and L. Beghin (eds.). SEMA SIMAI Springer[53] Cui, L., Hawkes, A. and Yi, H. 2020. An elementary derivation of moments of Hawkes processes.
Advances in Applied Probability 52(1), 102-137.
[54] Liu, A., Chen, J., Yang, S.Y. and Hawkes, A.G. 2020. The ﬂow of information in trading: an entropy
approach to market regimes. Entropy 22(9). Published online 22 Sept 2020.
[55] Liu, A., Chen, J., Yang, S.Y. and Hawkes, A.G. 2020. Information transition in trading and its
eﬀect on market eﬃciency: an entropy approach. Chapter 5 of Proceedings of the First International
Forum on Financial Mathematics and Fintech. Zhiyong Zheng (Ed.). Springer Verlag, Singapore.