Стохастичний аналіз процесів та полів за допомогою мартингальних методів: Автореф. дис… канд. фіз.-мат. наук / Я.О. Ольцік, Київ. ун-т ім. Т.Шевченк

Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea

IEUeOe?E ssi?ia Ieaenaiae??aia

OAeE 519.21

NOIOANOE*IEE AIAE?C I?IOeAN?A OA IIE?A

CA AeIIIIIAITH IA?OEIAAEUeIEO IAOIAe?A

01.01.05 – oai??y eiia??iinoae oa iaoaiaoe/ia noaoenoeea

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

Ee?a – 1999

Aeena?oaoe??th ? ?oeiien.

?iaioo aeeiiaii a Ee?anueeiio oi?aa?neoao? ?iai? Oa?ana Oaa/aiea.

Iaoeiaee ea??aiee: aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?,

I?OO?A THe?y Noaiai?aia,

eaoaae?a iaoaiaoe/iiai aiae?co

Ee?anueeiai oi?aa?neoaoo ?i. O.A.Oaa/aiea,

i?ioani?

Io?oe?ei? iiiiaioe: aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?

EIIIIA Iaaei Nieiiiiiae/,

?inoeooo e?aa?iaoeee IAI Oe?a?ie,

i?ia?aeiee iaoeiaee ni?a?ia?oiee;

aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

NA?UOE Aiaoie?e A?oae?eiae/,

?inoeooo iaoaiaoeee IAI Oe?a?ie,

i?ia?aeiee iaoeiaee ni?a?ia?oiee;

I?ia?aeia onoaiiaa: ?inoeooo i?eeeaaeii? iaoaiaoeee ? iaoai?ee IAI
Oe?a?ie, a?aeae?e oai??? eiia??iinoae oa iaoaiaoe/ii? noaoenoeee, i.
Aeiiaoeuee

Caoeno a?aeaoaeaoueny “24” o?aaiy 1999 ?ieo i 14.00 aiae. ia
can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee Ae 26.001.37 i?e Ee?anueeiio
oi?aa?neoao? ?iai? Oa?ana Oaa/aiea ca aae?anith:

252127, i.Ee?a — 127, i?iniaeo aeaae. Aeooeiaa, 6, Ee?anueeee
oi?aa?neoao ?iai? Oa?ana Oaa/aiea, iaoai?ei-iaoaiaoe/iee oaeoeueoao.

C aeena?oaoe??th iiaeia iciaeiieoenue o a?ae?ioaoe? Ee?anueeiai
oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea (aoe. Aieiaeeie?nueea, 58).

Aaoi?aoa?ao ?ic?neaii “15” ea?oiy 1999 ?ieo

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee Iieey/oe
I.I.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE.

Aeooaeuei?noue oaie.

Iaeiei c iaeaaaeeea?oeo iiiyoue noioanoe/iiai aiae?co aeiaaeeiaeo
i?ioean?a oa iie?a ? iiiyooy eieaeueiiai /ano. A?aeiia oi?ioea Oaiaea
aeey iiaoaeiae eieaeueiiai /ano aeey iaeiiia?aiao?e/iiai a?ioi?anueeiai
?ooo. O 1975 ?ioe? Iae?? iieacaa, ye aeei?enoiaoaaoe oi?ioeo Oaiaea aeey
iiaoaeiae eieaeueieo /an?a aeey iaeiiia?aiao?e/ieo ia?oeiaae?a c
iaae?iaeaeaiith iaia?a?aiith /anoeiith. I?ci?oa aoei iieacaii, ui o
aeiaaeeo /enoi ?ic?eaiiai ia?oeiaaeo oi?ioea Oaiaea ia i?aoeth?. Aann
iiaoaeoaaa aeinoaoi? oiiae ?nioaaiiy eieaeueiiai /ano aeey /enoi
?ic?eaieo iaeiiia?aiao?e/ieo nai?ia?oeiaae?a.

A aeaiia?aiao?e/iiio aeiaaeeo aeey iaia?a?aieo ia?oeiaae?a ieoaiiy
?nioaaiiy eieaeueiiai /ano ?icaeyiooi a ?iaioao Iane?iaa, Naioe, Aa??n.
Ieoaiiy, iia’ycai? c eieaeueiei /anii aeey aaaaoiia?aiao?e/ieo
iaia?a?aieo ia?oeiaae?a, ?icaeyioo? a ?iaio? ?ieaeea?a

Aei oeueiai /ano a e?oa?aoo?? ia c’yaeyeeny ?acoeueoaoe, ui noinothoueny
ieoaiiy ?nioaaiiy eieaeueiiai /ano aeey aeaiia?aiao?e/ieo /enoi
?ic?eaieo ia?oeiaae?a. A cai?iiiiiaai?e aeena?oaoe?ei?e ?iaio? aaoi?
ocaaaeueith? ia aeaiia?aiao?e/iee aeiaaeie ?acoeueoaoe, iaea?aeai?
Aannii aeey iaeiiia?aiaoe/ieo eieaeueieo /an?a, a naia, aoaeothoueny
aeinoaoi? oiiae ?nioaaiiy eieaeueiiai /ano aeey /enoi ?ic?eaieo neeueieo
ia?oeiaae?a ia ieiuei? oa aea/athoueny aeanoeaino? oaeiai eieaeueiiai
/ano, cie?aia, iaia?a?ai?noue oa ?nioaaiiy ne?i/aiieo iiiaio?a.

A iaeiiia?aiao?e/iiio aeiaaeeo eieaeueiee /an aoaeo?oueny ca aeiiiiiaith
c?inoath/iai ia?aaeaa/oaaiiai i?ioeano, anioe?eiaaiiai ?c iaiaaeaiei
iioaioe?aeii. Oiio a oeueiio aeiaaeeo ?nioaaiiy ne?i/aiieo iiiaio?a
aoaeue-yeiai ii?yaeeo aeey eieaeueiiai /ano aeieeaa? c oai?aie Aa?n?a.
Aea aeyaey?oueny, ui a aeaiia?aiao?e/iiio aeiaaeeo ?acoeueoao,
aiaeia?/iee oe?e oai?ai?, ia ia? i?noey. A aeena?oaoe?? aoaeothoueny
aeinoaoi? oiiae ?nioaaiiy ne?i/aiieo iiiaio?a aeey c?inoath/eo iie?a,
anioe?eiaaieo ?c aeaiia?aiao?e/ieie iioaioe?aeaie, a oaeiae aea/a?oueny
ieoaiiy ?nioaaiiy ne?i/aiieo iiiaio?a aeey aeaiia?aiao?e/ieo eieaeueieo
/an?a.

Aeooaeueiei ? ieoaiiy canoinoaaiiy ia?oeiaaeueieo iaoiae?a aei caaea/
o?iainiai? iaoaiaoeee. O aeene?aoiiio aeiaaeeo iiaoaeiaaia iiaeaeue
Eiena-?inna-?oa?iooaeia ooieoe?iioaaiiy ?eieo oe?iieo iaia??a. Aeae ?
Oioen iaea?aeaee oi?ioeo aeey ciaoiaeaeaiiy ni?aaaaeeeai? aa?oino?
iioe?ii?a, eiio?aeoe c yeeie a?aeaoaathoueny ia ?eieo, ui neeaaea?oueny
c aaie?anueeiai ?aooieo (aai iae?aaoe??) oa aeoe??, c iaia?a?aiei /anii.
Caaea/a a?aeooeaiiy no?aoaa?? oaaeaeoaaiiy ia iiaiiio o?iainiaiio ?eieo
?icaeyaeaeanue, cie?aia, a ?iaio? Oa???niia oa Ie?nee. A neeo iino?eieo
ci?i a?aenioeiaeo noaaie, ??aiy ?ioeyoe?? oiui ia o?iainiaiio ?eieo ?
aeueoa?iaoeai? iiaeeeaino? aeey iaea?aeoaaiiy i?eaooe?a. Oiio a aaeoc?
o?iainiai? iaoaiaoeee aeooaeueiith ? caaea/a iioei?caoe??

o?iainiaeo no?aoaa?e, cie?aia, caaea/a iioeiaeueii? coieiee aai
iioeiaeueiiai ia?aeeth/aiiy i?ae aeueoa?iaoeaieie o?iainiaeie
no?aoaa?yie.

Oai??y iioeiaeueieo iiiaio?a coieiee o canoinoaaii? aei aeayeeo
o?iainiaeo caaea/ aaaaeaia a ?iaio?. A?aeooeaiiy iioeiaeueiiai iiiaioo
coieiee (aac ia?aeeth/aiiy) aeaoaeueii ?icaeyaea?oueny a eiec?.
«Ia?aeaeueia» ia?iaea aeaio aeoea?a, a naia, aia?eeainueeee iioe?ii
eoi?ae? c ooieoe??th aeieaoe, yea aecia/a?oueny iaeneioiii c aeaio
aeoea?a, ?icaeyaea?oueny a noaoo?. A noaoo?

?icaeyiooi caaea/o iioeiaeueiiai ia?aeeth/aiiy aeey aeaiia?aiao?e/ieo
aeiaaeeiaeo i?ioean?a. Ieoaiiy, iia’ycai? ?c a?aeooeaiiyi iioeiaeueiiai
?iaanooaaiiy ia o?iainiaiio ?eieo oaeiae ?icaeyioo? a ?iaio?. A aeai?e
?iaio? i?iaeiaaeo?oueny aeine?aeaeaiiy ieoaiiy iioeiaeueii? coieiee oa
ia?aeeth/aiiy ia o?iainiaiio ?eieo, aaaaeaii oai??? ia?aeaiieo iiiaio?a
coieiee oa iioeiaeueieo iiiaio?a coieiee aeey oaeoi?eciaaieo i?ioean?a,
ca aeiiiiiaith yeeo ?ica’ycothoueny caaea/? a?aeooeaiiy iioeiaeueieo
iiiaio?a coieiee aeey i?ioean?a, ui caaeiaieueiythoue ??ci? oeie
noioanoe/ieo aeeoa?aioe?aeueieo ??aiyiue.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie.

Aeaio aeena?oaoe?eio ?iaioo aeeiiaii ca?aeii iaoeiai? i?ia?aie
«Noaoenoe/iee aiae?c aeiaaeeiaeo i?ioean?a ? iie?a oa eiai canoinoaaiiy»
eaoaae?e oai??? eiia??iinoae oa iaoaiaoe/ii? noaoenoeee
iaoai?ei-iaoaiaoe/iiai oaeoeueoaoo Ee?anueeiai oi?aa?neoaoo ?i.
O.A.Oaa/aiea, iiia? aea?ae?a?no?aoe?? 0197U003176.

Iaoa ? caaea/? aeine?aeaeaiiy.

Iaoith aeaii? aeena?oaoe?? ? ?ica’ycaiiy ca aeiiiiiaith ia?oeiaaeueieo
iaoiae?a oaeeo caaea/ noioanoe/iiai aiae?co ye aeiaaaeaiiy ?nioaaiiy
eieaeueiiai /ano aeey aeaiia?aiao?e/ieo /enoi ?ic?eaieo neeueieo
ia?oeiaae?a, ye? ? cao?aiiyie no?eeeo neiao?e/ieo iie?a; iiaoaeiaa
aeinoaoi?o oiia ?nioaaiiy ne?i/aiieo iiiaio?a aoaeue-yeiai ii?yaeeo aeey
aeaiia?aiao?e/iiai eieaeueiiai /ano oa aeiaaaeaiiy eiai iaia?a?aiino?;
a?aeooeaiiy iioeiaeueieo iiiaio?a coieiee aeey ??cieo oei?a aeiaaeeiaeo
i?ioean?a oa canoinoaaiiy iaea?aeaieo ?acoeueoao?a aei ??cieo caaea/
o?iainiai? iaoaiaoeee.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a.

An? iaea?aeai? a aeena?oaoe?? iaoeia? ?acoeueoaoe ? iiaeie. Iaea?aeaii
aeinoaoi? oiiae ?nioaaiiy eieaeueiiai /ano aeey aeaiia?aiao?e/ieo /enoi
?ic?eaieo ia?oeiaae?a, ye? ocaaaeueiththoue aiaeia?/iee ?acoeueoao,
iaea?aeaiee Aannii a iaeiiia?aiao?e/iiio aeiaaeeo. Aeiaaaeaii
iaia?a?ai?noue iiaoaeiaaiiai eieaeueiiai /ano oa iiaoaeiaaii aeinoaoi?
oiiae ?nioaaiiy eiai ne?i/aiieo iiiaio?a aoaeue-yeiai ii?yaeeo.
Iaea?aeaii e?eoa??e ?nioaaiiy iiiaio?a c?inoath/eo i?ioean?a,
anioe?eiaaieo ?c iaeiiia?aiao?e/ieie iioaioe?aeaie, yeee ?
oaeineiiaeaiiyi oai?aie Aa?n?a. Aea/aii aeanoeaino? aeaiia?aiao?e/ieo
iioaioe?ae?a, anioe?eiaaieo ?c c?inoath/eie ia?aaeaa/oaaieie iieyie.
Iiaoaeiaaii aeaa iiaeo iaoiaee (ia?aeaiieo iiiaio?a coieiee oa
oaeoi?ecaoe??) ye iaoiaee a?aeooeaiiy iioeiaeueieo iiiaio?a coieiee aeey
aeiaaeeiaeo i?ioean?a, ye? canoiniaaii aei ?ica’ycaiiy caaea/
iioeiaeueii? coieiee oa iioeiaeueiiai ia?aeeth/aiiy ia o?iainiaiio ?eieo
c aeueoa?iaoeaieie no?aoaa?yie aeey ??cieo oei?a noioanoe/ieo
aeeoa?aioe?aeueieo ??aiyiue.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao?a.

?acoeueoaoe aeena?oaoe?? iathoue oai?aoe/ia cia/aiiy. Aiie iiaeooue aooe
aeei?enoai? a oai??? aeaiia?aiao?e/ieo ia?oeiaae?a, iaeii- oa
aeaiia?aiao?e/ieo

iioaioe?ae?a oa c?inoath/eo i?ioean?a ? iie?a, oai??? iioeiaeueieo
iiiaio?a coieiee oa a o?iainia?e iaoaiaoeoe?.

Iniaenoee aianie caeiaoaa/a.

An? iaoeia? ?acoeueoaoe, aeeth/aieo a aeena?oaoe?th, iaea?aeaii
caeiaoaa/ai iniaenoi.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??.

Io?eiai? a aeena?oaoe?? ?acoeueoaoe aeiiia?aeaeenue ia nai?ia?? eaoaae?e
oai??? eiia??iinoae oa iaoaiaoe/ii? noaoenoeee iaoai?ei-iaoaiaoe/iiai
oaeoeueoaoo Ee?anueeiai oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea; ia Ae?oa?e
anaoe?a?inuee?e eiioa?aioe?? «No/ani? o?ceei-iaoaiaoe/i? aeine?aeaeaiiy
iieiaeeo iaoeiaoe?a aoc?a Oe?a?ie» (i. Ee?a, 1995 ??e); ia V
I?aeia?iaei?e iaoeia?e eiioa?aioe?? ?iai? aeaaeai?ea I. E?aa/oea (i.
Ee?a, 1996 ??e); ia XVIII nai?ia?? c

i?iaeai no?eeino? noioanoe/ieo iiaeaeae (i. Aeaa?aoeai, Oai?ueia, 1997
??e); ia Ae?oa?e Neaiaeeiaanueei-Oe?a?inuee?e eiioa?aioe?? c
iaoaiaoe/ii? noaoenoeee (i. Oiaa, Oaaoe?y, 1997 ??e).

Ioae?eaoe??.

Iniiai? ?acoeueoaoe aeena?oaoe?? iioae?eiaai? a 3 iaoeiaeo noaooyo, a
oaeiae o 4 oacao aeiiia?aeae iaoeiaeo eiioa?aioe?e. Nienie ?ia?o
iaaaaeaii ieae/a.

No?oeoo?a oa ia’?i ?iaioe.

Aeena?oaoe?eia ?iaioa neeaaea?oueny ?c anooio, aeaio ?icae?e?a,
aeniiae?a oa nieneo aeei?enoaieo aeaea?ae, yeee i?noeoue 59
iaeiaioaaiiy. Ianya ?iaioe neeaaea? 140 noi??iie iaoeiiieniiai oaenoo.

INIIAIEE CI?NO

O anooi? iaa?oioiao?oueny aeooaeuei?noue ? aaaeeea?noue ?acoeueoao?a,
iaea?aeaieo a aeena?oaoe??, i?iaiaeeoueny iaeyae aeecueeeo ca oaiaoeeith
?ia?o, oi?ioeththoueny caaea/? oa iaoa aeine?aeaeaiiy, aeaco?oueny
iiaecia ?acoeueoao?a.

Ia?oee ?icae?e i?enay/aii aeiaaaeaiith ?nioaaiiy eieaeueiiai /ano aeey
aeaiia?aiao?e/ieo /enoi ?ic?eaieo neeueieo ia?oeiaae?a, ye? ?
cao?aiiyie no?eeeo neiao?e/ieo iie?a. Aoaeothoueny aeinoaoi? oiiae
?nioaaiiy eieaeueiiai /ano, ye? ocaaaeueiththoue ia aeaiia?aiao?e/iee
aeiaaeie aiaeia?/i? aeinoaoi? oiiae Aanna ?nioaaiiy eieaeueiiai /ano
aeey iaeiiia?aiao?e/ieo /enoi ?ic?eaieo ia?oeiaae?a. Aeiaiaeeoueny
iaia?a?ai?noue iiaoaeiaaiiai eieaeueiiai /ano. Iaea?aeothoueny aeinoaoi?
oiiae ?nioaaiiy ne?i/aiieo iiiaio?a an?o ii?yaee?a oeueiai eieaeueiiai
/ano ca aeiiiiiaith aea/aiiy aeanoeainoae iaeii- oa aeaiia?aiao?e/ieo
iioaioe?ae?a, anioe?eiaaieo ?c c?inoath/eie ia?aaeaa/oaaieie i?ioeanaie
oa iieyie.

A i?ae?icae?e? 1.1 aaiaeyoueny iniiai? icia/aiiy. ?icaeyaea?oueny iiea
Xt, t(R2+, yea ? /enoi ?ic?eaiei neeueiei ia?oeiaaeii c eieaeueieie
oa?aeoa?enoeeaie (as,(s). Iaoae ua (((dh) – i??a Eaa? no?eeiai
neiao?e/iiai iiey c ?iaeaenii 1<(<2. I?eionoeii, ui aeeiiothoueny ianooii? oiiae: i.i.; 2) aeey aeayeiai 1<(<2 (s(dh)=(((dh), yeui |h|>1;

3) Xt ia? ia a?eueoa iaeiiai no?eaea acaeiaae aoaeue-yei? i?yii?,
ia?aeaeueii? a?nyi eii?aeeiao.

A i?ae?icae?e? 1.2 ?icaeyaeathoueny aeanoeaino? u?eueiinoae ?iciiae?eo
no?eeeo neiao?e/ieo i?ioean?a oa iie?a oa iaea?aeothoueny ioe?iee aeey
ooieoe?iiae?a, ii?iaeaeaieo oaeeie u?eueiinoyie.

A i?ae?icae?e? 1.3 aeiaiaeyoueny aeaye? aeanoeaino? /enoi ?ic?eaieo
eieaeueieo neeueieo ia?oeiaae?a, ?iceeaaee oa ioe?iee aeey ooieoe?iiae?a
iioaioe?aeueiiai oeio o i?eiouaii?, ui aeeiiothoueny oiiae (O1) oa

i.i.

?icaeyaea?oueny ooieoe?iiae

? aeiaiaeeoueny oaeee ?acoeueoao:

Oai?aia 1.3.5. I?e aeeiiaii? oiia (O1) oa (O2) iathoue i?noea
oaa?aeaeaiiy:

;

, oi

I?ae?icae?e 1.4 i?enay/aii iiaoaeia? iiey Lt(x), yea ? eieaeueiei /anii
aeey /enoi ?ic?eaiiai neeueiiai ia?oeiaaeo Xt. ?icaeyaea?oueny ?aaoey?ia
oiiaia eiia??i?noue Qt((,() a?aeiinii iioieo (-aeaaa? Ft, aeey yei?
aeiaiaeeoueny ?nioaaiiy iaa?ae’?iii? iaiaaeaii? u?eueiino? Vt((,x)( ()
a?aeiinii i??e Eaaaaa. Oiae? aeey an?o t(Q+2 iiea Ut((,x)=exp{-(1t1
-(2t2}Vt((,x) aeyaey?oueny iaa?ae’?iiei iaiaaeaiei noia?ia?oeiaaeii,
neaaeei noaia?oeiaaeii ? iioaioe?aeii aeey iaeaea an?o x.

I?e aeeiiaii? i?eiouaiue

(O3) 1) Aeey iaeaea an?o x a?aeiinii i??e Eaaaaa iiea { Ut((,x), Ft, P,
t( R+2} ? noia?ia?o?iaaeii ? neaaeei noaia?oeiaaeii;

2) aeey aeia?eueiiai s ( R+2 E Ut((,x) (E Us((,x), t (s, t>s, t ( R+2

aeiaiaeeoueny, ui iiea Ut((,x) aeiionea? ?iceeaae ia noio

Ut((,x)= Mt((,x)+Lt((,x),

aea Mt((,x) – neaaeee ia?oeiaae, Lt((,x) – c?inoath/a ia?aaeaa/oaaia
iiea. Ooeaiee eieaeueiee /an aoaeo?oueny ye

(1)

Iniiaiei ?acoeueoaoii oeueiai i?ae?icae?eo ?

Oai?aia 1.4.4. Iaoae Xt – /enoi ?ic?eaiee eieaeueiee neeueiee
ia?oeiaae, aeey yeiai aeeiiothoueny oiiae (O1), (O2) ? (O3). Oiae? ?nio?
noi?nii

aei??ia iaia?a?aia a Qt++ c?inoath/a iiea Lt(x), yea aecia/a?oueny
??ai?noth (1), ? iiiaeeia N3 oae?, ui P(N3)=0 ? yeui ((N3, B (
B(R), oi aeey an?o t( R+2

I?ae?icae?e 1.5 i?enay/aii aeiaaaeaiith iaia?a?aiino? iiaoaeiaaiiai
eieaeueiiai /ano Lt(x) ii t.

A ianooiieo i?ae?icae?eao ooeathoueny oiiae, i?e yeeo iiaoaeiaaiee
eieaeueiee /an ia? iiiaioe aoaeue-yeiai ii?yaeeo. A iaeiiia?aiao?e/iiio
aeiaaeeo, ia a?aei?io a?ae aeaiiai, ?nioaaiiy iiiaio?a eieaeueiiai /ano
aeieeaa? c

oai?aie Aa?n?a, a ye?e noaa?aeaeo?oueny, ui c iaiaaeaiino? iioaioe?aeo
aeieeaa? ?nioaaiiy ne?i/aiieo iiiaio?a anioe?eiaaiiai ?c iei
c?inoath/iai i?ioeano.

A i?ae?icae?e? 1.6 aaiaeyoueny icia/aiiy oa iaaiaeyoueny aeanoeaino?
aeaiia?aiao?e/ieo ia?aaeaa/oaaieo i?iaeoe?e, noioanoe/ieo ?ioaa?ae?a ia
ieiuei? oa iioaioe?ae?a, anioe?eiaaieo ?c c?inoath/eie ia?aaeaa/oaaieie
iieyie.

A i?ae?icae?e? 1.7 aeiaiaeeoueny ocaaaeueiaiiy oai?aie Aa?n?a, a naia,
e?eoa??e ?nioaaiiy ne?i/aiieo iiiaio?a c?inoath/iai i?ioeano,
anioe?eiaaiiai ?c

iaeiiia?aiao?e/iei iioaioe?aeii. Iaoae Xt, t ( R+ — iaeiiia?aiao?e/iee
iioaioe?ae, anioe?eiaaiee ?c ia?aaeaa/oaaiei c?inoath/ei i?ioeanii At.

Oai?aia 1.7.1. Ianooii? oaa?aeaeaiiy aea?aaeaioi?:

E??i oeueiai, iaaaaeai? i?eeeaaee, ye? iieacothoue iioeiaeuei?noue
oeueiai e?eoa??th. Oaeiae ca aeiiiiiaith oi?ioee ?oi aeiaiaeeoueny
aeinoaoiy oiiaa ?nioaaiiy ne?i/aiieo iiiaio?a c?inoath/iai i?ioeano,
anioe?eiaaiiai ?c iaia?a?aiei iioaioe?aeii.

A i?ae?icae?e? 1.8 ?icaeyaeathoueny aeanoeaino? aeaiia?aiao?e/ieo
iaiaaeaieo iioaioe?ae?a, anioe?eiaaieo ?c c?inoath/eie iieyie.
Iaea?aeothoueny ioe?iee aeey iiiaio?a aeayeeo ia?aoai?aiue a?ae oeeo
c?inoath/eo iie?a. Iiaoaeiaaii e?eoa??e ?nioaaiiy ne?i/aiieo iiiaio?a
c?inoath/iai iiey, a naia: iaoae {Xt, t( R+2 } — aeaiia?aiao?e/iee
iaa?ae’?iiee iaiaaeaiee iioaioe?ae, anioe?eiaaiee ?c

c?inoath/ei ia?aaeaa/oaaiei ?ioaa?iaaiei iieai At. Ia? i?noea

Oai?aia 1.8.2. Ianooii? oiiae aea?aaeaioi?:

Iaaaaeaii i?eeeaae, yeee iieaco?, ui a aeaiia?aiao?e/iiio aeiaaeeo
?acoeueoao, aiaeia?/iee oai?ai? Aa?n?a, ia ia? i?noey, oiaoi
iaiaaeaiino? aeaiia?aiao?e/iiai iioaioe?aeo ia aeinoaoiuei aeey
?nioaaiiy ne?i/aiieo iiiaio?a anioe?eiaaiiai c?inoath/iai iiey.

I?ae?icae?e 1.9 i?enay/aii ieoaiith ?nioaaiiy iiiaio?a eieaeueiiai /ano
Lt(x), iiaoaeiaaiiai a i?ae?icae?e? 1.4. Ine?eueee eieaeueiee /an
aoaeo?oueny ye ia?aoai?aiiy a?ae c?inoath/iai ia?aaeaa/oaaiiai iiey
Lt((,x), anioe?eiaaiiai ?c iaiaaeaiei iioaioe?aeii Ut((,x), oi ca
aeiiiiiaith iaea?aeaieo ?acoeueoao?a aeiaiaeeoueny

Oai?aia 1.9.3. Iaoae aeeiiothoueny oiiae (O1)-(O3), a oaeiae

Oiae? aeey an?o t( R+2 ? aeey an?o x Lt(x) ia? iiiaioe an?o ii?yaee?a.

Ae?oaee ?icae?e i?enay/aii iiaoaeia? aeinoaoi?o oiia ?nioaaiiy
iioeiaeueieo iiiaio?a coieiee aeey ??cieo aeae?a aeiaaeeiaeo i?ioean?a
ca aeiiiiiaith ia?oeiaaeueieo iaoiae?a. Iaea?aeai? ?acoeueoaoe
canoiniaothoueny aei caaea/ iioei?caoe?? o?iainiaeo no?aoaa?e c
aeueoa?iaoeaaie, eiee i?ioean, ui iieno? eai?oae ?iaanoi?a, ? ?ica’yceii
??cieo oei?a noioanoe/ieo aeeoa?aioe?aeueieo ??aiyiue.

A i?ae?icae?e? 2.1 ?icaeyaea?oueny ocaiaeaeaiee iaia?a?aiee aeiaaeeiaee
i?ioean (t, t( [0,T] ( R+, yeee caaeiaieueiy? oiiao

(2)

Aeey oeueiai i?ioeano ooea?oueny iioeiaeueiee iiiaio coieiee ca
aeiiiiiaith oai??? ia?aeaiieo iiiaio?a coieiee.

Icia/aiiy 2.1.11. Ia?aaeaa/oaaiee iiiaio coieiee (([0,T] aoaeaii
iaceaaoe ia?aeaiiei aeey i?ioeana (t, yeui ( caaeiaieueiy? ia??aiino?

i.i.

oa

i.i.

Oaeei /eiii, ia?aeaiiee iiiaio coieiee aaiaeeoueny ye oaeee iiiaio, aei
yeiai i?ioean (t ia? noaia?oeiaaeuei? aeanoeaino?, a i?ney yeiai
-noia?ia?oeiaaeuei?. Iniiaiei ?acoeueoaoii ?

Oai?aia 2.1.12. Iaoae {(t, Ft, t ( [0,T]} — aeiaeaoi?e iaia?a?aiee
i?ioean, yeee caaeiaieueiy? oiiao (2). sseui ?nio? ia?aeaiiee iiiaio
coieiee aeey i?ioeana (t, oi oeae iiiaio ? iioeiaeueiei.

Aiaeia?/ii, m-ia?aeaiiee iiiaio coieiee aecia/a?oueny ye iiiaio, a yeee
noia?ia?oeiaaeuei? aeanoeaino? i?ioeano (t ci?iththoueny ia
noaia?oeiaaeuei?, oa aeiaiaeeoueny, ui yeui aeey i?ioeano ?nio?
m-ia?aeaiiee iiiaio coieiee, oi oeae iiiaio ? m-iioeiaeueiei.

A i?ae?icae?e? 2.2 ?icaeyaea?oueny ?ioee iaoiae a?aeooeaiiy
iioeiaeueiiai iiiaioo coieiee o i?eiouaii?, ui i?ioean (t aeiionea?
oaeoi?ecaoe?eia i?aaenoaaeaiiy:

(t = (t (t , (2)

aea (t — aeayeee ia?oeiaae, (t – ocaiaeaeaiee aeiaaeeiaee i?ioean.
Iniiaiee ?acoeueoao oeueiai i?ae?icae?eo oaeee:

Oai?aia 2.2.1 (2.2.2). Iaoae aeeiiothoueny ianooii? oiiae:

1) (t — ??aiii??ii ?ioaa?iaaiee ia?oeiaae;

aeey aoaeue-yeiai

iiiaioo coieiee (([0,T].

Oiae? (* ((*) — iioeiaeueiee (m-iioeiaeueiee) iiiaio coieiee aeey
i?ioeano (t.

E??i oiai, ca aeiiiiiaith oai??? AII-ia?oeiaae?a (ia?oeiaae?a c
iaiaaeaiei a na?aaeiueiio eieeaaiiyi) aeiaiaeeoueny, ui oiiaa 1) oe???
oai?aie aeeiio?oueny, yeui (t ia? niaoe?aeueiee aeaeyae:

aea (s, (s — aeaye? iaiaaeai? ia?aaeaa/oaai? aeiaaeeia? ooieoe??,
s([0,T].

A ianooiieo i?ae?icae?eao iaea?aeai? ?acoeueoaoe canoiniaothoueny aei
??cieo o?iainiaeo caaea/.

A i?ae?icae?e? 2.3 ?icaeyaea?oueny caaea/a a?aeooeaiiy iioeiaeueiiai
iiiaioo coieiee aeey i?ioeano, ui iieno? eai?oae ?iaanoi?a, yeui eiai
o?iainiaee ii?ooaeue neeaaea?oueny c aeaio aeoea?a (aeoe?? oa
iae?aaoe??), aa?o?noue yeeo caaeiaieueiy?, a?aeiia?aeii, noioanoe/ia oa
cae/aeia aeeoa?aioe?aeuei? ??aiyiiy. I?e i?eiouaiiyo iaiaaeaiino? oa
ia?aaeaa/oaaiino? ia eiao?oe??ioe oeeo ??aiyiue oi?ioeth?oueny aeinoaoiy
oiiaa ?nioaaiiy iioeiaeueiiai iiiaioo coieiee.

A i?ae?icae?e? 2.4 ?icaeyaea?oueny caaea/a ia?aeeth/aiiy i?ae aeaiia
o?iainiaeie no?aoaa?yie. I?eionea?oueny, ui ?iaanoi? ia? iiaeeea?noue
iae?aoe i?ae aeaiia o?iainiaeie ii?ooaeyie, eiaeiee c yeeo neeaaea?oueny
c aeoe?? oa iae?aaoe??. I?e oeueiio i?ioean, ui iieno? eai?oae
?iaanoi?a, caaeiaieueiy?, a?aeiia?aeii, aeaa e?i?eieo noioanoe/ieo
aeeoa?aioe?aeueieo ??aiyiiy. I?eionea?oueny oaeiae, ui a iiiaio
ia?aeeth/aiiy (?aae?caoe?? aeoea?a ia?oiai ii?ooaeth ? aeeaaeaiiy
iaea?aeaieo eioo?a a ae?oaee ii?ooaeue) niea/o?oueny aeayea oe?ia ca
ia?aeeth/aiiy, i?iii?oe?eia aaee/ei? eai?oaeo a oeae iiiaio. I?e ??cieo
i?eiouaiiyo ia eiao?oe??ioe ??aiyiue, ui iienothoue eai?oae ?iaanoi?a,
aeey a?aeooeaiiy iioeiaeueiiai iiiaioo ia?aeeth/aiiy canoiniaothoueny
oai?aie 2.1.12 oa 2.2.1.

I?ae?icae?e 2.5 i?enay/aii canoinoaaiith oai??? m-ia?aeaiieo oa
m-iioeiaeueieo iiiaio?a coieiee aei o?iainiai? caaea/? ia?aeeth/aiiy o
aeiaaeeo, eiee i?ioean, ui iieno? eai?oae ?iaanoi?a, caaeiaieueiy?
iaa?iaia noioanoe/ia

aeeoa?aioe?aeueia ??aiyiiy. A naia, ooea?oueny iioeiaeueiee iiiaio
ia?aeeth/aiiy oaeei /eiii, uia a e?ioeaaee iiiaio /ano T aeinyaoe
caaeaiiai ??aiy aaee/eie eai?oaeo ( i?e i?i?iaeuei?e ii/aoia?e
?iaanoeoe?? x0. I?e oeueiio aaaaea?oueny, ui oe?ia, yea niea/o?oueny ca
ia?aeeth/aiiy, neeaaea?oueny c aeaio eiiiiiaio, iaeia c yeeo
i?iii?oe?eia aaee/ei? eai?oaeo a iiiaio ia?aeeth/aiiy, a ae?oaa ia
caeaaeeoue a?ae aaee/eie eai?oaeo.

A?aeooeaiiy iioeiaeueiiai iiiaioo ia?aeeth/aiiy aeey caaea/?,
aiaeia?/ii? aei ?icaeyiooi? a i?ae?icae?e? 2.4, aea o aeiaaeeo, eiee
oe?ia ca ia?aeeth/aiiy i?noeoue, ye ? a i?ae?icae?e? 2.5, eiiiiiaioo,
iacaeaaeio a?ae aaee/eie eai?oaeo, ?icaeyiooi a i?ae?icae?e? 2.6.
Caaea/a ?ica’yco?oueny ca aeiiiiiaith oi?ioee ?oi. Oaeiae
?icaeyaeathoueny aeiaaeee, eiee oe?ia ca ia?aeeth/aiiy ia niea/o?oueny,
yeui cia/aiiy aaee/eie eai?oaeo a iiiaio ia?aeeth/aiiy ? iaaeiaeaoi?i,
oa eiee oe?ia ca ia?aeeth/aiiy ia niea/o?oueny a e?ioeaaee iiiaio.

Inoaii?e i?ae?icae?e 2.7 i?enay/aii a?aeooeaiith iioeiaeueiiai iiiaioo
ia?aeeth/aiiy o i?eiouaii?, ui eai?oae ?iaanoi?a caaeiaieueiy? iae?i?eia
noioanoe/ia aeeoa?aioe?aeueia ??aiyiiy. Caaea/a ?ica’yco?oueny ca
aeiiiiiaith oai?aie ii??aiyiiy, yea ? iiaeeo?eaoe??th oai?aie ii??aiyiiy
Aaeue/oea

AENIIAEE

A aeena?oaoe?? aeiaaaeaii ?nioaaiiy eieaeueiiai /ano aeey
aeaiia?aiao?e/ieo

/enoi ?ic?eaieo neeueieo ia?oeiaae?a, eiai iaia?a?ai?noue oa ?nioaaiiy
ne?i/aiieo iiiaio?a aoaeue-yeiai ii?yaeeo. I?e oeueiio iiaoaeiaai?
aeinoaoi? oiiae ocaaaeueiththoue ia aeaiia?aiao?e/iee aeiaaeie
aiaeia?/i? aeinoaoi? oiiae Aanna ?nioaaiiy eieaeueiiai /ano aeey
iaeiiia?aiao?e/ieo /enoi ?ic?eaieo ia?oeiaae?a. Oaeiae aea/aii
aeanoeaino? iaeii- oa aeaiia?aiao?e/ieo iioaioe?ae?a, anioe?eiaaieo ?c
c?inoath/eie i?ioeanaie oa iieyie. A iaeiiia?aiao?e/iiio aeiaaeeo
iaea?aeaii e?eoa??e ?nioaaiiy iiiaio?a c?inoath/iai i?ioeano,
anioe?eiaaiiai ?c iaeiiia?aiao?e/iei iioaioe?aeii, ui ocaaaeueith?
a?aeiio oai?aio Aa?n?a. Iaea?aeaii ioe?iee aeey iie?a, ii?iaeaeaieo
c?inoath/eie ia?aaeaa/oaaieie iieyie oa iiaoaeiaaii aeinoaoi? oiiae
?nioaaiiy ne?i/aiieo iiiaio?a aeaiia?aiao?e/iiai eieaeueiiai /ano.

Aaeeeo oaaao i?eae?eaii aea/aiith oai??? iioeiaeueieo iiiaio?a coieiee

oa ?? canoinoaaiiyi aei o?iainiaeo caaea/ iioeiaeueii? coieiee oa
ia?aeeth/aiiy. Iiaoaeiaaii oai??th ia?aeaiieo iiiaio?a coieiee, yeo
canoiniaaii aei a?aeooeaiiy iioeiaeueieo iiiaio?a coieiee. Iiaoaeiaaii
aeinoaoi? oiiae ?nioaaiiy iioeiaeueiiai iiiaioo coieiee aeey i?ioeano,
ui aeiionea? oaeoi?ecaoe?eia i?aaenoaaeaiiy. Iaea?aeai? ?acoeueoaoe
canoiniaaii aei a?aeooeaiiy iioeiaeueiiai iiiaioo coieiee oa
iioeiaeueiiai iiiaioo ia?aeeth/aiiy i?ae aeaiia o?iainiaeie ii?ooaeyie,
o aeiaaeeao, eiee i?ioean, ui iieno? eai?oae, caaeiaieueiy? ??ci? oeie
noioanoe/ieo aeeoa?aioe?aeueieo ??aiyiue.

Iniiai? ?acoeueoaoe aeena?oaoe?? iioae?eiaaii a ianooiieo ?iaioao:

I?oo?a TH.N., Ieueoe?e ss.I. Iioaioe?aee oa eieaeuei? /ane, anioe?eiaai?
c aeaiia?aiao?e/ieie /enoi ?ic?eaieie neeueieie ia?oeiaaeaie // Oai?.
eiia??i. oa iaoai. noaoenoeea. — 1997. — aei. 56. — N. 133-144.

Ieueoe?e ss.I. Iaia?a?ai?noue eieaeueieo /an?a aeey aeaiia?aiao?e/ieo
ia?oeiaaeueieo iie?a // A?niee Ee?anueeiai oi?aa?neoaoo. — 1997. —
aei.2. — N. 76-82.

I?oo?a TH.N., Ieueoe?e ss.I. Ia?oeiaaeuei?, noia?ia?oeiaaeuei? oa
c?inoath/? iiey ia ieiuei? // Aeiiia?ae? IAI Oe?a?ie. — 1998. — N 3. —
C. 17-22.

I?oo?a TH.N., Ieueoe?e ss.I. ?nioaaiiy oa aeanoeaino? eieaeueieo /an?a
aeey /enoi ?ic?eaieo neeueieo ia?oeiaae?a ia ieiuei? // Oace V
I?aeia?iaeii? eiioa?aioe?? ?i. aeaaeai?ea I.E?aa/oea. — Ee?a (Oe?a?ia).
— 1996. — N. 288

Mishura Yu.S., Oltsik Ya.A. Two-parameter potentials, increasing fields,
and purely discontinuous strong martingales on the plane // Proc. XVIII
Seminar on stability problems of stochastic models. — Debrecen
(Hungary). — 1997. — P. 69.

Oltsik Ya.A. Local times for strong martingales and some properties of
asociated potentials // Proc. Second Scandinavian-Ukrainian conference
in mathem. statistics. — Umea (Sweden). — 1997. — P. 83.

Mishura Yu.S., Oltsik Ya.A. Optimal switching for alternative financial
strategies // Proc. Second Scandinavian-Ukrainian conference in mathem.
statistics. — Umea (Sweden). — 1997. — P. 70.

Ieueoe?e ss.I. Noioanoe/iee aiae?c i?ioean?a oa iie?a ca aeiiiiiaith
ia?oeiaaeueieo iaoiae?a. – ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.01.05 – oai??y
eiia??iinoae oa iaoaiaoe/ia noaoenoeea. – Ee?anueeee oi?aa?neoao ?iai?
Oa?ana Oaa/aiea, i. Ee?a, 1999.

Aeena?oaoe?th i?enay/aii ieoaiiyi ?nioaaiiy eieaeueieo /an?a oa
iioeiaeueieo iiiaio?a coieiee aeey aeiaaeeiaeo i?ioean?a oa iie?a.
Aeiaaaeaii ?nioaaiiy eieaeueiiai /ano aeey aeaiia?aiao?e/ieo /enoi
?ic?eaieo neeueieo ia?oeiaae?a oa ca aeiiiiiaith oai???
iioaioe?ae?aaeine?aeaeaii oae? eiai aeanoeaino? ye iaia?a?ai?noue oa
?nioaaiiy ne?i/aiieo iiiaio?a. Ca aeiiiiiaith ia?oeiaaeueieo iaoiae?a
iiaoaeiaaii aea? oai??? a?aeooeaiiy iioeiaeueieo iiiaio?a coieiee aeey
aeiaaeeiaeo i?ioean?a, ye? canoiniaaii aei ?ica’ycaiiy o?iainiaeo caaea/
iioeiaeueiiai ia?aeeth/aiiy aeey ??cieo oei?a noioanoe/ieo
aeeoa?aioe?aeueieo ??aiyiue.

Eeth/ia? neiaa: eieaeueiee /an, neeueiee ia?oeiaae, iioaioe?ae,
iioeiaeueiee iiiaio coieiee, noioanoe/ia aeeoa?aioe?aeueia ??aiyiiy.

Ieueoeee ss.I. Noioanoe/aneee aiaeec i?ioeannia e iieae n iiiiuueth
ia?oeiaaeueiuo iaoiaeia. – ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.05 – oai?ey
aa?iyoiinoae e iaoaiaoe/aneay noaoenoeea. – Eeaaneee oieaa?neoao eiaie
Oa?ana Oaa/aiei, a. Eeaa, 1999.

Aeenna?oaoeey iinayuaia aii?inai nouanoaiaaiey eieaeueiuo a?aiai e
iioeiaeueiuo iiiaioia inoaiiaee aeey neo/aeiuo i?ioeannia e iieae.
Aeieacaii nouanoaiaaiea eieaeueiiai a?aiaie aeey aeaoia?aiao?e/aneeo
/enoi ?ac?uaiuo neeueiuo ia?oeiaaeia, yaeythueony aiciouaieai onoie/eauo
neiiao?e/iuo iieae. Iino?iaiiua aeinoaoi/iua oneiaey nouanoaiaaiey
eieaeueiiai a?aiaie iaiauatho ia aeaoia?aiao?e/aneee neo/ae aiaeiae/iua
aeinoaoi/iua oneiaey nouanoaiaaiey eieaeueiiai a?aiaie aeey
iaeiiia?aiao?e/aneeo /enoi ?ac?uaiuo ia?oeiaaeia. N iiiiuueth oai?ee
iioaioeeaeia enneaaeiaaiu oaeea aai naienoaa eae iai?a?uaiinoue e
nouanoaiaaiea eiia/iuo iiiaioia. Aeey iino?iaiey oneiaee nouanoaiaaiey
eiia/iuo iiiaioia anao ii?yaeeia enneaaeiaaiu naienoaa iaeii- e
aeaoia?aiao?e/aneeo iioaioeeaeia, annioeee?iaaiiuo n i?aaeneacoaiuie
aic?anoathueie i?ioeannaie e iieyie. A iaeiiia?aiao?e/aneii neo/aa
iieo/aii iaiauaiea oai?aiu Aa?nea, a eiaiii, e?eoa?ee nouanoaiaaiey
eiia/iuo iiiaioia anao ii?yaeeia aeey i?aaeneacoaiiai aic?anoathuaai
i?ioeanna, annioeee?iaaiiiai n iaeiiia?aiao?e/aneei iioaioeeaeii. A
aeaoia?aiao?e/aneii neo/aa iieo/aiu ioeaiee aeey iiiaioia iaeioi?uo
ooieoeeiiaeia, naycaiiuo n i?aaeneacoaiuie aic?anoathueie iieyie,
annioeee?iaaiiuie n aeaoia?aiao?e/aneeie iioaioeeaeaie. Iieo/aiu
aeinoaoi/iua oneiaey nouanoaiaaiey iiiaioia anao ii?yaeeia aeey
aic?anoathuaai i?aaeneacoaiiai iiey, annioeee?iaaiiiai n
aeaoia?aiao?e/aneei iioaioeeaeii, i?e auiieiaiee eioi?uo aeieacaii
nouanoaiaaiea eiia/iuo iiiaioia iino?iaiiiai eieaeueiiai a?aiaie. N
iiiiuueth ia?oeiaaeueiuo iaoiaeia ?aoaaony caaea/a iouneaiey
iioeiaeueiuo iiiaioia inoaiiaee aeey neo/aeiuo i?ioeannia. Iino?iaia
oai?ey ia?aeiiiuo iiiaioia inoaiiaee, n iiiiuueth eioi?ie euaony
iioeiaeueiue iiiaio inoaiiaee aeey i?ioeannia, eciaiythueo a iaeioi?ue
iiiaio noaia?oeiaaeueiua naienoaa ia noia?ia?oeiaaeueiua e iaiai?io.
Oaeaea iino?iaia oai?ey iouneaiey iioeiaeueiuo iiiaioia inoaiiaee aeey
i?ioeannia, aeiioneathueo i?aaenoaaeaiea a aeaea i?iecaaaeaiey aeaoo
i?ioeannia, iaeei ec eioi?uo yaeyaony ia?oeiaaeii. Iaa iaoiaea
i?eiaiythony aeey iouneaiey iioeiaeueiuo iiiaioia inoaiiaee e
ia?aeeth/aiey a oeiainiauo caaea/ao i?e oneiaee, /oi i?ioeann,
iienuaathuee eaieoae eiaanoi?a, oaeiaeaoai?yao ?aciua oeiu
noioanoe/aneeo aeeooa?aioeeaeueiuo o?aaiaiee. ?anniio?aiu neo/ae
eeiaeiiai e iaeeiaeiiai, i?yiiai e ia?aoiiai noioanoe/aneiai
aeeooa?aioeeaeueiiai o?aaiaiey, a i?aaeiieiaeaiee oieaou iaeioi?ie oeaiu
ca ia?aeeth/aiea, eae i?iii?oeeiiaeueiie aaee/eia eaieoaea, oae e
niaea?aeauae eiiiiiaioo, ia caaenyuoth io aaee/eiu eaieoaea. A
iineaaeiai neo/aa caaea/a iouneaiey iioeiaeueiiai iiiaioa ia?aeeth/aiey
?aoaaony n iiiiuueth oi?ioeu Eoi. A neo/aa, eiaaea eaieoae
oaeiaeaoai?yao iaeeiaeiia noioanoe/aneia aeeooa?aioeeaeueiia o?aaiaiea,
caaea/a iouneaiey iioeiaeueiiai iiiaioa ia?aeeth/aiey ?aoaaony n
iiiiuueth oai?aiu n?aaiaiey, eioi?ay yaeyaony iiaeeoeeaoeeae oai?aiu
n?aaiaiey Aaeue/oea.

Eeth/aaua neiaa: eieaeueiia a?aiy, neeueiue ia?oeiaae, iioaioeeae,
iioeiaeueiue iiiaio inoaiiaee, noioanoe/aneia aeeooa?aioeeaeueiia
o?aaiaiea.

Oltsik Ya. A. Stochastic analysis of processes and fields by means of
martingale methods. – Manuscript.

Thesis for the degree of candidate of sciences in physics and
mathematics in speciality 01.01.05 – probability theory and mathematical
statistics. – Kyiv Taras Shevchenko University, Kyiv, 1999.

The dissertation is devoted to the problems of the existence of local
times and optimal stopping times for stochastic processes and fields.
The existence of the local time for twoparameter purely discontinuous
strong martingales is proved, and such their properties as continuity
and existence of finite moments are investigated by means of potential
theory. Two theories of finding of optimal stopping times for stochastic
processes are constructed and applied to the solving of financial
problems of optimal switching for different types of stochastic
differential equations.

Key words: local time, strong martingale, potential, optimal stopping
time, stochastic differential equation.

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?iaaein A., Neaioiae Ae., *ai E. Oai?ey iioeiaeueiuo i?aaee inoaiiaee.
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Press, 1992. — 300 p.

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assets // Mathem. Finance. — 1997. — V. 7, N 3. — P. 241-286.

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geometric Brownian motion // Finance Stochast. — 1998. — N 2. — P.
295-310.

Aaeue/oe E.E. Oai?aia n?aaiaiey aeey noioanoe/aneeo o?aaiaiee n
eioaa?aeaie ii ia?oeiaaeai e neo/aeiui ia?ai // Oai?. aa?iyoi. e aa
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