.

Оптимальне оцінювання та керування системами з швидкими випадковими осциляціями: Автореф. дис… канд. фіз.-мат. наук / Ю.М. Полшков, НАН України. Ін-

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IAOeIIIAEUeIA AEAAeAIIss IAOE OE?A?IE

IINOEOOO I?EEEAAeII? IAOAIAOEEE OA IAOAIIEE

IIEOEIA THEIAI IEEIEAEIAE*

OAeE 519.21

IIOEIAEUeIA IOeIITHAAIIss OA EA?OAAIIss NENOAIAIE

C OAEAeEEIE AEIAAeEIAEIE INOeEEssOeIssIE

01.01.05 – oai?iy eiiai?iinoae oa iaoaiaoe/ia noaoenoeea

Aaoi?aoa?ao aeena?oaoei? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa oiceei-iaoaiaoe/ieo iaoe

Aeiiaoeuee – 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia a Aeiiaoeueeiio aea?aeaaiiio oi?aa?neoao? I?i?noa?noaa

ina?oe Oe?a?ie.

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

Aiiaea??a Ai?en Aieiaeeie?iae/,

Aeiiaoeueeee aea?aeaaiee oi?aa?neoao,

caa?aeoaa/ eaoaae?e aeaaa?e oa oai???
iia??iinoae

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

Oo?a?i Aiaoie?e Oaaei?iae/,

?inoeooo iaoaiaoeee IAI Oe?a?ie (i.
Ee?a),

caa?aeoaa/ a?aeae?eo oai???
eiia??iinoae oa

iaoaiaoe/ii? noaoenoeee

eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe,
i?ioani?,

*ai? Ieaenaiae? Naiaiiae/,

?inoeooo i?eeeaaeii? iaoaiaoeee oa
iaoai?ee

IAI Oe?a?ie,

noa?oee iaoeiaee ni?a?ia?oiee
a?aeae?eo oai???

eiia??iinoae oa iaoaiaoe/ii?
noaoenoeee

I?ia?aeia onoaiiaa: ?inoeooo e?aa?iaoeee ?i. A.I.Aeooeiaa IAI

Oe?a?ie, a?aeae?e iaoaiaoe/ieo
iaoiae?a

aeine?aeaeaiiy iia?aoe?e, i. Ee?a.

Caoeno a?aeaoaeaoueny « 28 » 04 1999 ?. i 16
aiaeei? ia

can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee Ae11.193.01 ?inoeooo
i?eeeaaeii? iaoaiaoeee oa iaoai?ee IAI Oe?a?ie, 340114, Aeiiaoeuee, aoe.
?ice Ethenaiao?a, 74.

C aeena?oaoe??th iiaeia iciaeiieoenue o a?ae?ioaoe? ?inoeoooo
i?eeeaaeii? iaoaiaoeee oa iaoai?ee IAI Oe?a?ie, 340114, Aeiiaoeuee, aoe.
?ice Ethenaiao?a, 74.

Aaoi?aoa?ao ?ic?neaiee « 22 » 03 1999 ?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee________________ Eiaaeeanueeee I.A.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeueiinoue ?iaioe. A?aie/i? oai?aie oai??? eiia??iinoae
a?athoue iaeaaaeeea?oo ?ieue ye a i?aeoe/ieo, oae ? a oai?aoe/ieo
aeine?aeaeaiiyo. Iiney a?aie/iiai ia?aoiaeo, ye i?aaeei, c’yaeythoueny
ia’?eoe, ui e?aua i?aeaeathoueny e?euee?niiio aiae?co e a?eueo c?o/i?
aeey ia/eneaiue a?aeiia?aeieo oa?aeoa?enoee. A ca’yceo c oeei, iniaeeai
aeey ?ic?aooie?a, iaoe?eaieo ia i?aeoe/ia aeei?enoaiiy, ae?ae aaaeeeai
ciaeoe ioe?iee oaeaeeino? caiaeiinoi aei a?aie/iiai ia’?eoo. Ia?aiee a
aeena?oaoe?? iai?yiie aeine?aeaeaiue nie?a?oueny ia iaoiae iaeiiai
eiiai?iinoiiai i?inoi?o Nei?ioiaea. ?aeay Nei?ioiaea aeieac?a a?aie/ieo
oai?ai aeyaeeany iaaecae/aeii ie?aeiith ? i?eaa?ioea oaaao aaaaoueio
iaoaiaoee?a. A?aecia/eii ?iaioe Eiieioa, Iaei?a ? Oooiaaei, A??e?oa ?
Oieiia, Naoaiaiea, Oo?aa, *ieiia, Aiiaea??aa, Aiiaea??aa ? Oo?ei,
Aiiaea??aa ? Ei?ieueiaa oa ?i., i?enay/ai? aea/aiith oaeaeeino?
caiaeiinoi ia’?eo?a caniaaie iiaoaeiae ia iaeiiio eiiai?iinoiiio
i?inoi??. Caaea/a oaeaeeinoi caiaeiinoi aeooaeueia ye c i?aeoe/ii?, oae
i c oai?aoe/ii? noi??i aeine?aeaeaiiy. Ioe?iea oaeaeeino? caiaeiinoi
aeicaiey? ioe?ieoe eiia??iino? ciaoiaeaeaiiy ?ica’yceo aeo?aeii? nenoaie
a iaeano?, iienaoe eiiai?iinoii aeanoeainoi aeoiaeii? nenoaie,
iiaoaeoaaoe a?aeiia?aeia ea?oaaiiy, aeecueea aei iioeiaeueiiai,
iiaoaeoaaoe ioe?ieo iaa?aeiiiai ia?aiao?o oa ii.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. I?ioyaii
ia?iiaeo n?/aiue 1992 – a?oaeaiue 1995 caeiaoaa/ i?aoethaaa ia iinaaei
iieiaeoiai iaoeiaiai niia?iaioieea aea?aeathaeaeaoii? aeineiaeieoeueei?
oaie “Aeine?aeaeaiiy aaiethoe?eieo noioanoe/ieo nenoai” (iiia?
aea?aeaaii? ?a?no?aoei? 92-1/31). ?acoeueoaoe oei?? aeineiaeieoeueei?
?iaioe /anoeiai aaieoee a aeena?oaoeith.

Iaoa ?iaioe. Iaoith aeine?aeaeaiiy aeaii? aeena?oaoe?? ? ioe?iea
oaeaeeino? caiaeiinoi ?ica’yceia aeeoa?aioe?aeueieo ??aiyiue, iao?aieo
“o?ce/iei” a?eei ooiii, aei ?ica’yceia a?aeiia?aeieo ??aiyiue Ioi. Ia
i?aenoaa? oaeeo ioe?iie i?eionea?oueny iiaoaeoaaoe ioe?iee iaa?aeiieo
ia?aiao??a (e aea/eoe aeanoeaino? ioe?iie) a aeoiaeieo nenoaiao,
iiaoaeoaaoe ea?oaaiiy nenoaiaie, ui aeaaeooue aoaeo, aeecueeee aei
iioeiaeueiiai.

-aeinoaoi? ea?oaaiiy a nenoaiao iiae aieeaii “o?ce/iiai” a?eiai ooio,
ciaeaeaii ioe?iee eaaciiaeneiaeueii? i?aaaeiiiaeiaiinoi aeey iaa?aeiiiai
ia?aiao?o a ciin? aeoiaeiiai ??aiyiiy, aea/aii aeanoeaino? ioe?iie
iaa?aeiiiai ia?aiao?o, caeiaooeo iaoiaeii iaeiaioeo eaaae?ao?a,
anoaiiaeaii neooiinoue ? aneiioioe/io ii?iaeueiinoue oaeeo ioe?iie,
caeiaooi ioe?iee aeecueeino? ?ica’yceo caaea/? Eioi aeey ??aiyiiy c
/anoeiieie iio?aeieie ia?oiai ii?yaeeo, iao?aiiai “o?ce/iei” a?eei
ooiii, e ?ica’yceo a?aeiia?aeiiai ??aiyiiy Ioi, ciaeaeaii iioeiaeueia
ea?oaaiiy ?ica’yceii ??aiyiiy Ioi c /anoeiieie iio?aeieie ia?oiai
ii?yaeeo, iiaoaeiaaii ?iceeaae ooieoeiiiaeo yeino? ii noaiaiyi iaeiai
ia?aiao?o, ocaaaeueiaii iiaeaeue ?. Ia?oiia iioeiaeueiiai niiaeeaaiiy ?
?ici?uaiiy aeey ?eieo oe?iieo iaia??a, ?ic?aoiaaii iioeiaeueiee
ii?ooaeue aeoea?a, ciaeaeaii ioe?iee aeeneiioiaaii? ei?eniino?.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoaoia. An? ?acoeueoaoe
iaea?aeaii aeey aeecueeeo aei caaea/ i?aeoeee iiaeaeae aeiaaeeiaeo
aaiethoeie: ?icaeyaeathoueny ??aiyiiy, iao?ai? “o?ce/iei” a?eei ooiii.
?ic?iaeai? a aeena?oaoe?? iaoiaee iiaoaeiae iioeiaeueieo ea?oaaiiue,
ioe?iie iaa?aeiieo ia?aiao??a iiaeooue aooe canoiniaaii ia i?aeoeoe?.
I?eeiie, canoiniaai? i?e aeieac? iaea?aeaieo ?acoeueoao?a iiaeooue aooe
aeei?enoai? i?e ??oaii? aiaeia?/ieo caaea/.

Iniaenoee aianie caeiaoaa/a. Aeena?oaio iioae?eoaaa /ioe?e
iaoeiai noaoo? [1, 2, 3, 4], o?e c ieo [1, 2, 3] ?acii c i?ioani?ii
Aiiaea??aei A. A. Any oaoi?/ia ?iaioa, iia’ycaia c ocaaaeueiaiiyi
?acoeueoao?a ? aaeaioaoe??th aeieac?a, aeeiiaia caeiaoaa/ai.
Aeena?oaioii ciaeaeaia ? aeine?aeaeaia iioeiaeueia ea?oaaiiy ?ica’yceii
noioanoe/iiai ??aiyiiy Ioi, iaea?aeaiiai i?e ocaaaeueiaiii caaea/? ?.
Ia?oiia [1]. Eiio iaeaaeaoue ?aea? ? aeaaaeaiiy ??aiyiue o ?iaioi [2,
3].

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. ?acoeueoaoe aeena?oaoe??
aeiiia?aeaeeny ia Aocianueeie eiioa?aioe?? AeiiAeO (ea?oaiue 1995 ?., i.
Aeiiaoeuee), ia Ae?oa?e anaoe?a?inuee?e eiioa?aioe?? “No/ani?
oiceei-iaoaiaoe/i? aeinyaiaiiy iieiaeeo a/aieo aocia Oe?a?ie” (16 – 18
o?aaiy 1995 ?., i. Ee?a), ia Aeiiaoeueeiio Eieieaioii “Eiia??i?noue oa
noaoenoeea”, i?enay/aiiio 80-?i//th E.?. Aioiaia (24 – 27 o?aaiy 1998
?., i. Aeiiaoeuee), aaaaoi?aciai ia iaoeiaiio nai?ia?? eaoaae?e aeaaa?e
oa oai??? eiia??iinoae AeiiAeO, ia iaoeiaiio nai?ia?? a?aeae?eo oai???
eiia??iinoae IIII IAI Oe?a?ie.

No?oeoo?a aeena?oaoe??. Aeena?oaoeiy neeaaea?oueny ci anooio,
i’yoe ?icaeieia oa nieneo aeei?enoaieo aeaea?ae. Caaaeueiee ianya
aeena?oaoei? noaiiaeoue 147 noi?iiie, nienie aeei?enoaieo aeaea?ae
caeia? 10 noi?iiie oa iinoeoue 95 iaeiaioaaiue.

Ioaeieaoei?. Ca oaiith aeena?oaoei? iioaeieiaaii /ioe?e noaooi o
oaoiaeo aeaeaiiyo [1, 2, 3, 4] oa oace oinoueio eiioa?aioeie.

INIIAIEE CIINO

Ia?oee ?icaeie iinoeoue iaeyae eioa?aoo?e, ae?oaee iaa?oiooaaiiy
ia?aiiai iai?yieo aeineiaeaeaiiue. C ia?oiai oa ae?oaiai ?icaeieia
?iaeii aeniiaie i?i iaiaoiaeiicoue aea/aiiy ioe?iie oaeaeeino?
caiaeiinoi ?ica’yceia aeeoa?aioe?aeueieo ??aiyiue, iao?aieo “o?ce/iei”
a?eei ooiii aei ?ica’yceia a?aeiia?aeieo ??aiyiue Ioi. Nooiiiue iiaiioe
?ica’yceo aeacaii? caaea/i i?iiiio?ii ioeiithaaoe ca aeiiiiiaith
iiaoaeiae ioe?iie iaa?aeiieo ia?aiao??a (e aea/aiiy ?o aeanoeainoae) a
aeoiaeieo nenoaiao, iiaoaeiae ea?oaaiiue nenoaiaie, ui aeaaeooue aoaeo,
aeecueeee aei iioeiaeueiiai.

Iniiaii ?acoeueoaoe o?aoueiai ?icaeieo iieyaathoue o ianooiiiio.
A ia?oiio iiae?icaeieo i?eaaaeaii iaiaoiaeii icia/aiiy i ?acoeueoaoe, ui

noinothoueny ?ioaa?ae?a a?ae i?ioean?a c oaeaeeeie inoeeeyoeiyie.

aeeiio?oueny

caaeiaieueiy? oiia? ??aiii??ii neeueiiai ia?aiiooaaiiy (?.n.i.) aai
oiia? ia?aiiooaaiiy “ca Ia?aaiiiaei”, yeui

iaceaa?oueny eiao?oe?aioii ??aiii??ii neeueiiai ia?aiiooaaiiy.

caaeiaieueiy? oiia? aaniethoii? ?aaoey?iinoi (a. ?.), yeui

iaceaa?oueny eiao?oe?aioii a.?. Caoaaaeaii, yeui aeeiiaia oiiaa
?.n.i., oi aeeiio?oueny e oiiaa a.?.

Oai?aia 3.1. Iaoae aeeiiai? oiiae:

oaeei, ui

;

, ui aeeiiaia oiiaa E?aia?a

;

;

Oiae?

5) aoaea ai?iei ?iceeaae

– noaiaea?oiee aiia?ianueeee i?ioean e ai?ia ioeiiea

;

6) aoaea ai?iith ioeiiea

;

7) aoaea ai?iith ioeiiea

– noaea Aeea?a.

.

iaeeaaeaeeny aeineoue aei?noe? iaiaaeaiiy. Iiaeia aeae?eueea iineaaeoe
oiiao ia caeaaei?noue, aea iineeeoe aeiiae aei o?a?eoi??? e iaea?aeaoe
aiaeiae oaeeo ioe?iie (uii?aaaea o noaiaiaaiio aeaeyae?).

Oai?aia 3.2. Iaoae aeeiiaii oiiae:

aeeiio?ony

2) c eiiai?iinoth 1

;

i?e

;

.

Oiaei

5) aoaea ai?iei ?iceeaae

e ai?ia ioeiiea

;

6) aoaea ai?iith ioeiiea

;

7) aoaea ai?iith ioeiiea

– noaea Aeea?a.

.

O ae?oaiio i o?aoueiio iiae?icaeieao o?aoueiai ?icaeieo ciaeaeaii
ioe?iee oaeaeeino? caiaeiinoi ?ica’yceia noioanoe/ieo aeeoa?aioe?aeueieo
??aiyiue c oaeaeeeie aeiaaeeiaeie inoeeeyoeiyie aei ?ica’yceia
a?aeiia?aeieo ??aiyiue Ioi.

Aeoiaeia nenoaia ia? aeaeyae

?ica’ycie oaei? caaea/? ?nio? ? aii ?aeeiee (eaia 3.1). “Aeecueeeie” aei
aeoiaeii? nenoaie aoaeooue ??aiyiiy

aea

Oai?aia 3.3. Iaoae aeeiiaii oiiae oai?aie 3.1 oa eiaoioei?ioe
aeoiaeiiai ?iaiyiiy caaeiaieueiythoue oiiaai

Oiaei aoaea ai?iith ioeiiea

– a eaii 3.3,

Oai?aia 3.4. Iaoae aeeiiaii oiiae oai?aie 3.3, oiaei aoaea
ai?iith ioeiiea

Oai?aie 3.3, 3.4 aeieacaii o aeiaaeeo ?.n.i. Aiaeiai/ii
?acoeueoaoe iaea?aeaii o aeiaaeeo a.?. (oai?aie 3.5, 3.6).

-aeinoaoi? ea?oiiy a nenoaiao iiae aieeaii “o?ce/iiai” a?eiai ooio,
ciaeaeai? ioe?iee eaaciiaeneiaeueii? i?aaaeiiiaeiaiinoi aeey iaa?aeiiiai
ia?aiao?o a ciin? aeoiaeiiai ??aiyiiy, aea/ai? aeanoeaino? ioe?iie
iaa?aeiiiai ia?aiao?o, caeiaooeo iaoiaeii iaeiaioeo eaaae?ao?a.

caaea/o Eioi

– iiiaeeia i?eionoeieo ea?oaaiue. I?ae caaea/ath iioeiaeueiiai
ea?oaaiiy nenoaiith c ooieoeiiiaeii yeino?

oaei?, ui

aeey oe??? nenoaie ? anoaiiaeoe ni?aa?aeiioaiiy

,

-aeinoaoi?i ea?oaaiiyi aeey aeoiaeii? nenoaie.

“Aeecueeith” aei aeoiaeii? nenoaie aoaea nenoaia

Ioeiiee “aeecueeinoi” ciaeaeaii i?e aeieaci oai?aie 4.1 oa oai?aie 4.2
i?e oiiaao ?.n.i. oa a.?. aiaeiiaiaeii.

aeeiio?ony ia?iaiinoue

e ciae ?iaiinoi aeinyaa?ony i?e oiiai

,

– ?ica’ycie caaea/i

Oai?aia 4.4. Iaoae aeeiiaii oiiae: iniothoue

,

Oiaei aoaea ai?iei ianooiia:

e aeeiiothoueny oiiae oai?aie 3.3 i oai?aie 4.1.

-aeinoaoiii aeey aeoiaeii? nenoaie, i?e/iio aeeiio?ony

– ?ica’ycie caaea/i e

.

Aiaeiai/ia oai?aia (oai?aia 4.6) aeieacaia o aeiaaeeo a.?.

O iiae?icaeiei 4.3 ?icaeyaea?ony aeoiaeia nenoaia

. “Aeecueeeie” aoaeooue o?a?eoi?i? caaea/i

– ?ica’ycie caaea/i

iacaaii ioeiieo, yea iaaea? iaeneioi ooieoeiiiaeo uieueiinoi ii? ia
o?a?eoi?i? ?ica’yceo aeoiaeii? nenoaie. Iniiaiee ?acoeueoao oeueiai
iiae?icaeieo iieyaa? a ianooiiiio.

– ioeiiea eaaciiaeneiaeueii? i?aaaeiiiaeiaiinoi, oiaoi oaea, ui

,

ai?ii

. Iaoae aeeiiothoueny oiiae oai?aie 3.4.

Oiaei aoaea ai?iith ioeiiea

.

Oai?aia 4.7 aeieacaia i?e oiiai ?.n.i. ?? aiaeia o aeiaaeeo a.?.
– oai?aia 4.8.

caaea/o Eioi

Oaeiae ?icaeyiaii

, i?e yeiio ooieoeiiiae

aeeiio?oueny ia?iaiinoue

Oai?aia 4.9. Iaoae aeeiiothoueny ianooiii oiiae:

– noaoe?iia?iee o aocueeiio cia/aiii aeiaaeeiaee i?ioean c ioeueiaei
na?aaei?i, ui caaeiaieueiy? oiia? ?.n.i., aeey yeiai aeeiiothoueny oiiae
oai?aie 3.1 e

;

caaeiaieueiythoue oiiaai

ia iiiaeeii aeiaeaoii? ii?e Eaaaaa;

oaea, ui

.

i?e

.

.

e aeenia?nith

O oeueiio iiae?icaeiei oaeiae ?icaeyaeaeanue e?aeiaa caaea/a
Aei?ioe? c oaeei naiei aeiaaeeiaei aieeaii. Aeieacaia neooiinoue oa
aneiioioe/ia ii?iaeueiinoue ioeiiie (oai?aie 4.11, 4.12).

A ia?oiio iiae?icaeieo ?icaeieo 5 aoee caeiaooi ioe?iee
aeecueeino? ?ica’yceo caaea/? Eioi aeey ??aiyiiy c /anoeiieie iio?aeieie
ia?oiai ii?yaeeo, iao?aiiai “o?ce/iei” a?eei ooiii, e ?ica’yceo
a?aeiia?aeiiai ??aiyiiy Ioi.

?icaeyiaii caaea/o Eioi aeey ?iaiyiiy

c eiiai?iinoth 1.

Oai?aia 5.1. Iaoae

;

,

;

, oiaoi

;

.

Oiaei ?ica’ycie caaea/i Eioi inio? i aii ?aeeiee.

Ii?yae c aeoiaeiith caaea/ath ?icaeyiaii “aeecueeo” caaea/o Eioi

c eiiai?iinoth 1. I?e oiiai ?.n.i. aoea aeieacaia ianooiia oai?aia.

Oai?aia 5.2. Iaoae aeeiiothoueny oiiae oai?ai 5.1 i 3.1. Oiaei,
yeui

oi

I?e oiiai a.?. aiaeia iiia?aaeiuei? oai?aie – oai?aia 5.3.

A i. 5.2 ciaeaeaia iioeiaeueia ea?oaaiiy ?ica’yceii ??aiyiiy Ioi
c /anoeiieie iio?aeieie ia?oiai ii?yaeeo, iiaoaeiaai ?iceeaae
ooieoeiiiaeo yeino? ii noaiaiyi iaeiai ia?aiao?o.

?icaeyiaii caaea/o Eioi aeey ?iaiyiiy

aeey oei?? caaea/i c ooieoeiiiaeii yeinoi

Ii?yae c aeoiaeiith nenoaiith ?icaeyiaii caaea/o

aeeiio?oueny ia?iaiinoue

Ciae ?iaiinoi aeinyaa?oueny ia iioeiaeueiiio ea?oaaiii

– ?ica’ycie caaea/i

A oai?aii 5.4 aeieacaii, ui

.

c eiaeii? neii/aiii? iaeanoi aoaea ai?iei ?iceeaae

– ?ica’ycie caaea/i

aecia/a?oueny oi?ioeith

– ?ica’ycie ?iaiyiiy

A inoaiiueiio iiae?icaeieo ocaaaeueiaia iiaeaeue ?. Ia?oiia
iioeiaeueiiai niiaeeaaiiy ? ?ici?uaiiy aeey ?eieo oe?iieo iaia??a,
?ic?aoiaai iioeiaeueiee ii?ooaeue aeoea?a, ciaeaeai? ioe?iee
aeeneiioiaaiii? ei?eniino?.

– i?ioeaioia noaaea, ooieoeiy caaeiaieueiy? ?iaiyiith

.

Oeiia aeoei? caaeiaieueiy? iiaeaei I. Naioaeueniia “aeiiiii/iiai”
a?ioiianueeiai ?ooo

,

. Oeiia aeoei? caeaaeeoue aiae o?ueio ia?aiao?ia:

.

– /anoea eaiioaeo ia aaieianueeiio ?aooieo.

Aieueo ?aaeinoe/iith o?aaa aaaaeaoe iiaeaeue, aeey yei? oeiia
aeoei? caaeiaieueiy?

– noaoeiiia?iee aeiaaeeiaee i?ioean c ioeueiaei na?aaeiii.

Oeiia iiaanoeoeieiiai ii?ooaeth aoaea ?ica’yceii aeoiaeiiai
?iaiyiiy

Ooieoeiiiae yeinoi (aeeneiioiaaii? ei?eniinoi) aeey aeoiaeii? nenoaie
ia? aeaeyae

“Aeecueeei” aei aeoiaeiiai aoaea ?iaiyiiy

oae, uia

oaeei, ui

Oiaei aoaeooue ai?ieie oaa?aeaeaiiy:

oiaei i?e

aeeiio?oueny

– ?ica’ycie aeoiaeiiai ?iaiyiiy ca iioeiaeueieie ea?oaaiiyie
“aeecueeith” nenoaiith

aeeiio?oueny

– ?ica’ycie aeoiaeiiai ?iaiyiiy ca iioeiaeueieie ea?oaaiiyie
“aeecueeith” nenoaiith. O oeueiio aeiaaeeo

AENIIAEE

Iioi/iee noai ieoaiiy iieyaa? o ianooiiiio. Aeena?oaoe?th
i?enay/aii aea/aiith oaeaeeino? caiaeiinoi nenoai, iao?aieo i?ioeanaie
c? neaaeith caeaaei?noth. Iacaaaeath/e ia aaeeeo eieueeinoue
ioaeieaoeie, i?enay/aieo oei?e oaiaoeoe?, iciy? aaeeea, iaeaea ia
?ic?iaeaia iaeanoue oeueiai iai?yieo. Oey iaeanoue i?aeoe/iiai ieaio –
iioeiaeueia ioeiithaaiiy oa ea?oaaiiy nenoaiaie c oaeaeeeie aeiaaeeiaeie
inoeeeyoeiyie.

A aeena?oaoe?? iaea?aeaii ioe?iee oaeaeeino? caiaeiinoi ?ica’yceo
caaea/? Eioi, ia yeo aieeaa? inoeeeyoeaiee i?ioean, aei ?ica’yceo
a?aeiia?aeieo ??aiyiue Ioi. Inoeeeyoeaiee i?ioean caaeiaieueiy? oiiai
?.n.i. aai oiiai a.?. A?aecia/eii, ui oiiaa ?.n.i. iaeeaaea? ia
inoeeeyoeaiee i?ioean a?eueo aei?noe? iaiaaeaiiy, i?ae oiiaa a.?.
Uii?aaaea, ioe?iee, iaea?aeaii ca oiiae ?.n.i. aeniiiaioeiaeueiiai
aeaeyaeo. Oea caaacia/o? a?eueoo oaeaee?noue caiaeiinoi, iiae ca oiiae
a.?., aea iaea?aeoaai? ioe?iee aoaeooue noaiaiaaiai aeaeyaeo. C
?acoeueoao?a o?aoueiai ?icaeieo iiaeia c?iaeoe aeniiaie i?i
iaiao?aei?noue ?icaeyaeo nenoai ye ca oiiae ?.n.i., oae ? ca oiiae a.?.
Iaea?aeaii ioe?iee aeecueeino? ?ica’yceia a?aeiia?aeieo ??aiyiue
aeicaieeee ?ica’ycoaaoe aaaaoi caaea/ o ianooiieo ?icaeieao
aeena?oaoe??.

Ia caeii/aiiy o?aaa a?aecia/eoe, ui iaea?aeaii a aeena?oaoe??
?acoeueoaoe iiaeooue aooe canoiniaaii aei caaea/ i?aeoeee. Oea
iaa?oioiao?oueny oei, ui nenoaie c “o?ce/iei” a?eei ooiii a?eueo
?aaeuei?, iiae nenoaie ic cae/aeiei a?eei ooiii. ?ic?iaeai? iaoiaee
iiaoaeiae iioeiaeueieo ea?oaaiue, ioe?iie iaa?aeiieo ia?aiao??a iiaeooue
aeei?enoiaoaaoeny i?e ?ica’yceo ?ioeo caaea/.

NIENIE IIOAEIEIAAIEO AAOI?II I?AOeUe CA OAIITH

AeENA?OAOeI?

1. Aiiaea?aa A.A., Iieoeia TH.I. Ia iaeiie caaea/a ?. Ia?oiia //
Aanoiee Aeiiaoeeiai oieaa?neoaoa. Na?ey A. Anoanoaaiiua iaoee. – 1997. –
?1. – N. 9-20.

2. Aiiaea?aa A.A., Iieoeia TH.I. Ia iioeiaeueiii oi?aaeaiee
?aoaieai noioanoe/aneiai o?aaiaiey a /anoiuo i?iecaiaeiuo ia?aiai
ii?yaeea // Eeaa?iaoeea e nenoaiiue aiaeec. – 1998. – ?2. – N. 134-143.

3. Polshkov Yu.N. On estimate of unknown parameters in stochastic

systems with fast random oscillations // Random Oper. and Stoch. Equ. –
1996. – V. 4, N 2. – P. 119-132.

-sufficient control in stochastic systems with fast random oscillation
// Theory of Stochastic Processes. – 1998. – V. 4(20), N. 1-2. – P.
71-81.

5. Bondarev B.V., Polshkov Yu. N. On the some problem of R.
Merton // Abstracts of the Second Scandinavian-Ukrainian Conference in
Mathematical Statistics (Umea, Sweden, 8-13 June 1997). – Umea (Sweden),
1997. – P. 9.

6. Polshkov Yu.N. On optimal control for the solution of the
stochastic differential equation // Stochastic Dynamical Systems:
Theory and Applications (First Ukrainian-Scandinavian Conference,
Uzhgorod, Ukraine, September 30-October 6, 1995). – Oaeai?iae, 1995. –
P. 70.

Iieoeia TH.I. Iioeiaeueia ioeiithaaiiy oa ea?oaaiiy nenoaiaie c
oaeaeeeie aeiaaeeiaeie inoeeeyoeiyie.- ?oeiien.

Aeena?oaoeiy ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
oiceei-iaoaiaoe/ieo iaoe ca niaoeiaeueiinoth 01.01.05 – oai?iy
eiiai?iinoae oa iaoaiaoe/ia noaoenoeea.- Iinoeooo i?eeeaaeii? iaoaiaoeee
oa iaoaiiee IAI Oe?a?ie, Aeiiaoeuee, 1999.

-aeinoaoi? ea?oaaiiy a nenoaiao iiae aieeaii “o?ce/iiai” a?eiai ooio,
ciaeaeaii ioe?iee eaaciiaeneiaeueii? i?aaaeiiiaeiaiinoi aeey iaa?aeiiiai
ia?aiao?o a ciin? aeoiaeiiai ??aiyiiy, aea/aii aeanoeaino? ioe?iie
iaa?aeiiiai ia?aiao?o, caeiaooeo iaoiaeii iaeiaioeo eaaae?ao?a,
anoaiiaeaii neooiinoue ? aneiioioe/io ii?iaeueiinoue oaeeo ioe?iie,
caeiaooi ioe?iee aeecueeino? ?ica’yceo caaea/? Eioi aeey ??aiyiiy c
/anoeiieie iio?aeieie ia?oiai ii?yaeeo, iao?aiiai “o?ce/iei” a?eei
ooiii, e ?ica’yceo a?aeiia?aeiiai ??aiyiiy Ioi, ciaeaeaii iioeiaeueia
ea?oaaiiy ?ica’yceii ??aiyiiy Ioi c /anoeiieie iio?aeieie ia?oiai
ii?yaeeo, iiaoaeiaaii ?iceeaae ooieoeiiiaeo yeino? ii noaiaiyi iaeiai
ia?aiao?o, ocaaaeueiaii iiaeaeue ?. Ia?oiia iioeiaeueiiai niiaeeaaiiy ?
?ici?uaiiy ia ?eieo oe?iieo iaia??a, ?ic?aoiaaii iioeiaeueiee ii?ooaeue
aeoea?a, ciaeaeaii ioe?iee aeeneiioiaaii? ei?eniino?.

-aeinoaoi? ea?oaaiiy, ioeiiea eaaciiaeneiaeueii? i?aaaeiiiaeiaiinoi,
“oice/iee” aieee ooi, iioeiaeueiee ii?ooaeue oeiiieo iaia?ia.

Polshkov Yu.N. Optimal estimation and control in systems with
fast random oscillations.- Manuscript.

Thesis for a candidate degree by speciality 01.01.05 –
probability theory and mathematical statistics.- The Institute of
Applied Mathematics and Mechanics of National Academy of Science of
Ukraine, Donetsk, 1999.

The dissertation is devoted to investigation of the convergence
rate of differential equations solutions with “physical” white noise to
Ito equations

-sufficient control in systems with “physical” white noise; obtain the
quasi-maximal likelihood estimator of the unknown parameter in the drift
of equation; investigate the properties of the estimate of the least
squares method for the unknown parameter; prove consistency and
asymptotic normality of such estimates; obtain the estimates of the
convergence of the solution for the partial differential equation of the
first order with “physical” white noise to Ito equations solution;
obtain the optimal control of the solution for the partial differential
Ito equation of the first order; construct the expansion of the quality
functional; generalize the optimum consumption-portfolio Merton model
in security market; construct optimum portfolio; obtain the estimates of
discounted utility.

-sufficient control, quasi-maximal likelihood estimator, “physical”
white noise, optimum securities portfolio.

Iieoeia TH.I. Iioeiaeueiia ioeaieaaiea e oi?aaeaiea nenoaiaie n
auno?uie neo/aeiuie inoeeeeyoeeyie.- ?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.- Einoeooo i?eeeaaeiie
iaoaiaoeee e iaoaieee IAI Oe?aeiu, Aeiiaoee, 1999.

Aeenna?oaoeey iinayuaia eco/aieth nei?inoe noiaeeiinoe ?aoaiee
aeeooa?aioeeaeueiuo o?aaiaiee, aiciouaiiuo “oece/aneei” aaeui ooiii, e
?aoaieyi niioaaonoaothueo o?aaiaiee Eoi. Ia iniiaaiee yoeo ioeaiie auee
iieo/aiu ?acoeueoaou, ninoaaeaoea niaea?aeaiea aeaiiie aeenna?oaoeeiiiie
?aaiou. E iaeaieaa nouanoaaiiui ?acoeueoaoai ioiinyony neaaeothuea
iieiaeaiey:

auienaiu a yaiii aeaea;

-aeinoaoi/iua oi?aaeaiey a nenoaiao, aiciouaiiuo auno?uie neo/aeiuie
inoeeeeyoeeyie. Aeey oaeeo nenoai iaeuecy a iauai neo/aa iaienaoue
(?aoaaiua) o?aaiaiey oeia o?aaiaiee Aaeeiaia, naycuaathueo iioeiaeueiia
oi?aaeaiea e cia/aiea ooieoeeiiaea noieiinoe ii oie i?e/eia, /oi ?aoaiea
oaeeo o?aaiaiee a iauai neo/aa ia ia?eianeea e yaeythony neo/aeiuie
i?ioeannaie aanueia iauaai aeaea. Iaeaieaa i?eaieaiui i?e ?aoaiee yoie
caaea/e yaeyaony iooue, iniiauaathueeny ia naeeaeaiee ?aoaiee enoiaeiiai
o?aaiaiey n ?aoaieai niioaaonoaothuaai o?aaiaiey Eoi. Eniieuecoy eaeae
aieaa ?aiieo ?aaio e aeieacaa naienoai “aeaaeeinoe”, oaeaeinue
aeieacaoue, /oi eniieuecoaiia oi?aaeaiea iiaeao iaania/eoue iaiaoiaeeiue
o?iaaiue aeey ooieoeeiiaea noieiinoe;

, eioi?ay aeinoaaeyao iaeneioi ieioiinoe, a aeenna?oaoeee iacuaaaony
ioeaieie eaaceiaeneiaeueiiai i?aaaeiiiaeiaey;

– iaeei ec ia?aa?aoia aeenna?oaoeee iinayuai eco/aieth naienoa
ioeaiie iaecaanoiiai ia?aiao?a a niina o?aaiaiey, aiciouaiiiai
“oece/aneei” aaeui ooiii, iieo/aiiuo ii iaoiaeo iaeiaiueoeo eaaae?aoia.
Onoaiiaeaia ninoiyoaeueiinoue oaeeo ioeaiie e aneiioioe/aneay
ii?iaeueiinoue. Aiaeiae/iua ?acoeueoaou iieo/aiu aeey e?aaaie caaea/e
Aee?eoea n ioeaauie a?aie/iuie oneiaeyie i?e oao aea neo/aeiuo
aicaeaenoaeyo;

– ?anniio?aia caaea/a Eioe aeey o?aaiaiey n /anoiuie
i?iecaiaeiuie ia?aiai ii?yaeea n auno?uie neo/aeiuie inoeeeeyoeeyie.
I?eaaaeaiu aeinoaoi/iua oneiaey aeey nouanoaiaaiey e aaeeinoaaiiinoe
?aoaiey oaeie caaea/e. Onoaiiaeaia ioeaiea, oa?aeoa?ecothuay aeecinoue
?aoaiee enoiaeiie caaea/e e niioaaonoaothuae i?aaeaeueiie caaea/e;

– iaeaeaii iioeiaeueiia oi?aaeaiea ?aoaieai o?aaiaiey Eoi n
/anoiuie i?iecaiaeiuie ia?aiai ii?yaeea. Iino?iaii ?aceiaeaiea
ooieoeeiiaea noieiinoe ii noaiaiyi iaeiai ia?aiao?a;

-aeinoaoi/iinoe i?eiaiyaiiai oi?aaeaiey ii?ooaeai. Oaeei ia?acii, auea
enneaaeiaaia aaaeiay a i?aeoe/aneii ieaia caaea/a oeiainiaie iaoaiaoeee
– oi?aaeaiea ii?ooaeai oeaiiuo aoiaa.

-aeinoaoi/iia oi?aaeaiea, ioeaiea eaaceiaeneiaeueiiai i?aaaeiiiaeiaey,
“oece/aneee” aaeue ooi, iioeiaeueiue ii?ooaeue oeaiiuo aoiaa.

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