JournalofAppliedMathematicsandPhysics,2018,6,2219-2226
http://www.scirp.org/journal/jamp
ISSNOnline:2327-4379
ISSNPrint:2327-4352
SomeImportantPropertiesofMultipleG-Itˆo
IntegralintheG-ExpectationSpace
FangyuanLiu
UniversityofShanghaiforScienceandTechnology,Shanghai,China
Howtocitethispaper:Liu,F.Y.(2018)
SomeImportantPropertiesofMultipleG-Itˆo
IntegralintheG-ExpectationSpace.Journal
ofAppliedMathematicsandPhysics,6,2219-
2226.
https://doi.org/10.4236/jamp.2018.611186
Received:October20,2018
Accepted:November10,2018
Published:November13,2018
Copyright
c
2018byauthorandScientific Re-
search PublishingInc.
This work is licensed under the Creative Com-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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turientmontes,nasceturridiculusmus.Pel-
lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integermetus.Morbi pulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehicula purus.Fusce acnislaliquam ante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdum malesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbi dapibus mauris condimentum
nulla.Cum sociisnatoque penatibusetmagnis
disparturientmontes,nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesque aliquetsodalesurna.
Nullameleifendjustoin nisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
dui dignissim suscipit.Morbimagna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sed pede sit amet elitrhoncus ultrices.Nullam
atenimvitaepedevehiculaiaculis.
Classaptent tacitisociosquadlitoratorquen-
tperconubia nostra,perinceptoshymenaeos.
Aeneannonummyturpisidodio.Integereu-
ismodimperdietturpis.Utnecleonecdi-
amimperdietlacinia.Etiamegetlacuseget
miultriciesposuere.Inplacerattristiquetor-
tor.Sedportavestibulummetus.Nullaia-
culissollicitudinpede.Fusceluctustellusin
dolor.Curabiturauctorvelitasem.Mor-
bisapien.Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Donecadipiscingurnavehic-
ulanunc.Sedornareleoinleo.Inrhoncus
leoutdui.Aeneandolorquam,volutpatnec,
fringillaid,consectetuervel, pede.
Nullamalesuadarisusuturna.Aenean
pretiumvelitsitametmetus.Duisiaculis.In
hachabitasseplateadictumst.Nullammo-
lestieturpisegetnisl.Duisamassaidpede
dapibusultricies.Sedeuleo.Inatmaurissit
amettortorbibendumvarius.Phasellusjusto
risus,posuerein,sagittisac,variusvel,tor-
tor.Quisqueidenim.Phasellusconsequat,
liberopretium nonummy fringilla,tortor lacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitae ipsumac metusdictum tempus.Namut
wisi.Quisque tortor felis, interdum ac, sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donec tempusneque vitaeest.Aeneanegestas
odio sed risus ullamcorper ullamcorper.Sed in
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehicula eget,accumsaneu,tincidunta,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra, lacus nibh dignissim tellus, eu suscipit
risusantefringilladiam.Quisquealiberov-
elpedeimperdietaliquet.Pellentesquenunc
nibh,eleifenda,consequatconsequat,hen-
dreritnec,diam.Sedurna.Maecenaslaoreet
eleifendneque.Vivamuspurusodio,eleifend
non,iaculisa,ultricessitamet,urna.Mau-
risfaucibusodiovitaerisus.Innisl.Praesent
purus.Integeriaculis,semeuegestaslacinia,
lacuspedescelerisqueaugue,inullamcorper
dolorerosaclacus.Nuncinlibero.
Fuscesuscipitcursussem.Vivamusrisusmi,
egestasac,imperdietvarius,faucibusquis,
leo.Aeneantincidunt.Donecsuscipit.Cras
idjustoquisnibhscelerisquedignissim.Ali-
quamsagittiselementumdolor.Aeneancon-
sectetuerjusto inpede.Curabitur ullamcorper
ligula nec orci.Aliquam purus turpis, aliquam
id, ornarevitae, porttitor non,wisi.Maecenas
luctusportalorem.Donec vitaeligulaeu ante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclacinia auctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Ut quisnunc sedodio malesuadaegestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesque eu wisi.Proin turpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justo lectus, condimentum sit amet, posuere a,
fringillamollis, felis.Morbi nulla nibh,pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquam pulvinarblandit.Nunc sitamet
orci.Aliquameratelit,pharetranec,aliquet
a,gravidain,mi.Quisqueurnaenim,viverra
quis,suscipitquis,tinciduntut,sapien.Cras
placeratconsequatsem.Curabituracdiam.
Curabiturdiamtortor,molliset,viverraac,
tempusvel, metus.
Curabituraclorem.Vivamusnonjustoin
duimattisposuere.Etiamaccumsanligula
idpede.Maecenastinciduntdiamnecvelit.
Praesentconvallissapienacest.Aliquamul-
lamcorpereuismodnulla.Integermollisenim
veltortor.Nullasodalesplaceratnunc.Sed
tempusrutrumwisi.Duisaccumsangravi-
dapurus.Nuncnunc.Etiamfacilisisduieu
sem.Vestibulumsemper.Praesenteueros.
Vestibulumtellusnisl,dapibusid,vestibulum
sit amet,placerat ac, mauris.Maecenas et elit
uteratplaceratdictum.Namfeugiat,turpis
etsodalesvolutpat, wisiquamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisque consectetuer.In suscipit mauris a do-
lor pellentesque consectetuer.Mauris convallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiam fermentum convallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarcu libero, malesuadavitae, ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisseutpurus sedsem
tempor rhoncus.Utquam dui, fringillaat, dic-
tum eget,ultricies quis, quam.Etiam sem est,
pharetra non, vulputate in, pretium at, ipsum.
Nunc semper sagittis orci.Sedscelerisque sus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareante nuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisi ultricestellus,quislobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,pellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummy vulputate orci.Donec et velit ac ar-
cuinterdumsemper.Morbipedeorci,cursus
ac,elementumnon,vehiculaut,lacus.Cras
volutpat.Namvelwisiquisliberovenenatis
placerat.Aeneansedodio.Quisqueposuere
purusacorci.Vivamusodio.Vivamusvar-
ius,nullasitametsemperviverra,odiomau-
risconsequatlacus,atvestibulumnequearcu
eutortor.Doneciaculistincidunttellus.Ali-
quameratvolutpat.Curabiturmagnalorem,
dignissimvolutpat,viverraet,adipiscingnec,
dolor.Praesentlacusmauris,dapibusvitae,
sollicitudinsitamet,nonummyeget,ligula.
Crasegestasipsum anisl.Vivamus variusdo-
lorutdolor.Fuscevelenim.Pellentesque
accumsanligulaeteros.Crasidlacusnon
tortorfacilisisfacilisis.Etiamnislelit,cur-
sussed,fringillain,conguenec,urna.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.In-
tegeratturpis.Cumsociisnatoquepenati-
busetmagnisdisparturient montes,nascetur
ridiculusmus.Duisfringilla,ligulasedporta
fringilla,ligulawisi commodofelis,utadipisc-
ingfelisduiinenim.Suspendissemalesuada
ultricesante.Pellentesquescelerisqueaugue
sitameturna.Nullavolutpataliquettortor.
Crasaliquam,tellusataliquetpellentesque,
justosapiencommodoleo,idrhoncussapien
quamaterat.Nullacommodo,wisiegetsol-
licitudinpretium,orciorcialiquamorci,ut
cursusturpisjustoetlacus.Nullaveltor-
tor.Quisque eratelit, viverrasit amet,sagittis
eget,portasitamet,lacus.
In hac habitasse platea dictumst.Proin at est.
Curabitur tempusvulputate elit.Pellentesque
sem.Praesent eusapien.Duiselit magna,ali-
quet at,tempus sed, vehicula non, enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimet commodomalesuada, tortormetusel-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultricies accumsanmagna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesent tempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
teger rutrum sem egetwisi.Aenean eu sapien.
Quisqueornaredignissimmi.Duisaurnavel
risus pharetraimperdiet.Suspendissepotenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sed eros.Suspendisseconsequat.Mauris a dui
et tortormattis pretium.Sed nulla metus,vo-
lutpatid, aliquameget, ullamcorper ut,ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Utblandit egestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
ris ornarevelitid nibh pretium ultricies.Fusce
temporpellentesqueodio.Vivamusauguepu-
rus,laoreetin,scelerisquevel,commodoid,
wisi.Duisenim.Nulla interdum, nunc eusem-
pereleifend,enimdolorpretiumelit,utcom-
modoligulanislaest.Vivamusante.Nulla
leomassa,posuerenec,volutpatvitae,rhon-
cuseu,magna.
Quisquefacilisisauctorsapien.Pellentesque
gravidahendreritlectus.Maurisrutrumso-
dalessapien.Fuscehendreritsemvellorem.
Integerpellentesquemassavelaugue.Integer
elit tortor, feugiat quis, sagittis et, ornare non,
lacus.Vestibulumposuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quam quis quam nonmetus eleifend interdum.
Nam eget sapien ac mauris malesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibus a, odio.Ut eros erat, vulputate ut, in-
terdumnon,portaeu,erat.Crasfermentum,
felis inporta congue,velit leofacilisis odio,vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrum tellus,velposuere nibhlacus nec nibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestie dolornec metus.Vestibulum libero.Sed
quiserat.Sedtristique.Duispedeleo,fer-
mentumquis,consectetuereget,vulputatesit
amet,erat.
Donecvitaevelit.Suspendisseportafermen-
tummauris.Utvelnuncnonmaurisphare-
travarius.Duisconsequatliberoquisurna.
Maecenasatante.Vivamusvarius,wisised
egestastristique,odiowisiluctusnulla,lobor-
tisdictumdolorligulainlacus.Vivamusali-
quam, urna sed interdum porttitor,metus orci
interdum odio,sit ameteuismodlectus feliset
leo.Praesentacwisi.Namsuscipitvestibu-
lum sem.Praesent eu ipsumvitae pedecursus
venenatis.Duis sed odio.Vestibulum eleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravida ante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissim ac, venenatisvitae, consequat ut, lacus.
Nam nibh.Vestibulum fringilla arcu mollis ar-
cu.Sedetturpis.Donecsemtellus,volut-
pat et,varius eu,commodosed, lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus, nibh massa eleifendturpis, sed pellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpatduimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquamconsectetuer variusnul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravida velit at elit vehicula porttitor.Phasel-
lusnislmi, sagittisac,pulvinarid,gravidasit
amet,erat.Vestibulumest.Loremipsumdo-
lorsit amet,consectetueradipiscing elit.Cur-
abitur idsem elementumleo rutrum hendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sed turpis quam, sollicitudin a, hendrerit eget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodo nec, semper sit amet, elit.Nulla lec-
tus risus,condimentum ut,laoreet eget,viver-
ra nec, odio.Proin lobortis.Curabitur dictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorper tristique.Mauris enimsem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate, neque.Quisquelacus.Donec etipsum sit
amet elitnonummy aliquet.Sed viverra nislat
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolor sit amet, consectetuer adipiscing elit.Ut
pede sem, tempus ut, porttitor bibendum, mo-
lestieeu,elit.Suspendissepotenti.Sedid lec-
tus sitamet purus faucibus vehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverra ultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac, nunc.Maurismalesuada malesua-
da elit.Etiam lacus mauris, pretium vel, blan-
ditin, ultriciesid, libero.Phasellus bibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.Inhac habitasseplatea dictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristique quis,felis.Vestibulumanteip-
sumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Pellentesqueinterdum
quamsitametmi.Pellentesquemaurisdui,
dictuma,adipiscingac,fermentumsitamet,
lorem.
Utquiswisi.Praesentquismassa.Vivamus
egestasrisusegetlacus.Nunctincidunt,risus
quisbibendumfacilisis,lorempurusrutrum
neque,necportatortorurnaquisorci.Ae-
neanaliquet,liberosempervolutpatluctus,
pedeeratlaciniaaugue,quisrutrumsemip-
sumsitametpede.Vestibulumaliquet,nibh
sediaculissagittis,odiodolorblanditaugue,
egetmollisurnatellusidtellus.Aeneanali-
quetaliquamnunc.Nullaultriciesjustoeget
orci.Phasellustristiquefermentumleo.Sed
massametus,sagittisut,semperut,pharetra
vel,erat.Aliquamquamturpis,egestasvel,
elementumin,egestassitamet,lorem.Duis
convallis,wisisitametmollismolestie,libero
mauris portadui, vitae aliquamarcu turpisac
sem.Aliquamaliquetdapibusmetus.
Vivamuscommodoeroseleifenddui.Vestibu-
luminleoeuerattristiquemattis.Crasat
elit.Craspellentesque.Nullamidlacussit
ametliberoaliquethendrerit.Proinplacerat,
minonelementumlaoreet,eroselittincidun-
tmagna,arhoncussemarcuidodio.Nul-
laegetleoaleoegestasfacilisis.Curabitur
quisvelit.Phasellus aliquam,tortornecornare
rhoncus,purusurnaposuerevelit,etcommo-
dorisustellusquistellus.Vivamusleoturpis,
tempussitamet,tristiquevitae,laoreetquis,
odio.Proinscelerisquebibendumipsum.Eti-
amnisl.Praesentveldolor.Pellentesquevel
magna.Curabitururna.Vivamuscongueur-
na in velit.Etiam ullamcorper elementum dui.
Praesentnon urna.Sed placeratquam non mi.
Pellentesquediammagna,ultricieseget,ultri-
cesplacerat,adipiscingrutrum,sem.
Morbisem.Nullafacilisi.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Nullafacilisi.Mor-
bisagittisultriceslibero.Praesenteuligula
sedsapienauctorsagittis.Classaptenttaciti
sociosquadlitoratorquentperconubianos-
tra,perinceptoshymenaeos.Donecvelnunc.
Nuncfermentum,lacusidaliquamporta,dui
tortoreuismoderos,velmolestieipsumpurus
eulacus.Vivamuspede arcu,euismod ac,tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovelnibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapien urna, blanditnec, iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringillaturpisnon arcu.Donec inante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbi ipsummassa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueeget loremnec elitmalesuada vestibu-
lum.Quisquesollicitudinipsum vel sem.Nul-
laenim.Proinnonummy felisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjusto id orciiaculis varius.Quisque et
lacus.Phaselluselementum,justoat dignissim
auctor,wisi odio lobortisarcu, sedsollicitudin
felisfeliseuneque.Praesentatlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,portain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimperdietrutrum.Donec
ac odio nec neque vulputatesuscipit.Nam nec
magna.Pellentesquehabitantmorbitristique
senectus et netus et malesuada fames acturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisi neque fringilla sapien, vel commodo lorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesque nonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nullafacilisi.Quisque
posuerebibendumpurus.Nulla quammauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenas lorem.Vivamus viverra consequat enim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitursollicitudinhen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tis eget, scelerisqueac, aliquet in, dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo, nequesemlacinia nulla, quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectus ipsum, vulputate a, mollis ut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsan pulvinar justo.Nullam semper, duiut
dignissim auctor, orcilibero fringilla massa,b-
landitpulvinarpedetortoridmagna.Nunc
adipiscing justo sed velit tincidunt fermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristique senectuset netus etmalesuada fames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereodioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusin augue.Sedvestibulumdapibus
neque.
Mauris tempus eros at nulla.Sed quisdui dig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.Inlectus wisi,elementumeu, sollic-
itudinnec,imperdietquis,dui.Nullaviverra
nequeaclibero.Maurisurnaleo,adipiscing
eu,ultricesnon,blanditeu,dui.Maecenas
duineque,suscipit sitamet,rutruma,laoreet
in,eros.Uteunibh.Fuscenecerattempus
urnafringilla tempus.Curabituridenim.Sed
ante.Crassodalesenimsitametwisi.Nunc
fermentum consequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitant morbi tristiquesenectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lam laoreeturnased enim.Doneceget erosut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
eget eros a ligula commodo ultricies.Donec eu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpatfaucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aenean tincidunt laoreet dui.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendrerit ut,semper at,metus.Vivamus sapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque, dignissim in,facilisis nec, hendrerit vel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputate purus.Morbi varius.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Intempor.Phaselluscommodoporttitor
magna.Curabiturvehiculaodioveldolor.
Praesentfacilisis,augueaadipiscingvene-
natis,liberorisusmolestieodio,pulvinarcon-
sectetuerfeliseratacmauris.Namvestibu-
lumrhoncusquam.Sedveliturna,pharetra
eu,eleifendeu,viverraat,wisi.Maecenasul-
tricesnibhatturpis.Aeneanquam.Nulla
ipsum.Aliquamposuereluctuserat.Cur-
abiturmagnafelis,laciniaet,tristiqueid,ul-
tricesut,mauris.Suspendissefeugiat.Cras
eleifendwisivitaetortor.Phasellusleopurus,
mattissitamet,auctorin,rutrumin,magna.
Inhachabitasseplateadictumst.Phasellus
imperdietmetusinsem.Vestibulumacenim
nonsemultriciessagittis.Sedveldiam.
Integervelenimsedturpisadipiscingbiben-
dum.Vestibulumpededolor,laoreetnec,po-
suerein,nonummyin,sem.Donecimperdiet
sapienplaceraterat.Donecviverra.Aliquam
eros.Nunc consequat massaid leo.Sed ullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem, nonporttitorpurusodioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget, posuere a,felis.Quisque vellectus utme-
tustincidunteleifend.Duis utpede.Duis velit
erat,venenatisvitae,vulputate a,pharetra sit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid, mi.Nullam necnibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donec mattis ullamcorper tellus.Phasellus vel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nuncnulla pulvinarjusto, aeuismod
erossemnecnibh.Nullamsagittisdapibus
lectus.Nullamegetipsumeutortorlobortis
sodales.Etiampurusleo,pretiumnec,feu-
giatnon,ullamcorpervel,nibh.Sedvelelit
etquamaccumsanfacilisis.Nuncleo.Sus-
pendissefaucibuslacus.
Pellentesqueinterdumsapiensednulla.Proin
tincidunt.Aliquamvolutpatestvelmassa.
Seddolorlacus,imperdietnon,ornarenon,
commodoeu,neque.Integerpretiumsem-
perjusto.Proinrisus.Nullamidquam.
Namneque.Duisvitaewisiullamcorperdi-
amcongueultricies.Quisqueligula.Mauris
vehicula.
Curabiturnuncmagna,posuereeget,vene-
natiseu,vehiculaac,velit.Aeneanornare,
massaaaccumsanpulvinar,quamlorem
laoreetpurus, eusodalesmagna risusmolestie
lorem.Nunceratvelit,hendreritquis,male-
suadaut,aliquamvitae,wisi.Sedposuere.
Suspendisseipsumarcu,scelerisquenec,ali-
quameu,molestietincidunt,justo.Phasellus
iaculis.Sedposuereloremnonipsum.Pel-
lentesquedapibus.Suspendissequamlibero,
laoreeta,tincidunteget,consequatat,est.
Nullamutlectusnonenimconsequatfacili-
sis.Maurisleo.Quisquepedeligula,auctor
vel,pellentesquevel,posuereid,turpis.Cras
ipsumsem,cursuset,facilisisut,tempuseu-
ismod,quam.Suspendissetristiquedoloreu
orci.Maurismattis.Aeneansemper.Viva-
mustortormagna,facilisisid,variusmattis,
hendreritin,justo.Integerpurus.
Vivamusadipiscing.Curabitur imperdiet tem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursus arcu,vitaefacilisisodioauguevitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlobortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donecfelis elit,dignissimsed,sagit-
tisut,ullamcorpera,nulla.Aeneanpharetra
vulputateodio.
Quisqueenim.Proinvelitneque,tristiqueeu,
eleifendeget,vestibulumnec,lacus.Vivamus
odio.Duisodiourna,vehiculain,elementum
aliquam,aliquetlaoreet,tellus.Sedvelit.Sed
velmiacelitaliquetinterdum.Etiamsapi-
enneque,convalliset,aliquetvel,auctornon,
arcu.Aliquamsuscipitaliquamlectus.Proin
tinciduntmagnasedwisi.Integerblanditla-
cusutlorem.Sedluctusjustosedenim.
Morbimalesuadahendreritdui.Nuncmauris
leo, dapibus sitamet, vestibulum et,commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus, sitamet posuerelectus neque
velaugue.Cras consectetuerlibero aceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictum auctor.In eget risusluctus wisi conva-
llis pulvinar.Vivamus sapien risus,tempor in,
viverra in, aliquetpellentesque, eros.Aliquam
euismodliberoasem.
Nunc velit augue,scelerisque dignissim, lobor-
tiset,aliquamin,risus.Ineueros.Vestibu-
lumanteipsumprimisinfaucibusorciluctus
etultricesposuerecubiliaCurae;Curabitur
vulputateelitviverraaugue.Maurisfringilla,
tortorsitametmalesuadamollis,sapienmi
dapibusodio,acimperdietligulaenimeget
nisl.Quisquevitaepedeapedealiquetsus-
cipit.Phasellustelluspede,viverravestibu-
lum,gravidaid,laoreetin,justo.Cumsoci-
isnatoquepenatibusetmagnisdisparturient
montes,nasceturridiculusmus.Integercom-
modoluctuslectus.Maurisjusto.Duisvar-
iuseros.Sedquam.Craslacuseros,rutrum
eget,variusquis,convallisiaculis,velit.Mau-
ris imperdiet, metus at tristique venenatis, pu-
rusnequepellentesquemauris,aultriceselit
lacusnectortor.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentmalesuada.Nam
lacuslectus,auctorsitamet,malesuadavel,
elementumeget,metus.Duisnequepede,fa-
cilisiseget,egestaselementum,nonummyid,
neque.
Proinnonsem.Donecnecerat.Proinlibero.
Aliquamviverraarcu.Donecvitaepurus.
Donecfelismi,semperid,scelerisqueporta,
sollicitudinsed,turpis.Nullainurna.Integer
variuswisinonelit.Etiamnecsem.Mau-
risconsequat,risusnecconguecondimentum,
ligulaligula suscipiturna,vitae portaodioer-
at quissapien.Proinluctus leo iderat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
na eros, ornarenec, vulputate eget, cursus sed,
justo.Phasellus nec lorem.Nullam ligula ligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
Fuscevehicula,tortor et gravida porttitor, me-
tusnibhconguelorem,ut tempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sit amet, turpis.Quisque lobortis.Proin vene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Nam ipsumelit, imperdiet vitae,faucibus nec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
Craspretium.Nullamalesuadaipsumut
libero.Suspendissegravidahendrerittellus.
Maecenas quislacus.Morbifringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbi tristique senectus et netus et malesuada
famesacturpisegestas.Namdictumdictum
urna.
Phasellusvestibulumorcivelmauris.Fusce
quamleo,adipiscingac,pulvinareget,mo-
lestiesitamet,erat.Seddiam.Suspendisse
erosleo,tempuseget,dapibussitamet,tem-
puseu,arcu.Vestibulumwisimetus,dapibus
vel,luctussitamet,condimentumquis,leo.
Suspendissemolestie.Duisinante.Utso-
dales sem sit amet mauris.Suspendisse ornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeros elit,gravidaac, pharetrased,
lobortisin,massa.Proinatdolor.Duisac-
cumsanaccumsanpede.Nullamblanditelit
inmagnalaciniahendrerit.Utnonummyluc-
tuseros.Fusceegettortor.
Utsitametmagna.Crasaligulaeuurna
dignissimviverra.Nullamtemporleoporta
ipsum.Praesentpurus.Nullamconsequat.
Maurisdictumsagittisdui.Vestibulumsol-
licitudinconsectetuerwisi.Insitametdiam.
Nullammalesuadapharetrarisus.Proinlacus
arcu, eleifendsed, vehicula at,conguesit amet,
sem.Sedsagittispedeanisl.Sedtincidunto-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nulla ettellus nonfelis imperdi-
etmattis.Aliquameratvolutpat.
Vestibulumsodalesipsumidaugue.Integer
ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
lorem convallis hendrerit.Maecenas hendrerit.
Sedmagnaodio,aliquamimperdiet,portaac,
aliqueteget,mi.Cumsociisnatoquepenati-
busetmagnisdisparturient montes,nascetur
ridiculusmus.Vestibulumnislsem,dignissim
vel,euismodquis,egestasut,orci.Nuncvi-
taerisusvelmetuseuismodlaoreet.Crassit
ametnequeaturpislobortisauctor.Sedali-
quamsemacelit.Crasvelitlectus,facilisis
id,dictumsed,portarutrum,nisl.Namhen-
dreritipsumsedaugue.Nullamscelerisque
hendreritwisi.Vivamusegestasarcusedpu-
rus.Utornarelectussederos.Suspendisse
potenti.Maurissollicitudinpedevelvelit.In
hachabitasseplateadictumst.
Suspendisseeratmauris,nonummyeget,
pretiumeget,consequatvel,justo.Pellen-
tesqueconsectetuereratsedlacus.Nullam
egestasnullaacdui.Doneccursusrhoncus
ipsum.Nuncetsemeumagnaegestasmale-
suada.Vivamusdictummassaatdolor.Mor-
biestnulla,faucibusac,posuerein,interdum
ut,sapien.Proinconsectetuerpretiumurna.
Donecsit ametnibh necpurusdignissim mat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utcongue malesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titor pede, nec aliquam turpisdiam feugiat ar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulumelit.Nam ligulaarcu, volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisus atmetuspulvinar malesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
Vestibulumanteipsumprimis infaucibusorci
luctus et ultrices posuere cubilia Curae; Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemper.Crasquiswisi
utloremposueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdum porta.Nunc bibendum augueluctus
diam.Etiam nec sem.Sed eros turpis, facilisis
nec,vehiculavitae,aliquamsed,nulla.Cur-
abiturjustoleo,vestibulumeget,tristiqueut,
tempusat,nisl.
Nullavenenatisloremidarcu.Morbicur-
susurnaaipsum.Donecporttitor.Integer
eleifend,estnonmattismalesuada,minulla
convallismi,etauctorlectussapienutpu-
rus.Aliquamnullaaugue,pharetrasitamet,
faucibussemper,molestievel,nibh.Pellen-
tesquevestibulummagnaetmi.Sedfringilla
dolorveltellus.Nuncliberonunc,venenatis
eget,convallishendrerit,iaculiselementum,
mi.Nullamaliquam,felisetaccumsanvehic-
ula,magnajusto vehiculadiam,eu condimen-
tumnislfelis etnunc.Quisquevolutpatmauris
avelit.Pellentesquemassa.Integeratlorem.
Nammetuserat,laciniaid,convallisut,pul-
vinarnon,wisi.Crasiaculismaurisutneque.
Crassodales,semvitaeimperdietconsequat,
pede purussollicitudin urna,ac aliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orci sodales lorem, non ornare nulla lorem quis
nisl.
Sedatsemvitaepurusultricesvestibulum.
Vestibulumtinciduntlacusetligula.Pel-
lentesquevitaeelit.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;Duisornare,erateget
laoreetvulputate,lacusipsumsuscipitturpis,
etbibendumnislorcinonlectus.Vestibu-
lumnecrisusnecliberofermentumfringilla.
Morbinonvelitinmagnagravidahendrerit.
Pellentesquequislectus.Vestibulumeleifend
lobortisleo.Vestibulumnonaugue.Vivamus
dictumtempordui.Maecenasatligulaidfe-
liscongueporttitor.Nullaleomagna,egestas
quis,vulputatesitamet,viverra id,velit.
Utlectuslectus,ultriciessitamet,semper
eget,laoreetnon,ante.Proinatmassaquis
nuncrhoncusmattis.Aliquamlorem.Cur-
abiturpharetraduiatneque.Aliquameutel-
lus.Aeneantempus,felisvitaevulputateia-
culis,estdolorfaucibusurna,inviverrawisi
nequenonrisus.Fusceveldolornecsapien
pretiumnonummy.Integerfaucibusmassaac
nulla ornarevenenatis.Nullaquissapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisque nisl.In dignissim dapibus massa.Ae-
neansemmagna,scelerisquenec,ullamcorper
quis,porttitorut,lectus.Fuscedignissimfa-
cilisistortor.Vivamusgravidafelissitamet
nunc.Nampulvinarodiovelenim.Pellen-
tesquesitamet est.Vivamus pulvinarleonon
sapien.Aliquamerat volutpat.Utelementum
auctormetus.Maurisvestibulumnequevitae
eros.Pellentesquealiquamquam.Donecve-
nenatistristiquepurus.Innisl.Nullavelit
libero,fermentumat,portaa,feugiatvitae,
urna.Etiamaliquetornareipsum.Proinnon
dolor.Aeneannuncligula,venenatissuscipit,
porttitorsit amet, mattis suscipit,magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Nampedemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisse ut augue ac nibhullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusest idlacus.Vivamusloremdui, commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Cras sodales.Quisque vestibulum pulvinar di-
am.Phasellustincidunt, leovitae tristiquefa-
cilisis,ipsum wisiinterdumsem,dapibus sem-
pernulla velit vel lectus.Crasdapibusmauris
etaugue.Quisquecursusnullainlibero.Sus-
pendisseetloremsitametmaurismalesuada
mollis.Nullamidjusto.Maecenasvenenatis.
Doneclacusarcu,egestasac,fermentumcon-
sectetuer,tempuseu,metus.Proinsodales,
seminpretiumfermentum,arcusapiencom-
modomauris,venenatisconsequataugueur-
nainwisi.Quisquesapiennunc,variuseget,
condimentumquis,laciniain,est.Fuscefacil-
isis.Praesentnecipsum.
Suspendisseadolor.Namerateros,congue
eget,sagittisa,laciniain,pede.Maecenas
inelit.Proinmolestievariusnibh.Vivamus
tristiquepurussedaugue.Proinegestassem-
pertortor.Vestibulumanteipsumprimisin
faucibusorciluctusetultricesposuerecubil-
iaCurae;Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Vestibulumorcienim, sagittis
ornare,eleifendut,mattisat,ligula.Nulla
molestieconvallisarcu.Uterostellus,condi-
mentum at,sodalesin,ultricesvel,nulla.
Duismagnaante,bibendumeget,eleifend
eget,suscipitsed,neque.Vestibuluminmi
sedmassacursuscursus.Pellentesquepulv-
inarmollisneque.Fusceutenimvitaemauris
malesuadatincidunt.Vivamusaneque.Mau-
rispulvinar,sapienidcondimentumdictum,
quamarcurhoncusdui,idtemporlacusjusto
etjusto.Proinsitametorcieudiameleifend
blandit.Nunceratmassa,luctusac,fermen-
tumlacinia,tinciduntultrices,sapien.Prae-
sentsed orci vitaedolor sollicitudinadipiscing.
Crasaneque.Utrisusdui,interdumat,plac-
eratid,tristiqueeu,enim.Vestibulumante
ipsum primisin faucibus orciluctus et ultrices
posuerecubiliaCurae;Etiamadipiscingeros
vestibulumdolor.Pellentesquealiquam,diam
eget eleifendposuere,augue erosporttitorlec-
tus,acdignissimduimetusnecfelis.Quisque
lacinia.Vestibulumtellus.Suspendissenec
wisi.Aeneanacfelis.Aliquamultricesmetus
etnulla.
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Praesentrhoncus.Curabitursagittisestsit
ametneque.Sedcommodomalesuadalectus.
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aliquameu,quam.Aeneanquisquamquis
wisigravidavehicula.Pellentesqueamassa
aleopretiumrhoncus.Suspendisseultrices.
Doneclaciniamalesuadamassa.Classapten-
ttacitisociosquadlitoratorquentperconu-
bianostra,perinceptoshymenaeos.Donec
pretiumornaremauris.Phasellusauctorer-
ategetenim.Integerscelerisque,feliseucon-
sequatfringilla,loremwisiultriciesvelit,id
vehiculapurusnullaegetodio.Nullammat-
tis,diamarutrumfermentum,odiosapien
tristiquequam,idmollistellusquaminodi-
o.Mauriseusapien.Donecaliquamloremsit
ametlorempharetralobortis.
Donecacvelit.Sedconvallisvestibulumsapi-
en.Vivamustemporlacussedlacus.Nunc
utlorem.Utettortor.Nullamvariuswisiat
diam.Etiamultricies,dolorsitametfermen-
tumvulputate,nequeliberovestibulumorci,
vitaefringillanequearcualiquetante.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisquevenenatislobortisaugue.Sed
tempor,tellusiaculispellentesquepharetra,
pededuimalesuadamauris,velultricesurna
maurisacnibh.Etiamnibhodio,ultriciesve-
hicula,vestibulumvitae,feugiateleifend,fe-
lis.Vivamuspulvinar.Aliquam eratvolutpat.
Nullaegestasvenenatismetus.Namfeugiat
nuncquiselitegestassagittis.Sedvitaefe-
lis.Inliberoarcu,rhoncusin,commodoeget,
auctorin,enim.Vivamussuscipitest.Nul-
ladapibus,magnavelaliquetegestas,massa
massahendreritlacus,acrutrumtellustellus
sitametfelis.Crasviverra.
Suspendisseeununc.Aliquamdignissimurna
sit ametmauris.Cras commodo, urna ut port-
titor venenatis, arcu metussodales risus,vitae
gravidasapienligulainest.Donecvulputate
sollicitudinwisi.Donecvehicula,estidinter-
dumornare,nibhtellusconsectetuerjusto,a
ultrices felis erat at lectus.In est massa, male-
suadanon,suscipitat,ullamcorpereu,elit.
Namnullalacus,bibendumsitamet,sagit-
tissed,temporeget,libero.Praesentligula.
Suspendissenulla.Etiamdiam.Nullaante
diam,vestibulumet,aliquetac,imperdietvi-
tae,urna.Fuscetinciduntlacusvelelit.Mae-
cenasdictum,tortornoneuismodbibendum,
pedenibh pretiumtellus,at dignissimleo eros
egetpede.Nullavenenatiseleifenderos.Ae-
nean ut odio dignissim augue rutrum faucibus.
Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
cipitvulputate.Sedinterdumlacusnonvelit.
Vestibulumanteipsumprimis infaucibusorci
luctusetultricesposuerecubiliaCurae;
Vivamus vehicula leoa justo.Quisque necau-
gue.Morbimauriswisi,aliquetvitae,dignis-
simeget,sollicitudinmolestie,ligula.Indic-
tumenim sitametrisus.Curabiturvitae velit
eudiamrhoncushendrerit.Vivamusutelit.
Praesentmattisipsumquisturpis.Curabitur
rhoncusnequeeudui.Etiamvitaemagna.
Namullamcorper.Praesentinterdumbiben-
dummagna.Quisqueauctoraliquamdolor.
Morbieuloremetestporttitorfermentum.
Nuncegestasarcuattortorvariusviverra.
Fusce eunulla utnullainterdum consectetuer.
Vestibulumgravida.Morbimattisliberosed
est.
Abstract
IntheG-expectationspace,weproposethemultipleItˆointegral,
whichisdrivenbymulti-dimensionalG-Brownianmotion.Weprove
therecursiverelationshipof multiple G-ItˆointegralsbyG-Itˆoformu-
laandmathematicalinduction,andweobtainsomecomputational
formulasforakindofmultipleG-Itˆointegrals.
Keywords
G-BrownianMotion,MultipleG-ItˆoIntegrals,G-ItˆoFormula,
RecursiveRelationship
1.Introduction
With therapid developmentof financialmarkets, traditionallinear ex-
pectations cannotexplain itsuncertainty sometimes.In 2007, Peng [1]
introducedanewsublinearexpectation–G-expectation,andheintro-
ducedG-normaldistributionandG-BrownianmotionundertheG-
expectationframework.In2008,Peng[2]provedthelawoflarge
numbersandthecentrallimittheoremunderthesublinearexpecta-
tion, and he definedthe Itˆo integralabout G-Brownian motion.Later,
Peng[3]obtainedtheG-Itˆoformulaandprovedtheexistenceandu-
niquenessofsolutionforthestochasticdifferentialequationsdriven
by G-Brownian motion (G-SDEs for short) and the backward stochas-
ticdifferential equationsdriven by G-Brownian motion(G-BSDEsfor
short).
Sincethen,G-expectationspaceandtheapplicationsofG-Itˆoin-
tegralhavebeenextensivelystudiedbymanyresearchers.In2014,
Hu,Ji,PengandSong[4]studiedthecomparisontheorem,nonlinear
Feynman-KacformulaandGirsanovtransformationofG-BSDE.In
2016,Hu,WangandZheng[5]obtainedtheIto-Krylov formulaunder
theG-expectationframework.Thentheyprovedthereflectionprin-
cipleofG-Brownianmotion,andtheygotthereflectionprincipleof
G-Brownian motion by Krylov’sestimate in [6].[7] studied rough path
propertiesofstochastic integrals ofItˆo’stypeandStratonovich’stype
DOI:10.4236/jamp.2018.611186Nov.13,20182219JournalofAppliedMathematicsandPhysics
F.Y.Liu
withrespecttoG-Brownian motion.Then,Hu,JiandLiu[8]studied
thestrongMarkovpropertyforG-SDEsin2017.Wu[9]introduced
the multiple Itˆo integrals driven by one-dimensional Brownian motion
inG-expectationspace.Healsoobtainedtherelationshipbetween
HermitepolynomialsandmultipleG-Itˆointegral.In2012,Yin[10]
introducedtheStratonovichintegralwithrespecttoG-Brownian mo-
tion,andshealsoresearchedthepropertiesofG-Stratonovichinte-
grals.In2014,Sun[11]studiedmultiplestochasticintegralsunder
one-dimensionalG-BrownianmotionanddevelopedtheL
p
estima-
tionofmaximalinequalitiesforniteratedintegralsbytheproperty
ofHermitepolynomials.Themorecontentsaboutmultiplerandom
integralscanbefoundintheliterature[12].
AnaturequestionishowtodefineandcalculatethemultipleG-
Itˆointegralofmulti-dimensionalG-Brownianmotion.Thisproblem
willbesolvedinthispaper.WedefinemultipleItˆointegralsdriven
by multi-dimensional G-Brownian motion underG-expectation space.
AndweprovetherecursiverelationshipbetweenmultipleG-Itˆoin-
tegralsstrictlybyusingG-Itˆoformulaandmathematicalinduction
method.Then we obtainsomeimportantformulasforcalculatingmul-
tipleG-Itˆointegrals andmakesomepreparationsforfurtherstudyon
scientificcalculationofG-SDEs.
Theremainderofthispaperisorganizedasfollows:InSection2,
weintroducesomeconceptsandlemmassuchasG-Brownian motion,
G-Itˆoformula andso on.InSection 3,we define multiple Itˆointegrals
driven by multi-dimensional G-Brownian motion, and prove therecur-
sive relationshipbetween multiple G-Itˆo integrals.Thenwe give some
importantformulasforcalculatingmultipleG-Itˆointegrals.Finally,
severalconcludingremarksaregiveninSection4.
2.PreliminariesandNotation
Inthissection,wewillgivesomebasictheoriesaboutG-Brownian
motionandmulti-indices.Somemoredetailscanbefoundinlitera-
tures[1–3]and[12].LetΩbeagivenset,andletHbealinearspace
ofrealvaluedfunctionsdefinedonΩ.Foreachcwesupposethat
c∈H,and|X|∈HifX∈H.ThespaceHcanbeconsideredasthe
spaceofrandomvariables.
2.1.G-BrownianMotionandG-ItˆoFormula
Firstly,weintroducesomenotationsaboutG-Brownianmotion.
Definition1.[3]Ad-dimensionalprocess(B
t
)
t0
onasublinear
expectationspace(Ω,H,E)iscalledaG-Brownianmotionifthefol-
lowingpropertiesaresatisfied:
(i)B
0
(ω) = 0;
(ii)Foreacht,s0,theincremenB
t+s
B
t
N(0 ×s
P
)is
independentfrom{B
t
1
,B
t
2
,...,B
t
n
},foreachnN,0t
1
t
2
...t
n
.
LetG(·):S(d)→Rbeagivenmonotonicandsublinearfunc-
tion.WedenotebyS(d)thecollectionofalld×dsymmetricmatri-
ces.Thereexistsabounded,convexandclosedsubset
P
S
+
(d)=
{θS(d)0},suchthatG(A) =
1
2
sup
B
P
(A,B),AS(d).
Inthefollowingsections,wedenotebyΩ=C
d
0
(R
+
)thespaceof
allR
d
-valuedcontinuouspaths(ω
t
)
t∈R
+
,withω
0
= 0,equippedwith
DOI:10.4236/jamp.2018.6111862220JournalofAppliedMathematicsandPhysics
F.Y.Liu
the distanceρ(ω
1
2
) :=
P
i=1
2
i
[(max
t[0,i]
|ω
1
t
ω
2
t
|)1].For each
fixedT[0,),wesetΩ
T
:= {ω
·∧T
: ω},
L
ip
(Ω
T
) :=
{ϕ(B
t
1
T
,...,B
t
n
T
) : nN,t
1
,...,t
n
[0,)C
l,lip
(R
n
)},
L
ip
(Ω) :=
n=1
L
ip
(Ω
n
),
whereB
t
isacanonicalprocess,thatisB
t
(ω)=ω
t
.Foragiven
p1,wealsodenoteL
p
G
(Ω)thecompletionofL
ip
(Ω)underthe
normkXk
p
:= (E[|X|
p
])
1
p
.
Werecall someimportantnotions about G-Itˆoformula,product rule
andsoon(see[3]).
Definition2.[3]Wedenotethesetofsimpleprocess
M
p,0
G
([0,T]) :=
n
η
t
(ω) :=
N1
X
j=0
ξ
t
j
(ω)I
[t
j
,t
j+1
)
(t);ξ
t
j
(ω) L
p
G
(Ω
t
j
),
N1,0 = t
0
<...<t
N
= T,j= 0,1,...,N1
o
.
Andforeachp1,wedenotebyM
p
G
([0,T])thecompletionof
M
p,0
G
([0,T])underthenorm
kηk
M
p
G
([0,T])
= {E
Z
T
0
|η
t
|
p
dt}
1
p
.
Definition3.[3]For each ηM
2,0
G
([0,T]),wedefinetheItˆointegral
ofG-Brownianmotionisasfollows:
I(η) =
Z
T
0
η
t
dB
t
:=
N1
X
j=0
ξ
t
j
(ω)(B
t
j+1
B
t
j
).
Definition4.[3]Wefirstconsiderthequadraticvariationprocess
ofone-dimensionalG-Brownianmotion(B
t
)
t0
withB
1
.
=N({0
[σ
2
2
]).Letπ
N
t
,N=1,2,...,beasequenceofpartitionsof[0,t].
Weconsider
B
2
t
=
N1
X
j=0
(B
2
t
N
j+1
B
2
t
N
j
)
=
N1
X
j=0
2B
t
N
j
(B
t
N
j+1
B
t
N
j
)+
N1
X
j=0
(B
t
N
j+1
B
t
N
j
)
2
.
Asµ(π
N
t
)0,thefirsttermoftherightsideconvergesto
2
R
t
0
B
s
dB
s
inL
2
G
(Ω).Thesecondtermmustbeconvergent.Wede-
noteitslimitbyhBi
t
,i.e.,
hBi
t
:=lim
µ(π
N
t
)0
N1
X
j=0
(B
t
N
j+1
B
t
N
j
)
2
= B
2
t
2
Z
t
0
B
s
dB
s
.
Bytheaboveconstruction,(hBi
t
)
t0
isanincreasingprocesswith
hBi
0
= 0.WecallitthequadraticvariationprocessoftheG-Brownian
motionB.
Nowletusintroducethefollowingtwoimportantlemmas.
DOI:10.4236/jamp.2018.6111862221JournalofAppliedMathematicsandPhysics
F.Y.Liu
Lemma1.[9]WedenoteB
t
beam-dimensionalG-Browninamo-
tion.LetΦ C
2
(R
n
)beboundedwithboundedderivativesand
2
x
i
x
j
Φ
areuniformlyLipschitz.Lets[0,T]befixedandletX
i
t
bethei
(i= 1,...,d)-thcomponentofX
t
= (X
1
t
,...,X
d
t
)
>
satisfying
X
i
t
= X
i
0
+
Z
t
0
a
i
s
ds+
m
X
j=1
Z
t
0
η
i,j
s
dhB
j
i
s
+
m
X
j=1
Z
t
0
σ
i,j
s
dB
j
s
,
wherea
i
bethei-thofa= (a
1
,...,a
d
)
>
,η
i,j
andσ
i,j
isthelinesi-th
andj-thofη=(η
i,j
)
d×m
andσ=(σ
i,j
)
d×m
,andtheyarebounded
processonM
2
G
(0,T).Fort,s0,thenwehave
Φ(X
t
)Φ(X
s
) =
d
X
i=1
[
Z
t
s
x
i
Φ(X
u
)a
i
u
du+
m
X
j=1
Z
t
s
x
i
Φ(X
u
)σ
i,j
u
dB
j
u
]
+
Z
t
s
[
d
X
i=1
m
X
j=1
x
i
Φ(X
u
)η
i,j
u
+
1
2
d
X
i,l=1
m
X
j=1
2
x
i
x
j
Φ(X
u
)σ
i,j
u
σ
l,j
u
]dhB
j
i
u
.
Lemma2.[1,3]InG-expectationspace,thefollowingproductrule
isestablished:
dB
i
t
dB
j
t
= δ
ij
=
dhB
i
i
t
,i= j
0,i6= j
,
dtdt= 0,dtdhBi
t
= 0,dtdB
t
= 0,dhBi
t
dt= 0,dhBi
t
dhBi
t
= 0,
dhBi
t
dB
t
= 0,dB
t
dt= 0,dB
t
dhBi
t
= 0,dB
t
dB
t
= dhBi
t
.
2.2.Multi-Indices
Letusintroducesomenotationsaboutmulti-indicesforsimplifys-
tatementsandproof.Weshallcallarowvectorα=(j
1
,j
2
,...,j
l
),
where j
i
∈{−m,(m1),...,1,0,1,2,...,m},i∈{1,2,...,l}and
m,l= 1,2,3,....amulti-indexoflengthl:= l(α) ∈{1,2,...}.
Definition5.[12]Wedenotethesetofallmulti-indicesbyM,so
M= {(j
1
,j
2
,...,j
l
) : j
i
∈{−m,(m1),...,1,0,1,2,...,m},
i∈{1,2,...,l},l∈{1,2,3,...}}∪{v},
wherevisthemulti-indexoflengthzero.
Wewrite n(α) forthe number ofcomponentsof amulti-indexαthat
are equal to 0 and s(α) for the number of components of a multi-index
αthat areequal to 1.Moreover, we writeαfor the multi-index ob-
tained by deleting thefirst componentof αand αfor the multi-index
obtainedbydeletingthefirstcomponentofα.α(j)forthemulti-
index obtained by deleting the last component of α= (j
1
,j
2
,...,j
k
,j)
sowecangetthemulti-index(j
1
,j
2
,...,j
k
).Additionally,giventwo
multi-indicesα
1
=(j
1
,j
2
,...,j
k
)andα
2
=(i
1
,i
2
,...,i
l
),weintro-
ducetheconcatenationoperatoronMdefinedby
α
1
α
2
= (j
1
,...,j
k
,i
1
,...,i
l
),
whereα
1
2
∈M.Theoperatorallowsustocombinetwomulti-
indices.Forinstance,assumingm= 2oneobtains
l((0,1,1)) = 3,n((0,1,1,2,0)) = 2,s((0,1,1,2,0)) = 1,
(0,1,1)= (0,1),(0,1,1)(0,2) = (0,1,1,0,2).
DOI:10.4236/jamp.2018.6111862222JournalofAppliedMathematicsandPhysics
F.Y.Liu
3.MainResults
Inthissection,byacomponentj∈{1,2,...,m}ofamulti-index
wewilldenoteinamultiplestochasticintegraltheintegrationwith
respect to the j-th Wiener process.A component j= 0 will denote in-
tegrationwithrespecttotime.Lastly,acomponentj∈{−m,(m
1),...,1}refertoanintegrationwithrespecttoquadraticvaria-
tionprocess.Weshalldefinethreesetsofadaptedrightcontinuous
stochasticprocessesg= {g(t,ω),t[0,T]}withlefthandlimits.
H
v
= {g:sup
t[0,T]
E(|g(t,ω)|) <∞};
H
(0)
= {g: E(
Z
T
0
|g(t,ω)|ds) <∞};
H
(j)
= {g: E(
Z
T
0
|g(s,ω)|
2
ds) <∞},
wherej∈{1,2,...,m}.
Definition6.Let%andτbetwostoppingtimeswith0%τ
Ta.s..Thenforamulti-indexα∈Mandaprocessg(·)∈H
α
,we
definethemultipleGItˆointegralI
α
[g(·)]
%,τ
recursivelyby
I
α
[g(·)]
%,τ
=
g(τ),l= 0,
R
τ
%
I
α
[g(·)]
%,τ
dz,l1j
l
= 0,
R
τ
%
I
α
[g(·)]
%,τ
dB
j
l
z
,l1j
l
∈{1,2,...,m},
R
τ
%
I
α
[g(·)]
%,z
dhB
j
l
i
z
,l1j
l
∈{−m,(m1),...,1},
(1)
whereg(·) = g(·,v
1
,...,v
s(α)
).
We use the following example to illustrate Definition6:I
v
[g(·)]
0,t
=
g(t),I
(0)
[g(·)]
0,t
=
R
t
0
g(z)dz,
I
(1)
[g(·)]
ρ,τ
=
Z
τ
ρ
g(z)dB
z
,
I
(2,0)
[g(·)]
ρ,τ
=
Z
τ
ρ
Z
z
2
ρ
g(z
1
)dB
2
z
1
dz
2
,
I
(0,1)
[g(·)]
ρ,τ
=
Z
τ
ρ
Z
z
2
ρ
g(z
1
)dz
1
dhB
1
i
z
2
.
Foramulti-indexα=(j
1
,j
2
,...,j
l
)∈Mandl(α)>1,wedefine
thesetH
α
tobethetotalityofadaptedrightcontinuousprocesses
g={g(t),t0}withlefthandlimitssuchthattheintegralprocess
{I
α
[g(·)]
ρ,t
,t[0,T]}consideredasafunctionof tsatisfiesI
α
[g(·)]
ρ,·
H
(j
l
)
.ForconveniencewewriteI
α,t
=I
α
[1]
0,t
andB
0
t
=tforα
M,t0.
Now,wewillgiveourmaintheorems.
Theorem1.Formulti-indexα
n
= (j
1
,j
2
,...,j
n
),j
i
∈{1,2,...,m},
wherej
1
,j
2
,...,j
n
arenotequalwitheachother.ThesetC(α
n
)be
theallofthenlevelarrangementofα
n
.Wedefine
C(α
n
) = {(a
1
,a
2
,...,a
n
)|a
i
∈{j
1
,j
2
,...,j
n
},i= 1,...,n,2 nm},
suchthat
H
C(α
n
)
=
X
αC(α
n
)
I
α,t
=
n
Y
i=1
B
j
i
t
.
DOI:10.4236/jamp.2018.6111862223JournalofAppliedMathematicsandPhysics
F.Y.Liu
Proof.Forn=2,wehaveI
(i,j),t
+I
(j,i),t
=
R
t
0
R
s
0
dB
i
r
dB
j
s
+
R
t
0
R
s
0
dB
j
r
dB
i
s
= B
i
t
B
j
t
;
Forn=kwehaveH
C(α
k
)
=
P
αC(α
k
)
I
α,t
=
k
Y
i=1
B
j
i
t
.Weneedto
provethat
H
C(α
k+1
)
=
X
αC(α
k+1
)
I
α,t
=
k+1
Y
i=1
B
j
i
t
.
Actually,weonlyneedtoprovethat
k+1
X
l=1
Z
t
0
H
C(α
k+1
(j
l
)),t
dB
j
l
t
=
k+1
X
l=1
Z
t
0
k
Y
i=1,i6=l
B
j
i
t
dB
j
l
t
=
k+1
Y
i=1
B
j
i
t
.(2)
whereα=(j
1
,j
2
,...,j
k
,j)andα
k+1
(j
l
)forthek-indexobtained
bydeletingthelastcomponentj
l
ofα
k+1
.ApplyingG-Itˆoformula
andindependenceofBrownmotion,onehas
d
k+1
Y
i=1
B
j
i
t
=
k+1
X
l=1
k
Y
i=1,i6=l
B
j
i
t
dB
j
l
t
.(3)
TakingintegralonEquation(3)andcombinedwithEquation(2),
theproofiscompleted.
Example1.Fori,j,k∈{1,2,3,...,m},andi,j,karedifferentwith
eachother.UsingG-ItˆoformulaandTheorem1,wecanget
I
(i,j,k),t
+I
(j,i,k),t
+I
(k,i,j),t
+I
(i,k,j),t
+I
(k,j,i),t
+I
(j,k,i),t
=
Z
t
0
B
i
z
B
j
z
dB
k
z
+
Z
t
0
B
k
z
B
i
z
dB
j
z
+
Z
t
0
B
k
z
B
j
z
dB
i
z
=B
i
t
B
j
t
B
k
t
.
NowweshallprovetherecursiverelationshipbetweenmultipleG-
Itˆointegrals.
Theorem2.Letj
1
,...,j
l
∈{0,1,...,m}andα= (j
1
,...,j
l
) ∈M,
wherel= 1,2,3,...Thenfort0,
B
j
t
I
α,t
=
l
X
i=0
I
(j
1
,...,j
i
,j,j
i+1
,...,j
l
),t
+
l
X
i=1
I
{j
i
=j6=0}
I
(j
1
,...,j
i1
,j
i
,j
i+1
,...,j
l
),t
.
(4)
Proof.WeconsidermultilinearG-ItˆoprocessX={X
t
,t0},
whichdefinedasfollows
X
t
=(X
(0)
t
,...,X
(m)
t
,X
(j
1
,j
2
)
t
,...,X
(j
1
,j
2
,j
3
)
t
,...,X
(j
1
,...,j
l
)
t
)
>
=(I
(0),t
,...,I
(m),t
,I
(j
1
,j
2
),t
,...,I
(j
1
,j
2
,j
3
),t
,...,I
(j
1
,j
2
,...,j
l
),t
)
>
,
(5)
whereeachcomponentofX
t
isamultiG-Itˆointegral.Forβ-thcom-
ponentβ(j
0
1
,...,j
0
r
)thecoefficientsare
a
β
=
x
β
,j
0
r
= 0
0,otherwise
,b
β,j
=
x
β
,j= j
0
r
∈{1,...,m}
0,otherwise
,
c
β
=
x
β
,j= j
0
r
∈{−m,(m1)...,1}
0,otherwise
.
DOI:10.4236/jamp.2018.6111862224JournalofAppliedMathematicsandPhysics
F.Y.Liu
ByDefinition4andG-Itˆoformula,onehave
B
j
t
I
α,t
= I
(j),t
I
α,t
=
Z
t
0
I
α,s
dI
(j),s
+
Z
t
0
I
(j),s
I
α,s
dB
j
l
s
+I
{j
l
=j6=0}
Z
t
0
I
α,s
dhB
j
l
i
s
.
(6)
Forl= 1,Equation(6)isestablishedobviously.Forl2wehave
B
j
t
I
α,t
= I
(j
1
,...,j
l
,j),t
+
Z
t
0
I
(j),s
I
α,s
dB
j
l
s
+I
{j
l
=j6=0}
I
(j
1
,...,j
l1
,j
l
),t
.
Theproofiscompleted.
Example2.Particularly,forj=1=(0,1),fromTheorem2it
followsthat
B
1
t
I
(0,1),t
= I
(j,j
1
,j
2
),t
+I
(j
1
,j,j
2
),t
+I
(j
1
,j
2
,j,),t
+I
{j
2
=j=16=0}
·I
(j
1
,j
2
),t
= I
(1,0,1),t
+2I
(0,1,1),t
+I
(0,1),t
;
Forj= 2= (0,1,3),applyingtheTheorem2wecanget
B
2
t
I
(0,1,3),t
= I
(j,j
1
,j
2
,j
3
),t
+I
(j
1
,j,j
2
,j
3
),t
+I
(j
1
,j
2
,j,j
3
),t
+I
(j
1
,j
2
,j
3
,j),t
+I
{j
3
6=j}
·I
(j
1
,j
2
,j
3
),t
= I
(2,0,1,3),t
+I
(0,2,1,3),t
+I
(0,1,2,3),t
+I
(0,1,3,2),t
+0
= I
(2,0,1,3),t
+I
(0,2,1,3),t
+I
(0,1,2,3),t
+I
(0,1,3,2),t
.
4.ConcludingRemarksandFutureWork
Inthiswork,wedefineG-Itˆointegraldrivenbymulti-dimensional
G-BrownianmotioninG-expectationspace.AndweuseG-Itˆofor-
mulaandmathematicalinductiontoobtainakindofmultipleG-Itˆo
integrals.AsdiscussedinSection1,thiseffortfocusesonmulti-
pleG-Itˆointegralsdrivenbymulti-dimensionalG-Brownianmotion
ratherthanone-dimensionalG-Brownianmotion.Ourfutureefforts
willfocusonintroducingthepropertiesofStratonovichintegraldriv-
enbymulti-dimensionalG-Brownianmotion,andexploringtherela-
tionshipbetweenStratonovichintegralandG-Itˆointegralunderthe
G-expectationframework.
ConflictsofInterest
Theauthordeclaresnoconflictsofinterestregardingthepublication
ofthispaper.
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