Empirical study, modelling and applications of multiple limits trades in limit order books

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Empirical study, modelling and applications of multiple limits trades in limit order books

Fabrizio Pomponio

To cite this version:

Fabrizio Pomponio. Empirical study, modelling and applications of multiple limits trades in limit

order books. Other. Ecole Centrale Paris, 2012. English. �NNT : 2012ECAP0050�. �tel-00879857�

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j h`/2b@i?`Qm;? , 2KTB`B+H 7+ib M/ TTHB+iBQMb iQ H2/@H; jj

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8 *QKT`BbQM Q7 i`/2b@i?`Qm;? KB+`Qbi`m+im`H DmKTb rBi? K+`Qb+QTB+ DmKTb

U/2i2+i2/ #v biiBbiB+H i2bi `Q#mbi iQ KB+`Qbi`m+im`2 MQBb2V 38

8XR AMi`Q/m+iBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 3e 8Xk h2biBM; 7Q` DmKTb BM /Bb+`2i2Hv Q#b2`p2/ MQBbv T`Q+2bb H śi@a?HB M/ C+Q/ X 3d 8XkXR h?2 +b2 rBi? MQ KB+`Qbi`m+im`2 MQBb2 X X X X X X X X X X X X X X X X X X X X X X 3d 8XkXk h?2 +b2 rBi? KB+`Qbi`m+im`2 MQBb2 X X X X X X X X X X X X X X X X X X X X X X X X 33 8Xj 1KTB`B+H +QKT`BbQM Q7 ?B;? Q++m``2M+2 Q7 DmKTb- /2i2+i2/ #v i`/2b@i?`Qm;? M/

#v biiBbiB+H i2bi K2i?Q/ X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Ny 8XjXR 1KTB`B+H /i M/ T`K2i2`b mb2/ X X X X X X X X X X X X X X X X X X X X X X X Ny 8XjXk h2biBM; 7Q` DmKTb BM /BHv i`D2+iQ`B2b X X X X X X X X X X X X X X X X X X X X X X NR 8XjXj h2biBM; 7Q` DmKTb BM BMi`/v i`D2+iQ`B2b X X X X X X X X X X X X X X X X X X X X NR 8XjX9 *QKT`BM; i?2 BMi`/v /Bbi`B#miBQMb Q7 DmKTb Q#iBM2/ rBi? i`/2b@i?`Qm;?

M/ rBi? i?2 biiBbiB+H i2bi K2i?Q/ X X X X X X X X X X X X X X X X X X X X X X X Nk 8X9 *QM+HmbBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Nj

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HQ;@HBMûB`2V 2Mi`2 iQmb H2b i`/2b 2i H2b i`/2b i`p2`bMib- BM+HmMi H +QM/BiBQM bm`

H2 pQHmK2 BMBiBH /m i`/2 U`K2Mû m pQHmK2 /2 i`/2 KQv2M- MQiû JhoVX X X X X X ky RXj .û+`QBbbM+2 2M HQB@TmBbbM+2 /2 HǶ2t+ĕb /2 bT`2/ T`ĕb mM i`/2 i`p2`bMi bm` H2

biQ+F "LS S`B#b U2M û+?2HH2 HQ;@HQ;V X X X X X X X X X X X X X X X X X X X X X X X X X X ky RX9 *QKT`BbQM /2b `ûbmHiib- 2Mi`2 HǶH;Q`Bi?K2 JAa. 2i +2HmB /m KtBKmK /2 p`B@

b2K#HM+2- TQm` H +HB#`iBQM 2KTB`B[m2 /m MQvm /2b i`/2b i`p2`bMib /m +Ƭiû #B/

/m +`M2i /ǶQ`/`2b- TQm` H2 biQ+F "LS S`B#b 2Mi`2 CmBM 2i P+iQ#`2 kyRy URy ,yy ¨ R9 ,yyVX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X k9 RX8 *QKT`BbQM /2 H /Bbi`B#miBQM /2 H biiBbiB[m2 /2 i2bi MQ`KHBbû2 t p2+ N

_/BHv =

10000- 2Mi`2

T

= 1

DQm` Ui`Bi TH2BMV 2i T

= 15

KBM UTQBMiBHHûV- p2+ mM2 /Bbi`B#miBQM MQ`KH2 biM/`/ UxQM2 ;`Bbû2VX lM2 HB;M2 p2`iB+H2 BM/B[m2 H Kû/BM2 /2 H /Bbi`B@

#miBQMX 1im/2 K2Mû2 bm` HǶBM/B+2 6QQib22 U7mim`2 2i H2b jy biQ+Fb H2b THmb HB[mB/2bV 2M kyRRX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ke RXe U>miV o`BiBQMb BMi`/v /2 H Kû/BM2 /m i2bi t p2+ N

_/BHv = 10000

2i T

=

15

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S`B#b QM CmM2ĜP+iQ#2` kyRy URy XKX iQ k TXKXV QM i?2 #B/ bB/2 UmTV M/ i?2 bF bB/2 U/QrMV Q7 i?2 HBKBi Q`/2` #QQF UQM HQ;@HQ; b+H2VX X X X X X X X X X X X X X X X X dN 8XR h`BM;mH` F2`M2H ; +?Qb2M 7Q` i?2 `Q#mbiB}+iBQM Q7 i?2 biiBbiB+H i2bi BM T`2b2M+2 Q7

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MQ`KH /Bbi`B#miBQM Ub?/2/ `2VX UmTV .CA BM/2t Ujy biQ+Fb +QKTQbBiBQMV rBi?

N

_/BHv= 5000

c UKB//H2V .s BM/2t U7mim`2 M/ jy biQ+Fb +QKTQbBiBQMV rBi? N

_/BHv= 10000

c U/QrMV 6QQib22 BM/2t U7mim`2 M/ jy KQbi HB[mB/ biQ+Fb +QKTQbBiBQMV rBi?

N

_/BHv= 10000X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

N8 8Xj LQ`KHBx2/ biiBbiB+ /Bbi`B#miBQM +QKT`BbQM BM kyRR rBi? N

_/BHv = 10000- #2ir22M

T

= 1

/v UbQHB/ HBM2V M/ T

= 15

KBM U/Qii2/ HBM2V- rBi? biM/`/ MQ`KH /Bbi`B@

#miBQM Ub?/2/ `2VX p2`iB+H HBM2 BM/B+i2b i?2 K2/BM Q7 i?2 /Bbi`B#miBQMX UmTV .CA BM/2t Ujy biQ+Fb +QKTQbBiBQMV c UKB//H2V .s BM/2t U7mim`2 M/ jy biQ+Fb +QKTQbBiBQMV c U/QrMV 6QQib22 BM/2t U7mim`2 M/ jy KQbi HB[mB/ biQ+Fb +QKTQbBiBQMVX Ne 8X9 AMi`/v /Bbi`B#miBQM Q7 i?2 MQ`KHBx2/ biiBbiB+ K2/BM BM kyRR rBi? N

_/BHv = 10000

M/ T

= 15

KBMX UmTV .s BM/2t U7mim`2 M/ jy biQ+Fb +QKTQbBiBQMV c U/QrMV 6QQib22 BM/2t U7mim`2 M/ jy KQbi HB[mB/ biQ+Fb +QKTQbBiBQMVX X X X X X X X X X X X X X X X X X Nd 8X8 AMi`/v /Bbi`B#miBQM Q7 i?2 MQ`KHBx2/ biiBbiB+ #QtTHQi BM kyRR rBi? N

_/BHv = 10000

M/ T

= 15

KBMX UmTV .s BM/2t U7mim`2 M/ jy biQ+Fb +QKTQbBiBQMV c U/QrMV 6QQib22 BM/2t U7mim`2 M/ jy KQbi HB[mB/ biQ+Fb +QKTQbBiBQMVX X X X X X X X X X X X X X X X X X N3 8Xe J2M BMi`/v ?BbiQ;`Kb Q7 iBK2biKTb Q7 i`/2b@i?`Qm;? 7Q` i?2 jy KQbi HB[mB/

+QKTQM2Mib Q7 6.s M/ 66A BM/B+2b QM kyRR- mbBM; QM2@KBMmi2 #BMb Uk ;`T?b mTV M/ R8@KBMmi2b #BMb Uk ;`T?b /QrMV X X X X X X X X X X X X X X X X X X X X X X X X X X X NN

Rk

GBbi2 /2b i#H2mt

jXR h_h>Ƕb KQbi BKTQ`iMi i`/2b ~;b X X X X X X X X X X X X X X X X X X X X X X X X X X X je jXk .i bmKK`v i#H2 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X j3 jXj "bB+ biiBbiB+b QM kM/@HBKBi i`/2b@i?`Qm;? U2biBKiBQMb #b2/ QM J`+? kyRy /iV j3 jX9 "bB+ biiBbiB+b QM j`/@HBKBi i`/2b@i?`Qm;? U2biBKiBQMb #b2/ QM J`+? kyRy /iV jN jX8 ai#BHBiv Q7 i?2 /2}MBiBQM Q7 i`/2b@i?`Qm;? rBi? `2bT2+i iQ i?2 J2M h`/2 oQHmK2

U2biBKiBQMb #b2/ QM J`+? kyRy /iV X X X X X X X X X X X X X X X X X X X X X X X X X 9R jXe _`27+iBQM Q7 /i 7Q` 6`2M+? biQ+F BM i?2 2biBKiBQM Q7 +QM/BiBQMH T`Q##BHBiv Q7

i`/2b@i?`Qm;? X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 9j jXd .BHv HB[mB/Biv biiBbiB+b QM kM/ M/ j`/@HBKBi i`/2b@i?`Qm;? 7Q` CM@62# kyRR X X X 8j jX3 G2/@H; `2bmHib bmKK`v X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 88 jXN *QKT`BbQM Q7 #2bi KB/@iQ@KB/ `2im`Mb rBi? i`Mb+iBQM +Qbib X X X X X X X X X X X X 88 9XR P++m``2M+2b Q7 i`/2b@i?`Qm;? i #B/ M/ bF bB/2b 7Q` "LS S`B#bX X X X X X X X X X ek 9Xk *Hmbi2`BM; Q7 i`/2b@i?`Qm;? QM #B/ M/ bF bB/2b UQM "LS S`B#b /iVX X X X X X X ek 9Xj aiiBbiB+b bmKK`v 7Q` i?2 KtBKmK@HBF2HB?QQ/ 2biBKi2b Q7 i?2 bF bB/2 Q7 KQ/2H

U9XRNVX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X eN 9X9 aiiBbiB+b bmKK`v 7Q` i?2 KtBKmK@HBF2HB?QQ/ 2biBKi2b Q7 i?2 #B/ bB/2 Q7 KQ/2H

U9XRNVX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X eN 9X8 aiiBbiB+b bmKK`v 7Q` i?2 KtBKmK@HBF2HB?QQ/ 2biBKi2b Q7 i?2 bBKTHB}2/ KQ/2H

U9XkyV rBi? λ

^A₀

M/ λ

^B₀

+QMbiMiX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X dy 9Xe aiiBbiB+b bmKK`v 7Q` i?2 KtBKmK@HBF2HB?QQ/ 2biBKi2b Q7 i?2 bF bB/2 Q7 KQ/2H

U9XkyV rBi? λ

^A₀

M/ λ

^B₀

TB2+2rBb2@HBM2` +QMiBMmQmb 7mM+iBQMbX X X X X X X X X X X X X X dR 9Xd aiiBbiB+b bmKK`v 7Q` i?2 KtBKmK@HBF2HB?QQ/ 2biBKi2b Q7 i?2 #B/ bB/2 Q7 KQ/2H

U9XkyV rBi? λ

^A₀

M/ λ

^B₀

TB2+2rBb2@HBM2` +QMiBMmQmb 7mM+iBQMbX X X X X X X X X X X X X X dR 9X3 S2`7Q`KM+2 Q7 i?2 +HB#`iBQM Q7 i?2 >rF2b KQ/2HbX 6Q` 2+? KQ/2H- i?Bb i#H2

;Bp2b i?2 MmK#2` Q7 i`/BM; /vb UQmi Q7

109V r?2`24- 3- Q`2

Q` H2bb i2bib Qmi Q7 7Q`

r?2`2 bm++2bb7mHHv Tbb2/X h?2 7Qm` i2bib `2 irQ BM/2T2M/2M+2 GDmM;@"Qt i2bib M/

irQ EQHKQ;Q`Qp@aKB`MQp i2bib 7Q` i?2 2tTQM2MiBH /Bbi`B#miBQMX oHm2b BM T`2Mi?2b2b

`2 T2`+2Mi;2bX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X dk 9XN aiiBbiB+b bmKK`v 7Q` i?2 2tT2+iiBQM@KtBKBxiBQM 2biBKi2b Q7 i?2 #B/ bB/2 Q7

KQ/2H U9XkyV rBi? ky #BMbX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 3y 9XRy aiiBbiB+b bmKK`v 7Q` i?2 2tT2+iiBQM@KtBKBxiBQM 2biBKi2b Q7 i?2 bF bB/2 Q7

KQ/2H U9XkyV rBi? ky #BMbX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 3y

Rj

R9

*?TBi`2 R

AMi`Q/m+iBQM U6`2M+? p2`bBQMV

*2ii2 i?ĕb2 bǶBMb+`Bi /Mb HǶûim/2 /2 H KB+`Qbi`m+im`2 /2b K`+?ûb }MM+B2`b m MBp2m /m +`M2i /ǶQ`/`2bX 1HH2 pBb2 ¨ /û+`B`2- ¨ KQ/ûHBb2` 2i ¨ miBHBb2` 2M pm2 /ǶTTHB+iBQMb +2`iBMb ûpĕM2K2Mib T`iB+mHB2`b- /ûMQiûb +B@T`ĕb Ǵi`/2b i`p2`bMibǴ 2i +`+iû`Bbûb T` mM2 +QMbQKKiBQM ;`2bbBp2 /2 H HB[mB/Biû T`ûb2Mi2 /Mb H2 +`M2i /ǶQ`/`2bX

RXR *QMi2ti2 ;ûMû`H /Ƕûim/2- T`ûb2MiiBQM 2i mMBiû /2 H i?ĕb2

G KBb2 ¨ /BbTQbBiBQM /ǶBKK2Mb2b [mMiBiûb /2 /QMMû2b }MM+Bĕ`2b `2;`QmTMi H2b i`Mb+iBQMb 2i H2b +`M2ib /ǶQ`/`2 /ǶmM ;`M/ MQK#`2 /Ƕ+iB7b T2`KBb mM /ûp2HQTT2K2Mi BKTQ`iMi /2 H +QKT`û@

?2MbBQM /m 7QM+iBQMM2K2Mi /2b K`+?ûb }MM+B2`b m MBp2m /m +`M2i /ǶQ`/`2b /2b +iB7b +QMbB/û`ûbX m HB2m /2 KQ/ûHBb2` /B`2+i2K2Mi H2b bû`B2b /2 T`Btf`2M/2K2Mib T` /2b T`Q+2bbmb biQ+?biB[m2b- QM bǶ2bi THmiƬi ii+?û ¨ mM2 KBb2 2M `2HiBQM /2b /Bzû`2Mib T?ûMQKĕM2b KB+`Qb+QTB[m2b ¨ HǶQ`B;BM2 /2 H 7Q`KiBQM 2i /2 H KQ/B}+iBQM /2b T`BtX *2ii2 ûim/2 bvbiûKiB[m2 /2b /QMMû2b /2 +`M2i /ǶQ`/`2b KBb 2M HmKBĕ`2 /2b T`QT`Bûiûb 2KTB`B[m2b 7QM/K2MiH2b +QM+2`MMi HǶBMi2`/ûT2M/M+2 /2 HǶQz`2 2i /2 H +QMbQKKiBQM /2 H HB[mB/Biû /Mb H2 T?ûMQKĕM2 /2 7Q`KiBQM /2b T`BtX AH `2bi2 ûpB/2KK2Mi 2M+Q`2 /2b [m2biBQMb 2M bmbT2Mb }M /Ƕ++û/2` ¨ mM2 +QKT`û?2MbBQM +QKTHĕi2 /2b Kû+MBbK2b 2M D2m- BM+HmMi H 7Qm`MBim`2 2i H +QMbQKKiBQM /2 HB[mB/Biû- H /vMKB[m2 /2b +`M2ib /ǶQ`/`2- H2 K`F2i@

BKT+i- H pQHiBHBiû 2i H2 +QKTQ`i2K2Mi /2b /Bzû`2Mib +i2m`b 2M T`ûb2M+2X hQmi27QBb- +2 /QKBM2 ii2BMi mM +2`iBM MBp2m /2 Kim`Biû- +QKK2 HǶii2bi2 H2 MQK#`2 +`QBbbMi /2 `2pm2b /BbTQMB#H2b /m /QKBM2 UpQB` T` 2t2KTH2 +2HH2b /2 "Qm+?m/ 2i HX (kyyN) 2i /2 *?F`#Q`iB 2i HX (kyRR)VX

*2ii2 Kim`Biû b2 i`/mBi ¨ T`ûb2Mi T` H `2+?2`+?2 /2 MQmp2mt +?KTb /ǶTT`Q7QM/Bbb2K2Mib /Mb H +QKT`û?2MbBQM /m /QKBM2 ¨ HǶûim/2X S`KB +2mt@+B- /2mt b2K#H2Mi ûK2`;2` iQmi T`iB+m@

HBĕ`2K2Mi , H2b ûim/2b 2KTB`B[m2b #bû2b bm` /2b /QMMû2b T2`K2iiMi HǶB/2MiB}+iBQM /2b +i2m`b ¨ HǶƾmp`2- 2i +2HH2b pBbMi ¨ +QKT`2M/`2 /2b ûpĕM2K2Mib T`iB+mHB2`b H2b THmb bB;MB}+iB7b 2i H2b THmb BMbi`m+iB7b TQbbB#H2bX m }MH- bB H2b biiBbiB[m2b ;HQ#H2b +QM+2`MMi H KB+`Qbi`m+im`2 /2b K`+?ûb }MM+B2`b QMi /ǶQ`2b 2i /ûD¨ T2`KBb mM2 #QMM2 +QKT`û?2MbBQM /m /QKBM2- BH b2K#H2 [m2 H2b biiBb@

iB[m2b +QM/BiBQMM2HH2b U¨ HǶB/2MiBiû /2b +i2m`b Qm ¨ +2`iBMb ûpĕM2K2Mib T`iB+mHB2`bV QMi Qmp2`i mM +?KT /ǶBMp2biB;iBQM BMiû`2bbMi 2i T`QK2ii2m`X

*2ii2 i?ĕb2 bǶBMb+`Bi /Mb H i2MiiBp2 /2 K2BHH2m`2 +QKT`û?2MbBQM /2 H KB+`Qbi`m+im`2 /2b K`+?ûb }MM+B2`b #bû2 bm` HǶûim/2 /ǶûpĕM2K2Mib T`iB+mHB2`b [mB QMi i`Bi- /Mb MQi`2 +b- ¨ mM2 +QMbQKKiBQM ;`2bbBp2 /2 H HB[mB/Biû T`ûb2Mi2 /Mb H2 +`M2i /ǶQ`/`2bX .Mb H2 +?TBi`2 j- QM

`TT2HH2 /Ƕ#Q`/ +2`iBM2b T`QT`Bûiûb 2KTB`B[m2b 2bb2MiB2HH2b ¨ H +QKT`û?2MbBQM /2 H KB+`Qbi`m+im`2 /2b K`+?ûb }MM+B2`b 2i ¨ H /û}MBiBQM /m bmD2i /Ƕûim/2X G HB[mB/Biû BMbm{bMi2 T`ûb2Mi2 /Mb H2 +`M2i /ǶQ`/`2b T` `TTQ`i ¨ +2HH2 /ûbB`û2 T` H2b BMp2biBbb2m`b- +QmTHû2 ¨ H pQHQMiû /2 M2 Tb `ûpûH2`

mt mi`2b T`iB+BTMib b2b BMi2MiBQMb /ǶBMp2biBbb2K2Mib }M /2 M2 Tb /û+H2` H2 T`Bt /Mb H2 KmpBb

R8

b2Mb- KĕM2Mi ¨ mM /û+QmT;2 mbm2H /2b Q`/`2b K`+?û 2M mM2 bm++2bbBQM /2 T2iBib Q`/`2b /Tiûb ¨ H HB[mB/Biû /BbTQMB#H2 /Mb H2 +`M2i /ǶQ`/`2bX HǶBMp2`b2- BH ``Bp2 [m2 H2b BMp2biBbb2m`b M2 TmBbb2Mi Tb ûi2M/`2 H2m` Tû`BQ/2 /ǶBMp2biBbb2K2Mib m@/2H¨ /ǶmM2 +2`iBM2 /m`û2- KBb /QBp2Mi m +QMi`B`2

;B` `TB/2K2Mi 2i bMb +QMbB/û`iBQM /2 KBMBKBbiBQM /2 H2m` BKT+i /2 K`+?ûX m +ƾm` /2 +2 +QKT`QKBb- H2b i`/2b i`p2`bMib bQMi Dmbi2K2Mi +2mt [mB pB/2Mi +QKTHĕi2K2Mi THmbB2m`b HBKBi2b- 2M M2 HBbbMi Tb H2 i2KTb ¨ H K2BHH2m`2 HBKBi2 /m +`M2i /2 b2 `2KTHB` ¨ MQmp2m T` HǶ``Bpû2 /2 MQmp2mt Q`/`2b HBKBi2bX .Mb H2 T`2KB2` +?TBi`2- QM `ûHBb2 mM2 ûim/2 TT`Q7QM/B2 /2 H2m`b T`QT`Bûiûb 2KTB`B[m2b 2i QM H2b miBHBb2 /Mb H2 +/`2 /2 /2mt TTHB+iBQMb HBû2b m H2/@H; 2Mi`2 +iB7bfK`+?ûbX

G2 +?TBi`2 9 TQm`bmBi HǶûim/2 /ǶmM2 /2b T`QT`Bûiûb 2KTB`B[m2b KD2m`2b /2b i`/2b i`p2`bMib-

¨ bpQB` H2m` +Hmbi2`BM;X lM2 KQ/ûHBbiBQM /2b i2KTb /Ƕ``Bpû2b /2b i`/2b i`p2`bMib T` /2b T`Q@

+2bbmb biQ+?biB[m2b miQ@2t+Biûb UH2b T`Q+2bbmb /2 >rF2bV 2bi T`QTQbû2 2i b +HB#`iBQM 2bi pHB/û2 biiBbiB[m2K2MiX *2 KQ/ĕH2 T2`K2i H2 +H+mH /ǶmM BM/B+i2m` /ǶBMi2MbBiû 2bTû`û2 /2 HǶ``Bpû2 7mim`2 /2b i`/2b i`p2`bMib 2i bQM miBHBbiBQM /Mb mM2 bi`iû;B2 /ǶBMp2biBbb2K2MiX PM bǶii+?2 û;H2K2Mi

¨ Q#i2MB` H 7Q`K2 2KTB`B[m2 /m MQvm KQ/ûHBbMi H /û+`QBbbM+2 i2KTQ`2HH2 /2 HǶBKT+i T`ĕb mM i`/2 i`p2`bMi- ;`+2 ¨ mM2 Kûi?Q/2 /2 +HB#`iBQM MQM@T`Kûi`B[m2X

G2 +?TBi`2 8 `TT2HH2 mM2 Kûi?Q/2 ;ûMû`H2 /2 /ûi2+iBQM biiBbiB[m2 /2 bmib /Mb /2b bû`B2b i2KTQ`2HH2b /2 T`Btf`2M/2K2Mib [mB bQBi `Q#mbi2 m #`mBi /2 KB+`Qbi`m+im`2X PM ;ûMû`HBb2 H2b `û@

bmHiib 2KTB`B[m2b +QMMmb ¨ /2 MQmp2mt BM/B+2b }MM+B2`bX PM /Ti2 +2ii2 Kûi?Q/2 ¨ H /ûi2+iBQM biiBbiB[m2 /2 bmib /Mb H2b i`D2+iQB`2b BMi`/v }M /ǶQ#i2MB` H /Bbi`B#miBQM /2 H T`QTQ`iBQM /2 bmib /ûi2+iûb m +Qm`b /2 H DQm`Mû2X PM /ûi2`KBM2 H2b pH2m`b 2ti`ûKH2b 2i H2b THmb ;`M/2b p`BiBQMb /2 +2ii2 T`QTQ`iBQM /2 bmib /ûi2+iûb- 2i QM H2b `2HB2 ¨ H /Bbi`B#miBQM BMi`/v /2b i`/2b i`p2`bMibX 1M}M- QM T`QTQb2 mM2 2tTHB+iBQM TQbbB#H2 /2b `ûbmHiib T`û+û/2Mib #bû2 bm` mM2 KQ@

/B}+iBQM /2 H T`i `2HiBp2 /2 +?+mM2 /2b +QKTQbMi2b /2 bmib /Mb H i`D2+iQB`2 /2b +iB7b +QMbB/û`ûb- ¨ bpQB` H +QKTQbMi2 HBû2 mt KQmp2K2Mib +QMiBMmb 2i +2HH2 HBû2 mt KQmp2K2Mib /2 bmib Tm`bX

*2ii2 i?ĕb2 pBb2 /QM+ ¨ mM2 K2BHH2m`2 +QKT`û?2MbBQM /2 H KB+`Qbi`m+im`2 /2b K`+?ûb }MM+B2`b- pB HǶûim/2 /2 +2b ûpĕM2K2Mib T`iB+mHB2`b m b2BM /2b +`M2ib /ǶQ`/`2b [m2 bQMi H2b i`/2b i`p2`bMibX .2b biiBbiB[m2b 2KTB`B[m2b 2i H2m` KQ/ûHBbiBQM T` /2b T`Q+2bbmb /2 >rF2b K2ii2Mi 2M HmKBĕ`2 H2m`b T`BM+BTH2b T`QT`Bûiûb 2i T2`K2ii2Mi /2b TTHB+iBQMb HBû2b m H2/@H;- T` 2t2KTH2X 1M}M- BHb B/2Mi ¨ HǶBMi2`T`ûiiBQM /2b `ûbmHiib Q#i2Mmb [mM/ QM ;ûMû`HBb2 mM2 Kûi?Q/2 biiBbiB[m2 +QMMm2 /2 /ûi2+iBQM /2 bmib m +b /2 i`D2+iQB`2b BMi`/vX

PM /QMM2 ¨ T`ûb2Mi TQm` +?[m2 +?TBi`2 mM +QMi2ti2- mM2 T`Q#HûKiB[m2 /ûiBHHû2 2i H2b T`BM@

+BTmt `ûbmHiib Q#i2MmbX

RXk S`QT`Bûiûb 2KTB`B[m2b /2b i`/2b i`p2`bMib 2i /2mt TTHB+@

iBQMb m H2/@H;

PM bBi /2TmBb H2b i`pmt /2 "Qm+?m/ 2i HX (kyy9) 2i /2 GBHHQ M/ 6`K2` (kyy9) [m2 H2

~mt /2b i`/2b T`ûb2Mi2 mM2 miQ+Q``ûHiBQM /2b bB;M2b T`iB+mHBĕ`2K2Mi 7Q`i2X SHmb T`û+BbûK2Mi- H2b mi2m`b QMi T`QTQbû /2b KQ/ĕH2b /2 ivT2 HQB@TmBbbM+2 TQm` `2T`ûb2Mi2` H 7QM+iBQM 2KTB`B[m2 /ǶmiQ+Q``ûHiBQM /2b bB;M2b /2b i`/2b , C

(τ)∼

τ

^−γ

- T`ĕb mM i2KTb τ X G2b 2tTQbMib 2KTB`B[m2b γ Q#i2Mmb p`B2Mi b2HQM H2b K`+?ûb ûim/Bûb T` +2b mi2m`b- KBb `2bi2Mi +QKT`Bb 2Mi`2

0.2

2i

0.7-

+2 [mB i`/mBi /2b /û+`QBbbM+2b i2KTQ`2HH2b i`ĕb 7B#H2bX 1M /Ƕmi`2b i2`K2b- H2 bB;M2 /ǶmM i`/2 +QMb2`p2 mM BKT+i bB;MB}+iB7 bm` H2 bB;M2 /2b i`/2b bmBpMib m #Qmi /2 /m`û2b TQmpMi bǶûi2M/`2 Dmb[mǶ¨ /2mt b2KBM2bX

Re

SHmbB2m`b 2tTHB+iBQMb ¨ +2 T?ûMQKĕM2 QMi ûiû T`QTQbû2b UpQB` H `2pm2 /2 "Qm+?m/ 2i HX (kyyN)- 2i T`iB+mHBĕ`2K2Mi H2b i`pmt /2 G2"`QM M/ uKKQiQ (kyyd)- /2 GBHHQ 2i HX (kyy8) 2i /2 :2`B;

(kyyd)VX m }MH- H HB[mB/Biû BMbm{bMi2 [mB 2bi T`ûb2Mi2 /Mb H2 +`M2i /ǶQ`/`2b T` `TTQ`i ¨ +2HH2 /ûbB`û2 T` H2b BMp2biBbb2m`b- +QmTHû2 ¨ H pQHQMiû /2 M2 Tb `ûpûH2` mt mi`2b T`iB+BTMib b2b BM@

i2MiBQMb /ǶBMp2biBbb2K2Mib }M /2 M2 Tb /û+H2` H2 T`Bt /Mb H2 KmpBb b2Mb- T2`K2ii2Mi /2 `2M/`2 +QKTi2 /2 +2 /û+QmT;2 mbm2H /2b Q`/`2b 2M mM2 bm++2bbBQM /2 T2iBib Q`/`2b /Tiûb ¨ H HB[mB/Biû /BbTQMB#H2 /Mb H2 +`M2i /ǶQ`/`2bX

G2b `ûbmHiib /2 :2`B; (kyyd) QMi ûiû Q#i2Mmb ;`+2 ¨ HǶmiBHBbiBQM /2 /QMMû2b BM+HmMi /2b BM7Q`@

KiBQMb bm` HǶB/2MiBiû /2b +i2m`b 2M T`ûb2M+2- MQM Tb HǶB/2MiBiû BM/BpB/m2HH2 /2b BMp2biBbb2m`b +QKK2 +2 b2`Bi B/ûH2K2Mi H2 +b- KBb H2 ǴK2K#2`b?BT +Q/2Ǵ B/2MiB}Mi +2HmB [mB 2tû+mi2 H2 i`/2X *2ii2 BM7Q`KiBQM 2bi T`iB2HH2 +` ¨ mM ǴK2K#2`b?BT +Q/2Ǵ /QMMû +Q``2bTQM/2Mi TQi2MiB2HH2K2Mi THmbB2m`b BMp2biBbb2m`b BM/BpB/m2Hb }Mmt- KBb 2HH2 bǶ2bi pû`û2 bm{bMi2X G2b mi2m`b QMi +QM/BiBQMMû H2 +H@

+mH /2 H 7QM+iBQM 2KTB`B[m2 /ǶmiQ+Q``ûHiBQM /2b bB;M2b /2 i`/2b ¨ /2b i`/2b T`Qp2MMi /m KāK2 ǴK2K#2`b?BT +Q/2Ǵ- /2 /Bzû`2Mib ǴK2K#2`b?BT +Q/2bǴ Qm BM+QM/BiBQMM2HH2K2MiX G2 +QM/BiBQMM2K2Mi T` mM KāK2 ǴK2K#2`b?BT +Q/2Ǵ #QmiBi ¨ mM2 7QM+iBQM 2KTB`B[m2 /ǶmiQ+Q``ûHiBQM /2b bB;M2b /2 i`/2b mM Q`/`2 /2 ;`M/2m` THmb ;`M/ [m2 +2HH2 BM+QM/BiBQMM2HH2- iQmi 2M +QMb2`pMi mM2 /û+`QBb@

bM+2 2M HQB@TmBbbM+2 p2+ mM 2tTQbMi THmb T2iBiX HǶBMp2`b2- [mM/ QM +QM/BiBQMM2 T` /2b i`/2b T`Qp2MMi /2 ǴK2K#2`b?BT +Q/2bǴ /Bzû`2Mib- H 7QM+iBQM 2KTB`B[m2 /ǶmiQ+Q``ûHiBQM /2b bB;M2b /2 i`/2b /û+`Qŗi i`ĕb `TB/2K2Mi p2`b xû`Q 2i M2 bmBi THmb mM2 HQB@TmBbbM+2X aQmb HǶ?vTQi?ĕb2 [m2 H2b BMp2biBbb2m`b MǶmiBHBb2Mi [mǶmM 7B#H2 MQK#`2 /2 #`QF2`b- +2b biiBbiB[m2b +QM}`K2Mi HǶ2tTHB+iBQM T`û+û/2Mi2 /m2 ¨ GBHHQ 2i HX (kyy8)X

SQm`iMi- BH ``Bp2 T`7QBb [m2 H2b BMp2biBbb2m`b M2 TmBbb2Mi Tb ûi2M/`2 H2m` Tû`BQ/2 /ǶBMp2biBbb2K2Mib m@/2H¨ /ǶmM2 +2`iBM2 /m`û2- KBb /QBp2Mi m +QMi`B`2 ;B` `TB/2K2Mi 2i bMb +QMbB/û`iBQM /2 KBMBKBbiBQM /2 H2m` BKT+i /2 K`+?ûX AHb M2 T2mp2Mi b2 T2`K2ii`2 /2 b+MM2` H2 +`M2i /ǶQ`/`2b TQm` HBKBi2` H iBHH2 /2 H2m`b T2iBib Q`/`2b ¨ H HB[mB/Biû T`ûb2Mi2 2M K2BHH2m`2 HBKBi2- KBb bQMi HQ`b +QMi`BMib /ǶmiBHBb2` /2b i`/2b +`+iû`Bbûb T` mM2 +QMbQKKiBQM T`iB+mHBĕ`2K2Mi ;`2bbBp2 /2 H HB[mB/BiûX *2b i`/2b pB/2Mi +QKTHĕi2K2Mi THmbB2m`b HBKBi2b- bMb HBbb2` H2 i2KTb ¨ H K2BHH2m`2 HBKBi2 /m +`M2i /2 b2 `2KTHB` ¨ MQmp2m T` HǶ``Bpû2 /2 MQmp2mt Q`/`2b HBKBi2bX *2 bQMi +2b i`/2b T`iB+mHB2`b @ H2b i`/2b i`p2`bMib @ /QMi QM bǶTTHB[m2 ¨ /ûi2`KBM2` H2b T`BM+BTH2b T`QT`Bûiûb 2K@

TB`B[m2b /Mb +2 T`2KB2` +?TBi`2X

pMi +2H- `TT2HQMb [m2 H2b T`QT`Bûiûb biiBbiB[m2b /2b +`M2ib /ǶQ`/`2b QMi /ûD¨ ûiû ûim/Bû2b /Mb H2 /ûiBH- T` 2t2KTH2 /Mb "Qm+?m/ M/ SQii2`b (kyyj)- "Qm+?m/ 2i HX (kyyN) 2i H2b `û7û@

`2M+2b [mB v bQMi K2MiBQMMû2bX BMbB `2pBbBi2@i@QM /Mb +2 +?TBi`2 +2`iBM2b /2b biiBbiB[m2b /ûD¨

#B2M +QMMm2b /2b +`M2ib /ǶQ`/`2b- KBb /Mb H2 +b T`iB+mHB2` /m +QM/BiBQMM2K2Mi mt i`/2b i`@

p2`bMibX S` 2t2KTH2- mM2 [m2biBQM `û+m``2Mi2 ûim/Bû2 /Mb H HBiiû`im`2 +QM+2`M2 HǶBKT+i /2 K`+?û UB2 H `ûTQMb2 KQv2MM2 /m T`Bt mt i`/2bV- +7 H2b i`pmt /2 SQii2`b M/ "Qm+?m/ (kyyj)- /2 q2#2` M/ _Qb2MQr (kyy8)- /2 HK;`2M 2i HX (kyy8)- /2 >mib+? M/ >mM; (kyRk)- /2 1BbH2`

2i HX (kyRk) 2i /2 *QMi 2i HX (kyRR)X G2 +QM/BiBQMM2K2Mi /2 H biiBbiB[m2 /ǶBKT+i /2 K`+?û mt i`/2b i`p2`bMib b2` /QM+ /ûiBHHû /Mb H bmBi2X LQiQMb /ĕb ¨ T`ûb2Mi [m2 +2`iBMb i`pmt mH@

iû`B2m`b /2 KQ/ûHBbiBQM KB+`Qb+QTB[m2 /2 HǶBKT+i /2 K`+?û 2z2+imûb T` 1BbH2` 2i HX (kyRk) QMi +?QBbB /2 /Qm#H2` H +HbbB}+iBQM mbm2HH2 /2b ûpĕM2K2Mib 2M i`QBb +iû;Q`B2b Ui`/2b- Q`/`2b HBKBi2b 2i MMmHiBQMb /ǶQ`/`2bV T` mM +QM/BiBQMM2K2Mi bBKBHB`2 ¨ +2HmB [m2 MQmb T`ûb2MiQMb- ¨ bpQB` bB H2 T`Bt /2 HǶûpĕM2K2Mi +QMbB/û`û ûiû KQ/B}ûX .Mb H2 +b T`iB+mHB2` /2b i`/2b- +2H `2pB2Mi ¨ 7Q`K2` /2mt +iû;Q`B2b /2 i`/2b 2M 7QM+iBQM /2 H2m` +QMbQKKiBQM +QKTHĕi2- Qm Tb- /2 H HB[mB/Biû T`ûb2Mi2 2M K2BHH2m`2 HBKBi2X PM +QKT`2` THmb 2M /ûiBH /Mb H2 +Q`Tb /2 H i?ĕb2 H2b `ûbmHiib Bbbmb /2 +2ii2 /û}MBiBQM T`Q+?2 /2b i`/2b i`p2`bMib- KāK2 bB /Bzû`2Mi2X

Rd

G2b i`/2b i`p2`bMib bQMi ûpB/2KK2Mi `2HBûb mt ;`Qb i`/2b 2i mt ;`M/2b p`BiBQMb /2 T`BtX AH 2bi ¨ MQi2` [m2 /2b `2+?2`+?2b T`û+û/2Mi2b bǶûiB2Mi BMiû`2bbû2b ¨ /ûi2`KBM2` H +mb2 /2b ;`M/b KQmp2K2Mib /2b +`M2ib /ǶQ`/`2b U+7 6`K2` 2i HX (kyy9) 2i q2#2` M/ _Qb2MQr (kyye)V- 2Mi`2 KM[m2 /2 HB[mB/Biû 2i ;`M/ pQHmK2 /2 i`/2bX 6`K2` 2i HX (kyy9) QMi `2HBû H2b ;`M/b +?M;2@

K2Mib /2 T`Bt bm` H2 GQM/QM aiQ+F 1t+?M;2 ¨ /2b i`/2b [mB +QMbQKK2Mi +QKTHĕi2K2Mi H2 pQHmK2 /BbTQMB#H2 2M K2BHH2m`2 HBKBi2 U+QKK2 H2 7QMi H2b i`/2b i`p2`bMibV- +`ûMi BMbB mM /û+H;2 /2 T`Bt û;H ¨ HǶû+`i 2Mi`2 H K2BHH2m`2 HBKBi2 2i H bmBpMi2X *2H BM/B[m2 mM2 +QMi`B#miBQM BKTQ`iMi2 /2b ~m+imiBQMb /2 HB[mB/Biû /2b +`M2ib /ǶQ`/`2b /Mb H2b ;`M/b +?M;2K2Mib /2 T`BtX .2b `ûbmHiib bBKBHB`2b QMi ûiû Q#i2Mmb T` q2#2` M/ _Qb2MQr (kyye) bm` /2b /QMMû2b Bbbm2b /m La.Z 2i QMi +QM}`Kû [m2 H2b ~m+imiBQMb 2ti`āK2b /2 T`Bt ûiB2Mi T`BM+BTH2K2Mi `2HBû2b ¨ mM2 7B#H2 HB[mB/Biû /Mb H2 +`M2i /ǶQ`/`2bX

G /ûi2`KBMiBQM /2b `2HiBQMb /2 H2/@H; 2Mi`2 +iB7b 2ifQm K`+?ûb 2bi /ǶmM ;`M/ BMiû`āi T`iB[m2X PM T`H2 /2 `2HiBQM /2 H2/@H; 2Mi`2 /2mt +iB7b [mM/ H +QMMBbbM+2 /2b `2M/2K2Mib /ǶmM +iB7 /QMMû B/2 ¨ H T`û/B+iBQM /2b `2M/2K2Mib 7mim`b /ǶmM mi`2 +iB7X *2ii2 [m2biBQM `2Ïm mM2 ii2MiBQM T`iB+mHBĕ`2 /Mb H HBiiû`im`2 `û+2Mi2- +QKK2 2M iûKQB;M2Mi H2b i`pmt /2 #2`;2H M/ >mi? (kyRR)- /2 >QzKMM 2i HX (kyRR) 2i H2b `û7û`2M+2b [mB v bQMi K2MiBQMMû2bX lM2 TT`Q+?2 +HbbB[m2 /2 `ûbQHmiBQM /2 +2 T`Q#HĕK2 +QMbBbi2 ¨ ûim/B2` mM2 7QM+iBQM /2 +Q``ûHiBQM +`QBbû2 /2b

`2M/2K2Mib H;;mûb- 2i ¨ K2bm`2` mM2 bvKûi`B2 2Mi`2 H2b H;b TQbBiB7b 2i +2mt Mû;iB7b- Qm #B2M ¨ 2t?B#2` mM H; MQM MmH [mB KtBKBb2 H +Q``ûHiBQM +`QBbû2X LQi`2 T`Q#HûKiB[m2 2bi [m2H[m2 T2m /Bzû`2Mi2 TmBb[mǶQM +?2`+?2 ¨ /ûi2`KBM2` [m2H +iB7fK`+?û #Qm;2 2M T`2KB2`- B2 bB mM KQmp2K2Mi BKTQ`iMi vMi HB2m bm` HǶ+iB7fK`+?û R 2bi- 2M KQv2MM2- THmiƬi bmBpB Qm THmiƬi T`û+û/û T` mM KQmp2K2Mi BKTQ`iMi bm` HǶ+iB7fK`+?û kX PM i2`KBM2 +2 T`2KB2` +?TBi`2 T` mM2 b2+QM/2 TTHB@

+iBQM m T`Q#HĕK2 /m H2/@H;- THmb +HbbB[m2 2M T`iB[m2- [mB ûpHm2 H 7Q`+2 T`û/B+iBp2 /m bB;MH /2b i`p2`bMib /Mb mM2 bi`iû;B2 /2 H2/@H; 2Mi`2 +iB7bX

RXkXR _ûbmHiib Q#i2Mmb

PM `ûHBb2 mM2 ûim/2 TT`Q7QM/B2 /2b T`QT`Bûiûb 2KTB`B[m2b /2b i`/2b i`p2`bMibX PM bǶBMiû`2bb2 /Ƕ#Q`/ ¨ bpQB` bB H2b i`/2b i`p2`bMib `2T`ûb2Mi2Mi mM2 TQTmHiBQM biiBbiB[m2 bm{bKK2Mi BK@

TQ`iMi2 TQm` āi`2 ûim/Bû2X *QM+2`MMi H2m` HB[mB/Biû- H2b i`/2b i`p2`bMib bQMi /2b ûpĕM2K2Mib

`2HiBp2K2Mi ``2b- KBb BHb M2 bQMi Tb /m iQmi Mû;HB;2#H2b TmBb[mǶBHb T2mp2Mi +QMi2MB` [mbBK2Mi

20W /m pQHmK2 i`Biû TQm` H2 7mim`2 bm` BM/B+2 .s- T` 2t2KTH2X AHb `2T`ûb2Mi2Mi 2MpB`QM mM2

T`QTQ`iBQM /2

5W /2b i`/2b /2b biQ+Fb /m **9y- +2 [mB +QM/mBi TQm` "LS@S`B#b Qm _2MmHi ¨

mM2 Q++m``2M+2 DQm`MHBĕ`2 /2

250@300

ûpĕM2K2MibX

AH 2bi BMimBiBp2K2Mi +HB` [m2 THmb H2 pQHmK2 /ǶmM i`/2 /QMMû 2bi BKTQ`iMi 2i THmb b T`Q##BHBiû /Ƕāi`2 mM i`/2 i`p2`bMi 2bi ;`M/2X PM pû`B}2 biiBbiB[m2K2Mi +2ii2 bBimiBQM 2M i`ÏMi H +Qm`#2 /2 T`Q##BHBiû /Ƕāi`2 mM i`/2 i`p2`bMi 2M 7QM+iBQM /m pQHmK2 /m i`/2- +Qm`#2 [mB 2bi #B2M 2KTB`B[m2K2Mi +`QBbbMi2X AH 2bi û;H2K2Mi BMiû`2bbMi /2 MQi2` [mǶBH 2tBbi2 +2`iBMb i`/2b /2 i`ĕb

;`M/b pQHmK2b [mB M2 bQMi Tb /2b i`/2b i`p2`bMib- +2 [mB `2~ĕi2 H T`ûb2M+2 2t+2TiBQMM2HH2 2M K2BHH2m`2 HBKBi2 /ǶmM2 [mMiBiû #B2M bmTû`B2m`2 ¨ H MQ`KH2X

lM2 T`QT`Bûiû 2KTB`B[m2 BKTQ`iMi2 /2b i`/2b i`p2`bMib- bm` H[m2HH2 MQmb `2pB2M/`QMb THmb 2M /ûiBH /Mb H2 T`Q+?BM +?TBi`2- +QM+2`M2 H2m` +Hmbi2`BM;X G2b i`/2b i`p2`bMib QMi i2M/M+2 ¨

``Bp2` T` ;`TT2b c mi`2K2Mi /Bi- QM ii2M/ 2M KQv2MM2 KQBMb H2 T`Q+?BM i`/2 i`p2`bMi [mM/

QM pB2Mi /Ƕ2M Q#b2`p2` mM- THmiƬi [m2 [mM/ QM pB2Mi /ǶQ#b2`p2` mM i`/2 [m2H+QM[m2X 1i +2 `ûbmHii 2bi p`B [m2 H2 /ûHB 2Mi`2 H2b i`/2b bQBi K2bm`û 2M MQK#`2 /2 i`/2b Ui2KTb ûpûM2K2MiB2HV Qm 2M b2+QM/2b Ui2KTb T?vbB[m2VX

G2b i2KTb /Ƕ``Bpû2 /2b i`/2b i`p2`bMib QMi û;H2K2Mi i2M/M+2 ¨ b2 `2;`QmT2` ¨ /2b ?2m`2b

R3

10 12 14 16

0.20.40.60.81.0

Hour of the day (Paris time)

Histogram

9 10 11 12 13 14 15 16 17

6B;m`2

RXR Ĝ >BbiQ;`KK2 BMi`/v KQv2M /2b i2KTb /Ƕ``Bpû2 /2b i`/2b i`p2`bMib TQm` H2b jy +QKTQbMi2b H2b THmb HB[mB/2b /2 HǶBM/B+2 6QQib22 2M kyRR Up2+ /2b BMi2`pHH2b /2 i2KTb /2 R KBMmi2VX

T`û+Bb2b /2 H DQm`Mû2- +QKK2 H2 `ûpĕH2 H2m` /Bbi`B#miBQM BMi`/v- T`iB+mHBĕ`2K2Mi K`[mû2 T` H T`ûb2M+2 /2 /2mt TB+b ¨ /2b ?2m`2b T`û+Bb2b /2 H DQm`Mû2 UR9 ,jy 2i Re ,yy

^R

V- HBûb ¨ H iQK#û2 /2 MQmp2HH2b K+`Q@û+QMQKB[m2b la T`iB+mHBĕ`2K2Mi bmBpB2b- 2i T` mM2 ?mbb2 /2 `û;BK2 ¨ R8 ,jy +QMbû+miBp2 ¨ HǶQmp2`im`2 /2b K`+?ûb Kû`B+BMb Lua1 2i La.ZX PM `2T`Q/mBi ¨ H };m`2 RXR HǶ?BbiQ;`KK2 KQv2M /2b i2KTb /Ƕ``Bpû2 /2b i`/2b i`p2`bMib /2b jy biQ+Fb H2b THmb HB[mB/2b /2 HǶBM/B+2 6QQib22 TQm` BHHmbi`2` +2b `ûbmHiibX

^k

GǶBKT+i /2 K`+?û /2b i`/2b i`p2`bMib 2bi bmTû`B2m` ¨ +2HmB /2b i`/2b +HbbB[m2b- 2i +2 KāK2

¨ pQHmK2 +QKT`#H2- +QKK2 BM/B[mû /Mb H };m`2 RXkX 1M `ûbmKû- H2b i`/2b i`p2`bMib T`ûb2Mi2Mi mM +QMi2Mm BM7Q`KiBQMM2H THmb ;`M/X

G };m`2 RXj KQMi`2 H `2HtiBQM /m bT`2/ 2M i2KTb T?vbB[m2 T`ĕb mM i`/2 i`p2`bMi- p2+

mM2 /û+`QBbbM+2 2M HQB TmBbbM+2 /Ƕ2tTQbMi

0.25

TQm` H2 biQ+F "LS@S`B#bX

T`ĕb +2b /Bzû`2Mi2b T`QT`Bûiûb 2KTB`B[m2b /2b i`/2b i`p2`bMib- QM T`ûb2Mi2 /2mt TTHB+iBQMb [mB TQ`i2Mi bm` H2 KāK2 bmD2i /Ƕûim/2 UH2 H2/@H;V- KBb [mB /Bzĕ`2Mi b2MbB#H2K2Mi /Mb H2m` 2bT`BiX G T`2KBĕ`2 TTHB+iBQM pBb2 ¨ miBHBb2` H2b i`/2b i`p2`bMib TQm` `ûTQM/`2 ¨ H [m2biBQM /2 bpQB`

[m2H +iB7fK`+?û #Qm;2 2M T`2KB2`X .Mb +2ii2 QTiB[m2- QM +QKK2M+2 T` /û}MB` /2mt K`+?ûb 2i

¨ 2M 2ti`B`2 H2b +iB7b H2b THmb `2T`ûb2MiiB7b TQbbB#H2b UH2b THmb HB[mB/2b- T` 2t2KTH2VX PM K2i 2M

`2HiBQM H2b i`/2b i`p2`bMib H2b THmb `TT`Q+?ûb 2Mi`2 H2b /2mt ;`QmT2b /Ƕ+iB7bX SmBb- QM K2bm`2 H2 /û+H;2 i2KTQ`2H TQm` +?[m2 TB`2 /2 i`/2b i`p2`bMib BMbB KBb 2M `2HiBQM- pMi /Ƕ2M i`+2`

H /Bbi`B#miBQMX 1M +QKT`Mi H2b +mKmHû2b /2b T`iB2b TQbBiBp2b 2i Mû;iBp2b /2 H /Bbi`B#miBQM /2 +2 /û+H;2 i2KTQ`2H 2Mi`2 H2b i`/2b i`p2`bMib /2b /2mt ;`QmT2b /Ƕ+iB7b- QM T2mi /ûi2`KBM2` [m2H K`+?û 2bi 2M pM+2 bm` HǶmi`2X 1M miBHBbMi +2ii2 i2+?MB[m2- QM KQMi`2 T` 2t2KTH2 [m2 H2 K`+?û +iBQMb la 2bi 2M pM+2 bm` H2 K`+?û +iBQMb 2m`QTû2M- 2i [m2 H2 biQ+F 7`MÏBb aQ+Bûiû :ûMû`H2 2bi 2M pM+2 bm` "LS@S`B#bX

ZmMi ¨ H b2+QM/2 TTHB+iBQM- 2HH2 K2bm`2 H 7Q`+2 /m bB;MH /2b i`/2b i`p2`bMib /Mb H2 +/`2 /ǶmM2 bi`iû;B2 /ǶBMp2biBbb2K2Mi /2 ivT2 H2/@H;X PM +QMbB/ĕ`2 mM2 TB`2 /Ƕ+iB7b /QMMû2 [mB bQBi bmb+2TiB#H2 /2 7Q`K2` mM +QmTH2 H2/2`@H;;2` BMiû`2bbMiX

^j

PM miBHBb2 HQ`b H2b i`/2b i`p2`bMib

RX ?2m`2 /2 S`Bb

kX G KāK2 biiBbiB[m2 ûiû 7Bi2 bm` /2b biQ+Fb 2m`QTû2Mb- /QMi 7`MÏBb 2i HH2KM/b- BMbB [m2 bm` /2b 7mim`2b 2m`QTû2Mb KD2m`b p2+ /2b +QM+HmbBQMb bBKBHB`2bX

jX PM T2mi T` 2t2KTH2 /ûi2`KBM2` +2ii2 TB`2 /Ƕ+iB7b 2M +?QBbBbbMi +2HH2 /QMi H +Q``ûHiBQM /2b `2im`Mb H;;mûb 2bi 2KTB`B[m2K2Mi ûH2pû2X

RN

1e03 1e01 1e+01 1e+03

0.00.20.40.60.81.01.2

Lag (in seconds)

Market impact (in ticks)

0 < V < MTV/2 MTV/2 < V < MTV MTV < V < 2 MTV 2 MTV < V Unconditional Conditional

6B;m`2

RXk Ĝ *QKT`BbQM /2b 7QM+iBQMb /2 `ûTQMb2 ;HQ#H2 TQm` H2 biQ+F "LS S`B#b U2M û+?2HH2 HQ;@HBMûB`2V 2Mi`2 iQmb H2b i`/2b 2i H2b i`/2b i`p2`bMib- BM+HmMi H +QM/BiBQM bm` H2 pQHmK2 BMBiBH /m i`/2 U`K2Mû m pQHmK2 /2 i`/2 KQv2M- MQiû JhoVX

5 10 20 50 100 200

0.150.200.250.300.350.40

Lag since last tradethrough (physical time in seconds)

Excess spread (in ticks)

Empirical Powerlaw fit

6B;m`2

RXj Ĝ .û+`QBbbM+2 2M HQB@TmBbbM+2 /2 HǶ2t+ĕb /2 bT`2/ T`ĕb mM i`/2 i`p2`bMi bm` H2 biQ+F "LS S`B#b U2M û+?2HH2 HQ;@HQ;V

ky

+QKK2 mM BM/B+i2m` HQ+H /ǶmM2 `2HiBQM /2 H2/@H;X G bi`iû;B2 /ǶBMp2biBbb2K2Mi +QMbBbi2 ¨ T`2M/`2 TQbBiBQM bm` HǶ+iB7 H;;2` T`ĕb +?[m2 i`/2 i`p2`bMi bm` HǶ+iB7 H2/2`- 2i +2 /Mb H2 KāK2 b2Mb [m2 +2 /2`MB2`- +2 [mB 2M 7Bi mM 2t2KTH2 /2 bi`iû;B2 ǶKQK2MimKǶX G2b `ûbmHiib Q#i2Mmb KQMi`2Mi /2b pH2m`b KQv2MM2b /2 `ûTQMb2 2M `2M/2K2Mi KB/T`B+2@iQ@KB/T`B+2 HHMi Dmb[mǶ¨

1.4

#T T`ĕb

0.5

b2+QM/2b TQm` H2b 7mim`2b 2i Dmb[mǶ¨

1.2

#T T`ĕb

10

b2+QM/2b TQm` H2b biQ+Fb c m+mM bB;MH M2 T`pB2Mi ¨ /ûTbb2` H2b +Qȿib /2 i`Mb+iBQM- +2 [mB Q#HB;2 ¨ M2 Tb HBKBi2` H bi`iû;B2 ¨ HǶmiBHBbiBQM /ǶQ`/`2b K`+?ûX

RXj JQ/ûHBbiBQM /2b i`/2b i`p2`bMib T` /2b T`Q+2bbmb /2 >rF2b

G2b T`Q+2bbmb /2 >rF2b bQMi /2b T`Q+2bbmb miQ@2t+Biûb BMBiBH2K2Mi BMi`Q/mBib T` >rF2b (RNdR)X PM T`H2 /2 T`Q+2bbmb miQ@2t+Biûb [mM/ HǶ``Bpû2 /ǶmM ûpĕM2K2Mi m;K2Mi2 H T`Q##BHBiû /Ƕ``Bpû2 /ǶmM MQmp2H ûpĕM2K2MiX GǶûim/2 /2 >rF2b (RNdR) b2 +QM+2Mi`2 bm` H2 +b 7QM/K2MiH /ǶmM2 m;K2MiiBQM BMbiMiMû2 2i /ǶmM2 /û+`QBbbM+2 2tTQM2MiB2HH2 /2 H T`Q##BHBiû /Ƕ``Bpû2 /ǶmM ûpĕM2K2Mi 7mim`- T`ĕb HǶ``Bpû2 /ǶmM ûpĕM2K2Mi /QMMûX

.Mb mM +b bBKTH2 UmMB/BK2MbBQMM2H ¨ T`Kĕi`2b +QMbiMib p2+ mM mMB[m2 MQvmV- HǶBMi2MbBiû /ǶmM T`Q+2bbmb /2 >rF2b 2bi /û+`Bi2 +QKK2 bmBi ,

λ(t) = λ

₀+ Z t

0

αe

^−β(t−s)

dN

_s=

λ

₀+X

ti<t

αe

^{−β(t−ti)}

G2 T`Q+2bbmb

(Nt)t≥0

/m MQK#`2 +mKmHû /2 bmib pMi HǶBMbiMi t 2bi /û}MB T` bQM BMi2MbBiû biQ@

+?biB[m2

(λ(t))t≥0

X HǶBMbiMi t >

0- +2ii2 BMi2MbBiû BMbiMiMû2

λ(t) 2bi H bQKK2 /ǶmM2 BMi2MbBiû /Ƕ``Bĕ`2@THM λ

₀

>

0

2i /2 H +mKmHû2 `ûbB/m2HH2 U+QMi`2 H2 MQvm 2tTQM2MiB2HV /2b ?mbb2b /ǶBMi2MbBiû +QMbû+miBp2b mt ûpĕM2K2Mib Tbbûb vMi 2m HB2m mt BMbiMib t

_i

< tX GǶmiQ@2t+BiiBQM pB2Mi /m 7Bi [mǶmM ûpĕM2K2Mi ``BpMi ¨ HǶBMbiMi t

^∗

p T`QpQ[m2` mM2 ?mbb2 BMbiMiMû2 /2 HǶBMi2MbBiû /ǶmM2 pH2m` α >

0X *2ii2 ?mbb2 /2 HǶBMi2MbBiû p /û+`Qŗi`2 m +Qm`b /m i2KTb p2`b xû`Q b2HQM H2 MQvm 2t@

TQM2MiB2H e

^{−β(t−t∗)}

X G2 T`Kĕi`2 β >

0

;ĕ`2 /QM+ H pBi2bb2 /2 /û+`QBbbM+2 i2KTQ`2HH2 /2 HǶBKT+i /ǶmM ûpĕM2K2Mi Tbbû bm` HǶBMi2MbBiû +im2HH2X

aB H2b T`Q+2bbmb /2 >rF2b QMi i`Qmpû mM /QKBM2 /ǶTTHB+iBQM ûpB/2Mi 2M bBbKQHQ;B2 U/Mb H T`ûpBbBQM /2b i`2K#H2K2Mib /2 i2``2

⁹

V- H }MM+2 [mMiBiiBp2 û;H2K2Mi iB`û ;`M/ T`Q}i /2 HǶBMi`Q/m+iBQM /2 +2b MQmp2mt QmiBHb /2 KQ/ûHBbiBQMX LQK#`2 /2 +QMi`B#miBQMb `û+2Mi2b QMi bQmHB;Mû H2b T`QT`Bûiûb BMiû`2bbMi2b /2 +2b T`Q+2bbmb miQ@ 2i Kmim2HH2K2Mi 2t+BiûbX AHb bQMi- T` 2t2KTH2- ¨ H #b2 /2b i`pmt /2 G`;2 (kyyd) 2i /2 JmMB hQF2 (kyRR) TQm` KQ/ûHBb2` H2b i2KTb /Ƕ``Bpû2b /2b Q`/`2b /Mb /2b KQ/ĕH2b /2 +`M2i /ǶQ`/`2bX PM H2b `2i`Qmp2 /Mb HǶûim/2 /2 "Qrb?2` (kyyd)- ++QK@

T;Mûb /ǶmM +/`2 û+QMQKûi`B[m2 /Tiû- TQm` +HB#`2` /2b ``Bpû2b /2 i`/2b 2i /2 +?M;2K2Mib /2 KB/[mQi2bX SHmb `û+2KK2Mi- H .v`B 2i HX (kyRR) H2b miBHBb2 TQm` KQ/ûHBb2` /2b ``Bpû2b /2 i`/2bX 1M}M- H2b i`pmt /2 "+`v 2i HX (kyRk) KQMi`2Mi [mǶBHb T2`K2ii2Mi mM2 KQ/ûHBbiBQM KB+`Qb+QTB[m2 bBKTH2 /2 HǶ2z2i 1TTb

⁸

X

.Mb H2 +/`2 /ǶmM2 KQ/ûHBbiBQM /2b i`/2b i`p2`bMib- +2ii2 miQ@2t+BiiBQM /2b T`Q+2bbmb /2

>rF2b BM+Bi2 ¨ H2b miBHBb2` TQm` `2M/`2 +QKTi2 /m +Hmbi2`BM; 2KTB`B[m2 Q#b2`pûX 1M v DQmiMi H TQbbB#BHBiû /2 ;ûMû`HBbiBQM m +b KmHiB@/BK2MbBQMM2H- HǶBMi2`T`ûi#BHBiû T?vbB[m2 /2b /Bzû`2Mib T`Kĕi`2b miBHBbûb- H /BbTQMB#BHBiû /ǶmM2 Kûi?Q/2 T2`7Q`KMi2 /2 +HB#`iBQM T` KtBKmK /2 p`Bb2K#HM+2 2i 2M}M- H2 +H+mH TQbbB#H2 T` i`Mb7Q`Kû2 /2 GTH+2 /ǶmM BM/B+i2m` /ǶBMi2MbBiû

9X Qɍ HǶQM bBi #B2M [m2 H bm`p2Mm2 /ǶmM i`2K#H2K2Mi /2 i2``2 BMBiBH 2bi ;ûMû`H2K2Mi ++QKT;Mû2 /2 i`2K#H2K2Mib /2 i2``2 mHiû`B2m`b- TT2Hûb `ûTHB[m2bX

8X /û+Q``ûHiBQM 2KTB`B[m2 /2b `2M/2K2Mib /2 /2mt +iB7b mt û+?2HH2b KB+`Qb+QTB[m2bX

kR

7mim`2 2bTû`û2- H2b T`Q+2bbmb /2 >rF2b bQMi /QM+ /2b +M/B/ib T`BpBHû;Bûb ¨ H KQ/ûHBbiBQM /2b i`/2b i`p2`bMibX

RXjXR _ûbmHiib Q#i2Mmb

.Mb H2 +?TBi`2 9- QM TT`Q7QM/Bi /Ƕ#Q`/ HǶûim/2 2KTB`B[m2 /m +Hmbi2`BM; /2 HǶ``Bpû2 /2b i`/2b i`p2`bMibX PM bBi /ûD¨ [m2 H2b i`/2b i`p2`bMib QMi i2M/M+2 ¨ ``Bp2` T` ;`TT2bX PM b2 /2@

KM/2 ¨ T`ûb2Mi bB +2 `ûbmHii `2bi2 p`B [mM/ QM iB2Mi û;H2K2Mi +QKTi2 /Mb HǶûim/2 /m +Ƭiû /m +`M2i [mB 2bi iQm+?ûX PM Q#b2`p2 HQ`b [mǶmM +Hmbi2`BM; 2bi ¨ HǶƾmp`2 bB QM `2;`/2 H2 KāK2 +Ƭiû /m +`M2i c BH v #B2M miQ@2t+BiiBQM /Ƕ``Bpû2 /2b i`/2b i`p2`bMib bm` mM +Ƭiû /QMMû /m +`M2i- T`ĕb HǶ``Bpû2 /ǶmM i`/2 i`p2`bMi bm` +2 KāK2 +ƬiûX 1M `2pM+?2- HǶ2t+BiiBQM 2KTB`B[m2 Kmim2HH2 /2b /2mt +Ƭiûb /m +`M2i T`ĕb mM i`/2 i`p2`bMi 2bi THmb 7B#H2X

PM `TT2HH2 2MbmBi2 H /û}MBiBQM 2i H2b T`QT`Bûiûb T`BM+BTH2b /2b T`Q+2bbmb /2 >rF2b miBH2b ¨ H bmBi2 /2 HǶûim/2 U+QM/BiBQM /2 biiBQMM`Biû- 2biBKiBQM T`Kûi`B[m2 `û+m`bBp2 T` KtBKmK /2 p`Bb2K#HM+2 2i i2bib /2 H +HB#`iBQM Q#i2Mm2VX G2 KQ/ĕH2 #B/BK2MbBQMM2H /Ƕ``Bpû2 /2b i`/2b i`p2`bMib m #B/ 2i ¨ HǶbF [m2 HǶQM T`QTQb2 2bi H2 bmBpMi ,

λ

^A(t) =

λ

^A₀ + Z t

0

α

_AA

e

^{−βAA(t−s)}

dN

_s^A+ Z t

0

α

_AB

e

^{−βAB(t−s)}

dN

_s^B

, λ

^B(t) =

λ

^B₀ +

Z t 0

α

BA

e

^{−βBA(t−s)}

dN

_s^A+ Z t

0

α

BB

e

^{−βBB(t−s)}

dN

_s^B

.

S` 2t2KTH2- HǶBMi2MbBiû /m T`Q+2bbmb /Ƕ``Bpû2 /2b i`/2b i`p2`bMib ¨ HǶbF λ

^A(t)

2bi bQKK2 /ǶmM2 BMi2MbBiû /Ƕ``Bĕ`2@THM λ

^A₀

- /2 H +QKTQbMi2 /ǶmiQ@2t+BiiBQM /m2 mt i`/2b i`p2`bMib T`û@

+û/2Mib ¨ HǶbF

(Rt

0

α

AA

e

^{−βAA(t−s)}

dN

_s^A)

2i /2 H +QKTQbMi2 /Ƕ2t+BiiBQM Kmim2HH2 /m2 mt i`/2b i`p2`bMib T`û+û/2Mib m #B/

(Rt

0

α

_AB

e

^{−βAB(t−s)}

dN

_s^B)X

G +HB#`iBQM /2b /Bzû`2Mib T`Kĕi`2b T` KtBKmK /2 p`Bb2K#HM+2 +QM/mBi ¨ /2b pH2m`b i`ĕb /Bzû`2Mi2b 2Mi`2 H2b T`Kĕi`2b /ǶmiQ@2t+BiiBQM 2i +2mt /Ƕ2t+BiiBQM Kmim2HH2X aB QM +QKT`2 H2m` BM~m2M+2 +mKmHû2- `2T`ûb2Miû2 T` H2 `iBQ

^α_β

û;H ¨ HǶBMi2MbBiû iQiH2 TQm` H2 MQvm 2tTQM2MiB2H

R_+∞

0

αe

^−βu

du- QM Q#iB2Mi /2b pH2m`b KQv2MM2b #B2M THmb 7B#H2b /Mb H2 +b /2 HǶ2t+BiiBQM Kmim2HH2 U

^α_β^AB

AB = 0.017- ^α_β^BA

BA = 0.038V [m2 /Mb H2 +b /2 HǶmiQ@2t+BiiBQM U^α_β^AA

AA = 0.111- ^α_β^BB

BB = 0.120VX

*2 `ûbmHii +QM}`K2 H2b biiBbiB[m2b 2KTB`B[m2b /2 +Hmbi2`BM; T`û+û/2KK2Mi K2MiBQMMû2b 2i MQmb KĕM2 ¨ +QKT`2` H2 KQ/ĕH2 +QKTH2i ¨ mM KQ/ĕH2 bBKTHB}û- bMb 2t+BiiBQM Kmim2HH2 ,

λ

^A(t) =

λ

^A₀ + Z t

0

α

_AA

e

^{−βAA(t−s)}

dN

_s^A

, λ

^B(t) =

λ

^B₀ +

Z t 0

α

BB

e

^{−βBB(t−s)}

dN

_s^B

.

1M bmBpMi H Kûi?Q/2 /2 "Qrb?2` (kyyd)- QM /ûiBHH2 /2mt i2bib biiBbiB[m2b /2 pHB/iBQM /2 H +HB#`iBQM /2b KQ/ĕH2b /2 >rF2b mt /QMMû2b 2KTB`B[m2bX :`+2 ¨ mM2 T`QT`Bûiû /2 +?M;2K2Mi /2 i2KTb TQm` H2b T`Q+2bbmb /2 >rF2b- QM T2mi +H+mH2` mM2 BMi2MbBiû BMiû;`û2 2KTB`B[m2- +2Mbû2 /û+`B`2 H2b BMi2`pHH2b /2 i2KTb /ǶmM T`Q+2bbmb /2 SQBbbQM ?QKQ;ĕM2X AH `2bi2 HQ`b ¨ pû`B}2` bB +2b BMi2MbBiûb BMiû;`û2b bQMi #B2M 2tTQM2MiB2HH2K2Mi /Bbi`B#mû2b U;`+2 ¨ mM i2bi biM/`/ /2 EQHKQ;Q`Qp@aKB`MQp- T` 2t2KTH2V 2i bB 2HH2b bQMi #B2M BM/ûT2M/Mi2b Up2+ mM i2bi /2 GDmM;@"Qt- T` 2t2KTH2VX

AH v /QM+ 9 i2bib ¨ 2z2+im2` m iQiH , /2mt TQm` H2b p`B#H2b 2KTB`B[m2b +H+mHû2b ¨ T`iB` /2 H KQ/ûHBbiBQM m #B/- /2 KāK2 TQm` +2HH2b Bbbm2b /2 H KQ/ûHBbiBQM ¨ HǶbFX PM KQMi`2 HQ`b [m2

kk

+2b 9 i2bib bQMi iQmb pHB/ûb /Mb

76W /2b +b- bB QM BM+Hmi HǶ2t+BiiBQM Kmim2HH2 /Mb H2 KQ/ĕH2X PM

`2bi2 bBMQM ¨ mM MBp2m /2

71W /2 +b pHB/ûb- +2 [mB +QM}`K2 H 7B#H2bb2 /2 HǶ2t+BiiBQM Kmim2HH2

/2b i`/2b i`p2`bMib +QKT`û2 ¨ H2m` miQ@2t+BiiBQMX aB QM Qz`2- 2M Qmi`2- H TQbbB#BHBiû mt T@

`Kĕi`2b λ

^A₀

2i λ

^B₀

/2 p`B2` m +Qm`b /m i2KTb pB mM T`Q}H BMi`/v HBMûB`2 T` KQ`+2mt- QM T`pB2Mi KāK2 ¨ pHB/2` biiBbiB[m2K2Mi +2b 9 i2bib /Mb THmb /2

87W /2b +bX .ĕb HQ`b- BH MǶv THmb

m+mM TTQ`i biiBbiB[m2 /2 H +QKTQbMi2 /Ƕ2t+BiiBQM Kmim2HH2 /Mb H KQ/ûHBbiBQM- T` `TTQ`i

¨ +2HH2 /ǶmiQ@2t+BiiBQMX

AH 2bi Mim`2H /2 pQmHQB` miBHBb2` H KQ/ûHBbiBQM /2b /QMMû2b }MM+Bĕ`2b T` /2b T`Q+2bbmb /2

>rF2b /Mb H2 +/`2 /2 bi`iû;B2b /ǶBMp2biBbb2K2Mib UpQB`- T` 2t2KTH2- H MQi2 /2 *`HbbQM 2i HX (kyyd)- /Mb H[m2HH2 mM bB;MH /Ƕ+?i@p2Mi2 2bi #bû bm` H2 `iBQ /ǶBMi2MbBiûb /2b i`/2b bB;MûbVX A+B- QM T`Q}i2 /2 H bi`m+im`2 2tTQM2MiB2HH2 /m MQvm TQm` +H+mH2` mM BM/B+i2m` /ǶBMi2MbBiû 7mim`2 KQv2MM2 /Ƕ``Bpû2 /2b i`/2b i`p2`bMib- /Mb H2 +/`2 /m KQ/ĕH2 bMb 2t+BiiBQM Kmim2HH2 pHB/û T`û+û/2KK2MiX :`+2 ¨ mM2 i`Mb7Q`KiBQM /2 GTH+2- QM Q#iB2Mi HǶ2tT`2bbBQM /2 HǶBM/B+i2m` `2@

+?2`+?û- bQmb H 7Q`K2 /ǶmM2 bQKK2 /2 HǶBMi2MbBiû KQv2MM2 /2 bmib Uλ

₀×_β−^β_α

V Hû;ĕ`2K2Mi T2`im`#û2- 2i /ǶmM i2`K2 /ǶBKT+i /ȿ mt bmib T`û+û/2Mib [mB /û+`Qŗi 2tTQM2MiB2HH2K2Mi pBi2 ,

E(λ(t

+

δ)

|Ft^N) =

λ

₀×

β

−

αe

^{−(β−α)δ}

β

−

α

+

e

^{−(β−α)δ} Z t

0

αe

^−β(t−u)

dN

u

PM KQMi`2 [m2 HǶmiBHBbiBQM /2 +2i BM/B+i2m` T2`K2i mM2 KûHBQ`iBQM b2MbB#H2 /m `2M/2K2Mi KQv2M Q#i2Mm /Mb mM2 bi`iû;B2 bBKTH2 /2 ivT2 ǶKQK2MimKǶ #bû2 bm` H2b i`/2b i`p2`bMib- +2 [mB +QM}`K2 HǶBMiû`āi /ǶmM2 KQ/ûHBbiBQM T2`K2iiMi /2 `2M/`2 +QKTi2 /m +Hmbi2`BM; /2 +2b ûpĕM2K2MibX PM bǶbbm`2 û;H2K2Mi [m2 H +Hbb2 /2 KQ/ĕH2b T`QTQbû2 T`û+û/2KK2Mi ¨ H +HB#`iBQM- 2i [mHB}û2 /2 Ǵ:2M2`HBx2/ HBM2` KQ/2HbǴ /Mb "Qrb?2` (kyyd)- 2bi /Tiû2 2i [mǶBH MǶ2tBbi2 Tb /Ƕ2z2i BM?B#Bi2m` T`ĕb HǶ``Bpû2 /ǶmM i`/2 i`p2`bMiX PM TQm``Bi 2M 2z2i BK;BM2` mM2 bBim@

iBQM Qɍ HǶ``Bpû2 /ǶmM ivT2 /ǶûpĕM2K2Mib /QMMû /BKBMm2 H T`Q##BHBiû /ǶQ++m``2M+2 /ǶmM mi`2 ivT2 /ǶûpĕM2K2Mib- T?ûMQKĕM2 [mB M2 TQm``Bi āi`2 BMiû;`û m +/`2 /2 KQ/ûHBbiBQM T`û+û/2MiX G +@

HB#`iBQM /2 /2mt KQ/ĕH2b /ǶBMi2MbBiû BMbTB`ûb /2b T`Q+2bbmb /2 >rF2b 2i T2`K2iiMi H KBb2 2M HmKBĕ`2 /ǶmM 2z2i BM?B#Bi2m` UmM KQ/ĕH2 HQ;@HBMûB`2 2i mM KQ/ĕH2 ¨ b2mBHV KQMi`2 [m2 +2i 2z2i MǶ2bi Tb bB;MB}+iBp2K2Mi T`ûb2Mi /Mb H2b /QMMû2b /2 i`/2b i`p2`bMibX

1M}M- QM T`QHQM;2 HǶûim/2 /2 H 7Q`K2 2KTB`B[m2 /m MQvm /2 /û+`QBbbM+2 i2KTQ`2H /ǶBKT+iX +2ii2 }M- QM miBHBb2 mM2 +HB#`iBQM MQM@T`Kûi`B[m2 /2 ivT2 1J UǴ1tT2+iiBQM JtBKBxiBQMǴV TQm` H2b T`Q+2bbmb /2 >rF2b- Bbbm2 /2b i`pmt /2 J`bM M/ G2M;HBM2 (kyy3) 2i TT2Hû2 JAa.

UǴJQ/2H AM/2T2M/2Mi aiQ+?biB+ .2+Hmbi2`BM;ǴVX *2HH2@+B K2i 2M ûpB/2M+2- bm` H };m`2 RX9- mM2 /û+`QBbbM+2 THmb 7Q`i2 [m2 H HQB 2tTQM2MiB2HH2 T`û+û/2KK2Mi miBHBbû2- 2i /pMi;2 T`Q+?2 /ǶmM2 HQB@TmBbbM+2X

RX9 *QKT`BbQM 2Mi`2 H2b i`/2b i`p2`bMib 2i H Kûi?Q/2 bi@

iBbiB[m2 /2 śi@a?HB 2i HX (kyRk) TQm` H /ûi2+iBQM /2 bmib /Mb H2b i`D2+iQB`2b BMi`/v

.2 MQK#`2mt T`Q#HĕK2b T`Qp2MMi /2 HǶBM/mbi`B2 }MM+Bĕ`2- HHMi /2 H KQ/ûHBbiBQM /Ƕ+iB7b }MM+B2`b ¨ H2m` +Qmp2`im`2- 2M TbbMi T` H2 ;2biBQM /2b `Bb[m2b- bQMi 7Q`i2K2Mi /ûT2M/Mib /2 H T`ûb2M+2 /2 bmib /Mb H2b +iB7b +QMbB/û`ûbX *2H 2tTHB[m2 T`Q##H2K2Mi [mǶmM2 [m2biBQM KD2m`2 /Mb H HBiiû`im`2 /2 H }MM+2 [mMiBiiBp2 bQBi /2 /ûi2`KBM2`- ¨ T`iB` /ǶQ#b2`piBQMb 2KTB`B[m2b /2 i`D2+iQB`2b- bB +2b /QMMû2b bQMi +QKTiB#H2b p2+ mM T`Q+2bbmb bQmb@D+2Mi +QMiBMm- Qm #B2M bB mM2 +QKTQbMi2 /2 bmib /QBi āi`2 BM+Q`TQ`û2 ¨ H KQ/ûHBbiBQMX .Mb H `ûHBiû- QM M2 /BbTQb2 [m2

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02468

Delta timestamp (in seconds)

Kernel value

● Nonparametric fit Exponential fit

6B;m`2

RX9 Ĝ *QKT`BbQM /2b `ûbmHiib- 2Mi`2 HǶH;Q`Bi?K2 JAa. 2i +2HmB /m KtBKmK /2 p`B@

b2K#HM+2- TQm` H +HB#`iBQM 2KTB`B[m2 /m MQvm /2b i`/2b i`p2`bMib /m +Ƭiû #B/ /m +`M2i /ǶQ`/`2b- TQm` H2 biQ+F "LS S`B#b 2Mi`2 CmBM 2i P+iQ#`2 kyRy URy ,yy ¨ R9 ,yyVX

/ǶQ#b2`piBQMb /Bb+`ĕi2b [mB bQMi +QKTQbû2b 2MiBĕ`2K2Mi /2 bmib- T` +QMbi`m+iBQMX AH 7mi HQ`b /B`2 bB +2b bmib 2KTB`B[m2b bQMi /ȿb ¨ HǶû+?MiBHHQMM;2 /ǶmM T`Q+2bbmb +QMiBMm- Qm bǶBHb `2~ĕi2Mi H T`ûb2M+2 /2 p`Bb bmib /2 HǶ+iB7X

SHmbB2m`b TT`Q+?2b QMi ûiû T`QTQbû2b TQm` /ûi2+i2` H T`ûb2M+2 /2 bmib /Mb H2b i`D2+iQB`2b /Bb+`ĕi2b Q#b2`pû2b bm` H2b K`+?ûb }MM+B2`bX śi@a?HB (kyyk) i2bi2 mM2 BMû;HBiû /2 +QMiBMmBiû

#bû2 bm` H 7QM+iBQM /2 i`MbBiBQM /ǶmM T`Q+2bbmb bmTTQbû bmBp`2 mM2 /BzmbBQMX G2 i2bi /2 *`` M/

qm (kyyj) 2bi #bû bm` mM2 T`QT`Bûiû /2b QTiBQMb Kim`Mi bQmb T2m c [mM/ H /m`û2 pMi û+?ûM+2 /2 HǶQTiBQM /û+`Qŗi p2`b xû`Q- H pH2m`@i2KTb /2 HǶQTiBQM +QMp2`;2 p2`b xû`Q ¨ mM2 pBi2bb2 /ûT2M/Mi

¨ H 7QBb /2 H bi`m+im`2 /m T`Q+2bbmb /m bQmb@D+2Mi U2M +HB`- /2 H T`ûb2M+2 TQi2MiB2HH2 /2 bmibV 2i /2 H TQbBiBQM T` `TTQ`i ¨ H KQMMB2 U/Mb H KQMMB2 Qm ?Q`b /2 H KQMMB2VX "`M/Q`z@LB2Hb2M M/ a?2T?`/ (kyy9) QMi BMBiBû mM2 MQmp2HH2 bû`B2 /2 i2bib `2TQbMi bm` /2b ;ûMû`HBbiBQMb /2 H p@

`BM+2 `ûHBbû2- TT2Hû2b p`BiBQMb #B@TmBbbM+2b 2i- THmb ;ûMû`H2K2Mi- p`BiBQMb KmHiB@TmBbbM+2bX CBM; M/ PQK2M (kyy3) QMi /ûp2HQTTû mM i2bi [mB bǶTTmB2 bm` HǶ2``2m` /2 `ûTHB+iBQM +mKmHû2 /ǶmM brT /2 p`BM+2 +Qmp2`i /2 KMBĕ`2 /Bb+`ĕi2X G22 M/ JvFHM/ (kyy3) 2i G22 M/ >MMB; (kyRy) +H+mH2Mi- [mMi ¨ 2mt- mM [mQiB2Mi /2 i2bi +QKT`Mi H2b `2M/2K2Mib 2i H2b pQHiBHBiûb BMbiMiMû2b U2biBKû2b p2+ /2b p`BiBQMb #B@TmBbbM+2b `ûHBbû2b TQm` HǶmM- 2i /2b p`BiBQMb [m/`iB[m2b i`QM@

[mû2b TQm` HǶmi`2VX 1M}M- śi@a?HB M/ C+Q/ (kyyN) /ûp2HQTT2Mi mM i2bi ¨ T`iB` /m [mQiB2Mi /2 p`BiBQMb TmBbbM+2b û+?MiBHHQMMû2b ¨ /2b 7`û[m2M+2b /Bzû`2Mi2bX

G2 +/`2 /Ƕ2biBKiBQM mbm2H 2M biiBbiB[m2 bbm`2 mM2 K2BHH2m`2 +QMp2`;2M+2 /2b 2biBKi2m`b [mM/ H 7`û[m2M+2 /Ƕû+?MiBHHQMM;2 m;K2Mi2X JBb H2b /QMMû2b }MM+Bĕ`2b QMi +2+B /2 T`iB+mHB2`

[mǶmM2 m;K2MiiBQM /2 H 7`û[m2M+2 /Ƕû+?MiBHHQMM;2 p2`b H2 /QKBM2 /2 H ?mi2@7`û[m2M+2 KĕM2

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