Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence
In the metric $\rho(X, Y) = (\sup\limits_{0 \leq t \leq T} M|X(t) - Y(t)|^2)^{1/2} $ for an ordinary stochastic differential equation of order $p \geq 2$ with small parameter of the higher derivative, we establish an estimate of the rate of convergence of its solution to a solution of stochastic equ...
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| Дата: | 2006 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2006
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3557 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509667810082816 |
|---|---|
| author | Bondarev, B. V. Kovtun, E. E. Бондарев, Б. В. Ковтун, Е. Е. Бондарев, Б. В. Ковтун, Е. Е. |
| author_facet | Bondarev, B. V. Kovtun, E. E. Бондарев, Б. В. Ковтун, Е. Е. Бондарев, Б. В. Ковтун, Е. Е. |
| author_sort | Bondarev, B. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:57:46Z |
| description | In the metric $\rho(X, Y) = (\sup\limits_{0 \leq t \leq T} M|X(t) - Y(t)|^2)^{1/2} $
for an ordinary stochastic differential equation of
order $p \geq 2$ with small parameter of the higher derivative, we establish an estimate of the rate of convergence of its solution to a solution of stochastic equation of order $p - 1$. |
| first_indexed | 2026-03-24T02:44:45Z |
| format | Article |
| fulltext |
UDK 519.21
B. V. Bondarev, E. E. Kovtun (Doneck. nac. un-t)
PONYÛENYE PORQDKA SYSTEMÁ
STOXASTYÇESKYX DYFFERENCYAL|NÁX URAVNENYJ
S MALÁM PARAMETROM PRY STARÍEJ PROYZVODNOJ.
OCENKA SKOROSTY SXODYMOSTY
In the metric ρ( , ) sup ( ) ( )( ) /X Y X t Y t
t T
= −
≤ ≤0
2 1 2M for an ordinary stochastic differential equation of
order p ≥ 2 with small parameter of the higher derivative, we establish an estimate of the rate of
convergence of its solution to a solution of stochastic equation of order p – 1.
U metryci ρ( , ) sup ( ) ( )( ) /X Y X t Y t
t T
= −
≤ ≤0
2 1 2M vstanovleno ocinku ßvydkosti zbiΩnosti roz-
v’qzku zvyçajnoho stoxastyçnoho dyferencial\noho rivnqnnq porqdku p ≥ 2 z malym para-
metrom pry starßij poxidnij do rozv’qzku stoxastyçnoho rivnqnnq porqdku p – 1.
Vvedenye. V stat\e rassmatryvagtsq uravnenyq, opys¥vagwye dvyΩenye ças-
tyc v b¥stroperemenn¥x sluçajn¥x polqx. Takym budet dvyΩenye otdel\n¥x
molekul Ωydkosty y haza, zarqΩenn¥x çastyc v yonyzyrovannoj plazme, dvy-
Ωenye πlektronov v krystallyçeskoj reßetke y t. p. Stoxastyçeskye uravne-
nyq v ukazann¥x zadaçax matematyçeskoj fyzyky, kak pravylo, soderΩat vto-
rug yly bolee v¥sokug proyzvodnug po vremeny. Odnako v teoryy sluçajn¥x
processov dlq yzuçenyq qvlenyj dyffuzyy, t. e. dvyΩenyq çastyc¥ pod vlyq-
nyem sluçajn¥x vzaymodejstvyj, yspol\zugtsq stoxastyçeskye uravnenyq bo-
lee nyzkoho porqdka. ∏ty uravnenyq poluçen¥ yz πvrystyçeskyx soobraΩenyj,
no ony suwestvenno prowe. Na svqz\ zapys¥vaem¥x uravnenyj dynamyky, soder-
Ωawyx vtorug proyzvodnug po vremeny, s markovskymy processamy vperv¥e
ukazano v rabote N. M. Kr¥lova y N. N. Boholgbova [1]. Voznykaet vopros:
moΩno ly pry kakyx-lybo uslovyqx toçnoe uravnenye matematyçeskoj fyzyky,
soderΩawee vtorug yly bolee v¥sokug proyzvodnug po vremeny, zamenyt\ sto-
xastyçeskym uravnenyem, soderΩawym proyzvodnug po vremeny porqdkom ny-
Ωe? Naprymer, sluçajn¥j process X t
dX t
dt
( ), ( ){ } opys¥vaet dvyΩenye çastyc¥
mass¥ m pod dejstvyem polq syl F x( ) v neprer¥vnoj srede, svojstva kotoroj
xarakteryzugt parametr¥ A y T. ∏tot process moΩet b¥t\ opysan y uravne-
nyem LanΩevena [2]
md dX
dt
A dX
dt
dt+ = AT dW t mF X t dt( ) ( ( ))+ , (1)
hde W t( ) — vynerovskyj process. Oboznaçyv ε = m A/ , yz (1) poluçym stoxas-
tyçeskoe dyfferencyal\noe uravnenye s mal¥m parametrom pry starßej pro-
yzvodnoj
© B. V. BONDAREV, E. E. KOVTUN, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12 1587
1588 B. V. BONDAREV, E. E. KOVTUN
εd dX
dt
= –
dX
dt
dt T
A
dW t F X t dt+ +( ) ( ( ))ε . (2)
Prymerom dvyΩenyq, opys¥vaemoho s pomow\g uravnenyq tret\eho porqdka, qv-
lqetsq dvyΩenye πlektrona [3 – 5]
εm d X
dt
3
3 = – m d X
dt
f X dX
dt
t lE
2
2 +
+, , , (3)
hde ε = ≅ −2
3
10
2
2
22l
mc
c, m — nablgdaemaq massa, c — skorost\ sveta v vakuu-
me, l — zarqd πlektrona, f x y t( , , ) — vneßnye syl¥, E — xaotyçeskye koleba-
nyq πlektryçeskoho polq vakuuma. V [4, 5] yzuçalos\ predel\noe povedenye re-
ßenyj uravnenyq vyda (3) vtoroho porqdka, tam Ωe moΩno najty ss¥lky na bo-
lee rannye rabot¥. V rabote [4] ustanovlen¥ ßyrokye uslovyq, obespeçyvag-
wye pry ε → 0 prevrawenye uravnenyq vtoroho porqdka v uravnenye pervoho
porqdka, a takΩe analytyçeskaq svqz\ meΩdu koπffycyentamy do predel\noho
y predel\noho uravnenyj dlq sluçaq A = aI , hde a — nekotoraq postoqnnaq.
V rabote [5] rezul\tat¥ [4] perenesen¥ na sluçaj uravnenyj porqdka p ≥ 3 s
ves\ma obwymy koπffycyentamy y mal¥m parametrom pry starßej proyzvod-
noj:
ε εd
d X t
dt
p
p
−
−
1
1
( )
= – A t X t
dX t
dt
dX t
dt
d X t
dt
dt
p
p
p
pε ε
ε ε ε, ( ),
( )
, ,
( ) ( )…
−
−
−
−
2
2
1
1 +
+ U t X t
dX t
dt
dX t
dt
dt B t X t
dX t
dt
dX t
dt
dW t
p
p
p
pε ε
ε ε
ε ε
ε ε, ( ),
( )
, ,
( )
, ( ),
( )
, ,
( )
( )…
+ …
−
−
−
−
2
2
2
2 ,(4)
hde X tε( ), Uε — vektor¥, Aε , Bε — matryc¥, W t( ) — mnohomern¥j vynerov-
skyj process. V rabotax [2 – 5] dokazana slabaq sxodymost\ sluçajnoho proces-
sa X tε( ), t T∈[ , ]0 , pry ε → 0 k reßenyg uravnenyq bolee nyzkoho porqdka.
Naprymer, v [5] pokazano, çto pry opredelenn¥x uslovyqx reßenye (4) slabo
sxodytsq pry ε → 0 k reßenyg stoxastyçeskoho dyfferencyal\noho uravne-
nyq
d
d X t
dt
p
p
−
−
2
0
2
( )
= A U dt q dt A B dW tt t t t t
− −+ +1 1 ( ), (5)
v kotorom vektor qt opredelqetsq πlementamy matryc At y Bt . Cel\ dannoj
rabot¥ — ustanovyt\ ocenku skorosty sxodymosty reßenyq (4) pry p ≥ 2 k re-
ßenyg (5) v metryke ρ( , ) sup ( ) ( )( ) /x y X t Y t
t T
= −
≤ ≤0
2 1 2M . Sleduet otmetyt\, çto
pry p = 2 analohyçn¥j sluçaj [4] (koπffycyent A tε( ) ne zavysyt ot vremeny,
ot fazovoj peremennoj y proyzvodn¥x po vremeny ot nee) rassmotren v [6], a
ymenno poluçena ocenka skorosty sxodymosty reßenyq (4) dlq p = 2 k reße-
nyg (5) v metrykax ρi
t T
i iX Y X t Y t( , ) sup ( ) ( )( ) /= −
≤ ≤0
1M , i = 1, 2, pry A aε ε= ,
hde aε — nekotoraq postoqnnaq. V πtom sluçae dokazatel\stvo znaçytel\no
prowe, y v rqde sluçaev poluçagtsq bolee syl\n¥e utverΩdenyq, çem dlq ob-
weho vyda koπffycyenta A t X t
dX t
dt
dX t
dt
p
pε ε
ε ε, ( ),
( )
, ,
( )…
−
−
2
2 , kohda v snose pre-
del\noho uravnenyq poqvlqetsq dopolnytel\n¥j koπffycyent qt . Krome to-
ho, v sluçae, kohda rassmatryvaetsq uravnenye (4), a koπffycyent ne zavysyt ot
proyzvodnoj
dX t
dt
p
p
ε
−
−
2
2
( )
, v snose predel\noho uravnenyq πtot dopolnytel\n¥j
koπffycyent ne poqvlqetsq.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
PONYÛENYE PORQDKA SYSTEMÁ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1589
Osnovn¥e rezul\tat¥. V dal\nejßem budem sçytat\ 0 1< <ε y rassmat-
ryvat\ skalqrnoe uravnenye porqdka p ≥ 2 vyda
ε εdX tp−1( ) = – A t X t X t X X t dtp p
ε ε ε( ), ( ), ( ), , ( )0
1 2 1… − − +
+ U t X t X t X dt B t X t X t X dW tp p
ε ε ε ε( ) ( ), ( ), ( ), , , ( ), ( ), , ( )0
1 2
0
1 2… + …− −
(6)
pry naçal\n¥x uslovyqx
M Yε( )0 12 < K
ε6 < + ∞ , M Xε( )0 12 < K < + ∞ ,
(6 ′ )
M Xl
ε( )0
12
< K < + ∞ , l = 1, … , p – 2
(naçal\n¥e uslovyq Y X X Xp
ε ε ε ε( ), ( ), , ( ), ( )0 0 0 02 1− … mohut b¥t\ y sluçajn¥my,
tohda budem predpolahat\ yx nezavysymost\ ot vynerovskoho processa W t( ), t ≥
≥ 0 ). Zapyßem (6) v vyde sledugwej system¥:
X t Y t
dX t Y t dt
dX t X t dt
dX t A t X t X t X t dY t
A t X
p
p
p p
ε ε
ε ε
ε ε
ε ε ε ε ε
ε ε
ε
−
−
− − −
−
=
=
=
= − … +
+
1
2
1
2 1
0
1 2
1
( ) ( ),
( ) ( ) ,
. . . . . . . . . . . . . . . . .
( ) ( ) ,
( ) , ( ), ( ), , ( ) ( )
,
( )
( (( ), ( ), , ( ) , ( ), ( ), , ( )
, ( ), ( ), , ( ) , ( ), ( ), , ( ) ( )
) ( )
( ) ( )
t X t X t U t X t X t X t dt
A t X t X t X t B t X t X t X t dW t
p p
p p
0
1 2
0
1 2
1
0
1 2
0
1 2
… … +
+ … …
− −
− − −
ε ε ε ε
ε ε ε ε ε ε
(6 ″ )
pry naçal\n¥x uslovyqx (6 ′ ).
Teorema. Pust\ v¥polnen¥ uslovyq:
1) 0 1 2< ≤ … ≤ < +∞−c A t x x x Cp( , , , , ) , B t z U t z Cε ε( , ) ( , )+ ≤ < +∞,
B t z B t z U t z U t zε ε ε ε( , ) ( , ) ( , ) ( , )1 2 1 2− + − ≤ C z z1 2− ,
∂ + ∂ + ∂ + ∂ + ∂ + ∂A
t
A
x
A
x
A
t x
A
x x
A
x xi i i j i
ε ε ε ε ε ε
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
2 2 2
≤ C < + ∞ ,
i, j = 1, … , p – 2,
vo vsej oblasty opredelenyq;
2) esly z x x xp= … −{ , , , }1 2 , to
A t z A t z A t z A t zε ε( , ) ( , ) ( , ) ( , )− + ∇ − ∇0
2
0
2 +
+ B t z B t z U t z U t zε ε( , ) ( , ) ( , ) ( , )− + −0
2
0
2 ≤ δ ε2( ) R,
z ≤ R < + ∞ , δ ( ε ) → 0, ε → 0;
3) funkcyy U t zε( , ) , B t zε( , ) neprer¥vn¥ po t,
M Yε( )0 12 < K
ε6 , M Xε( )0 12 < K < + ∞ ,
M Xi
ε( )0
12
< K < + ∞ , i = 1, … , p – 2;
4)
i
p i iX X=
−∑ −
0
2
0
2
0 0M ε ( ) ( ) ≤ r0( )ε → 0, ε → 0 ( zdes\ y dalee sçytaem,
çto X t X tε ε
0( ) ( )= , X t X t0
0
0( ) ( )= ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1590 B. V. BONDAREV, E. E. KOVTUN
Tohda spravedlyva ocenka
sup ( ) ( )
0 0
2
0
2
≤ ≤ =
−
∑ −
t T i
p
i iX t X tM ε ≤ χ ε( ) D → 0, ε → 0 (7)
( postoqnnaq D pryvedena v pryloΩenyy ) . Zdes\
χ ε ε ε δ ε( ) max{ ( ), , ( )}= r0 ,
dX t A t X t X t X t U t X t X t X t dtp p p
0
2
0
1
0 0
1
0
2
0 0 0
1
0
2− − − −= … …( ) , ( ), ( ), , ( ) , ( ), ( ), , ( )( ) ( ) +
+ A t X t X t X t
A
x
t X t X t X tp
p
p
0
3
0 0
1
0
2 0
2
0 0
1
0
2− −
−
−… …( ) ( ), ( ), ( ), , ( ) , ( ), ( ), , ( )
∂
∂
×
× B t X t X t X t dtp
0
2
0 0
1
0
2( ), ( ), ( ), , ( )… − +
+ A t X t X t X t B t X t X t X t dW tp p
0
1
0 0
1
0
2
0 0 0
1
0
2− − −… …( ) ( ), ( ), ( ), , ( ) , ( ), ( ), , ( ) ( ),
dX t X t dt0 0
1( ) ( )= , … , dX t X t dtp p
0
3
0
2− −=( ) ( ) ,
s naçal\n¥my uslovyqmy X X X p
0 0
1
0
20 0 0( ), ( ), , ( )… − .
Pry dokazatel\stve teorem¥ budem pryderΩyvat\sq sledugwej sxem¥:
1)NNzapys¥vaq poslednee uravnenye system¥ (6 ″ ) v yntehral\nom vyde y yspol\-
zuq formulu Yto, yntehryruem po çastqm pervoe slahaemoe pravoj çasty;
2)NNzapys¥vaem slahaem¥e, ne stremqwyesq k nulg y stremqwyesq k nulg pry
ε → 0; 3) ustanavlyvaem ocenku velyçyn¥ ε ε
6 12M Y t( ) ; 4) poluçaem for-
mal\no predel\noe uravnenye — v predele poluçytsq uravnenye (5); 5) stroym
sootvetstvugwee neravenstvo Hronuolla dlq uklonenyq Z tε( ) ot Z t0( ) v met-
ryke ρ ( X, Y ) = ( )sup ( ) ( ) /
0
2 1 2
≤ ≤
−
t T
X t Y tM ; 6) zapys¥vaem ocenku dlq uklonenyq
vektorov Z tε( ), Z t0( ) v metryke ρ( , ) sup ( ) ( )( ) /X Y X t Y t
t T
= −
≤ ≤0
2 1 2M .
Dokazatel\stvu teorem¥ predpoßlem neskol\ko lemm.
Lemma/1. V uslovyqx teorem¥ spravedlyvo predstavlenye
X tp
ε
−2( ) = X
U s
A s
dsp
t
ε
ε
ε
− + ∫2
0
0( )
( )
( )
+
+
0
3
2
2
0
1
2
t
p
t
A s
A
x
s B s ds
B s
A s
dW s∫ ∫
−
+ +
ε
ε
ε
ε
ε
∂
∂
ψ ε
( )
( ) ( )
( )
( )
( ) ( ) , (8)
hde
ψ ε( ) = ε ε ∂
∂
∂
∂ε ε ε ε
ε
ε
ε
ε
ε
εA Y A t Y t
A t
A
x
t Y t
A
A
x
Y
p p
− −
− −
−[ ] + −
1 1 2
3
2
2
3
2
20 0 1 1
0
0 0( ) ( ) ( ) ( )
( )
( ) ( )
( )
( ) ( ) –
– ε ∂
∂
ε ∂
∂
∂
∂ε
ε
ε
ε
ε
ε ε
ε
ε
ε ε
0
2
0
2
1
2
1
3
11 1 1
t t
i
p
i
i
A
A
s
s Y s ds
A
A
x
s X s Y s
A
A
x
s X s Y s∫ ∫ ∑− +
=
−
+( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ds –
– ε ∂
∂
ε ∂
∂ε
ε
ε ε
ε
ε ε ε
ε0
3
2 0
2
2
1 1
t
p
t
pA s
A
x
s Y s U s ds
A
A
x
s
B s Y s
A s
dW s∫ ∫
− −
−
( )
( ) ( ) ( ) ( )
( ) ( )
( )
( ) –
– ε ∂
∂ε
ε
ε2
0
2
3
2
1
t
p
Y s
d
ds A s
A
x
s ds∫
−
( )
( )
( ) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
PONYÛENYE PORQDKA SYSTEMÁ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1591
∂
∂
εA
s
s( ) ,
∂
∂
εA
x
s( ) ,
∂
∂
εA
x
s
i
( ) , i = 1, … , p – 2 , — çastn¥e proyzvodn¥e ot funkcyy
A t x x xp( , , , , )1 2… − po sootvetstvugwej peremennoj, vzqt¥e v toçke ( , ( ),s X sε
X s X sp
ε ε
1 2( ), , ( ))… −
.
Dokazatel\stvo. Uslovyj, pryvedenn¥x v teoreme, dostatoçno dlq suwe-
stvovanyq y edynstvennosty syl\noho reßenyq uravnenyq (6), zapysannoho v vy-
de system¥ (6 ″ ):
X t X A s X s X s X s dY sp p
t
p
ε ε ε ε ε ε εε− − − −= − …∫2 2
0
1 1 20( ) ( ) , ( ), ( ), , ( ) ( )( ) +
+
0
1 1 2 1 2
t
p pA X s X s X s U X s X s X s dss s∫ − − −… …ε ε ε ε ε ε ε ε( ) ( ), ( ), ( ), , ( ) , ( ), ( ), , ( ) +
+
0
1 1 2 1 2
t
p pA X s X s X s B X s X s X s dW ss s∫ − − −… …ε ε ε ε ε ε ε ε( ) ( ), ( ), ( ), , ( ) , ( ), ( ), , ( ) ( ). (9)
Yntehryruq po çastqm vtoroe slahaemoe v (9), poluçaem
–
0
1 1 2
t
pA s X s X s X s dY s∫ − −…ε ε ε ε ε ε( ), ( ), ( ), , ( ) ( ) =
= – ε εε ε ε ε ε ε ε ε ε εA t X t X t X t Y t A X X X Yp p− − − −… − …1 1 2 1 1 20 0 0 0 0( ) ( ), ( ), ( ), , ( ) ( ) , ( ), ( ), , ( ) ( ) –
– ε ∂
∂
ε ∂
∂
∂
∂ε
ε
ε
ε
ε
ε ε
ε
ε
ε ε
0
2
0
2
1
2
1
3
11 1 1
t t
i
p
i
i
A
A
s
s Y s ds
A
A
x
s X s Y s
A
A
x
s X s Y s ds∫ ∫ ∑− +
=
−
+( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) –
– ε ∂
∂ε
ε
ε
0
2
2
21
t
pA
A
x
s Y s ds∫
−
( ) ( ) . (10)
Zapyßem uravnenye (6) v vyde
dY t A t Y t dt U t dt B t dW tε ε ε ε εε ε ε
( ) ( ) ( ) ( ) ( ) ( )= − + +1 1 1 . (11)
Yspol\zuq formulu Yto, ymeem
dY t Y t A t dt Y t U t dt B t dt B t Y t dW tε ε ε ε ε ε ε εε ε ε ε
2 2
2
22 2 1 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )= − + + + ,
sledovatel\no,
ε ε ε εε
ε
ε
ε ε
ε
ε
ε
ε ε
ε
Y t dt
A t
dY t
Y t U t
A t
dt
B t
A t
dt
B t Y t
A t
dW t2
2
2
2
2 2
( )
( )
( )
( ) ( )
( )
( )
( )
( ) ( )
( )
( )= − + + + .
Tohda, podstavlqq poslednee ravenstvo v (10), poluçaem
–
0
1 1 2
t
pA s X s X s X s dY s∫ − −…ε ε ε ε ε ε( ), ( ), ( ), , ( ) ( ) =
= – ε εε ε ε ε ε ε ε ε ε εA t X t X t X t Y t A X X X Yp p− − − −… − …1 1 2 1 1 20 0 0 0 0( ) ( ), ( ), ( ), , ( ) ( ) , ( ), ( ), , ( ) ( ) –
– ε ∂
∂
ε ∂
∂
∂
∂ε
ε
ε
ε
ε
ε ε
ε
ε
ε ε
0
2
0
2
1
2
1
3
11 1 1
t t
i
p
i
i
A
A
s
s Y s ds
A
A
x
s X s Y s
A
A
x
s X s Y s ds∫ ∫ ∑− +
=
−
+( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) –
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1592 B. V. BONDAREV, E. E. KOVTUN
–
0
2
2
21
2
t
pA
A
x
s
Y s U s
A s
ds
B s
A s
ds
B s Y s
A s
dW s∫
−
+ +
ε
ε ε ε
ε
ε
ε
ε ε
ε
∂
∂
ε ε( )
( ) ( )
( )
( )
( )
( ) ( )
( )
( ) +
+
0
2
2
2
21
2
t
pA
A
x
s
A s
dY s∫
−ε
ε
ε
ε
∂
∂
ε( )
( )
( ) . (12)
Yntehryruq po çastqm poslednyj yntehral v (12), naxodym
ε ∂
∂
ε ∂
∂
∂
∂ε
ε
ε
ε
ε
ε
ε
ε
ε
2
0
3
2
2 2
3
2
2
3
2
21 1 1
0
0 0
t
p p pA s
A
x
s dY s
A t
A
x
t Y t
A
A
x
Y∫
− − −
= −
( )
( ) ( )
( )
( ) ( )
( )
( ) ( ) –
–
0
2
3
2
1
t
p
Y s d
ds A s
A
x
s ds∫
−
ε
ε
ε∂
∂
( )
( )
( ) . (13)
Podstavlqq (13) v (12), a zatem poluçenn¥j rezul\tat v (9), poluçaem (8).
LemmaN1 dokazana.
Zameçanye. Vezde v dal\nejßem ocenky sverxu dlq fyhuryrugwyx v for-
mulax konstant moΩno najty v pryloΩenyy.
DokaΩem sledugwee utverΩdenye.
Lemma/2. V uslovyqx teorem¥ spravedlyv¥ ocenky
M Y t
Q
ε ε
( ) 4 12
3
2≤ , M Y t
Q
ε ε
( ) 8 12
23
4≤ , M Y t
Q
ε ε
( ) 12 12
6≤ , (14)
sup ( )
,0 0 2
4
≤ ≤ ≤ ≤ −
≤
t T i p
iX t QM ε , (15)
hde Q12 , Q — koneçn¥e postoqnn¥e, zavysqwye ot K, C, c, T.
Dokazatel\stvo. Yspol\zuq formulu Yto, yz (11) ymeem
dY t Y t A t dt Y t U t dtε ε ε ε εε ε
12 12 1112 12( ) ( ) ( ) ( ) ( )= − + +
+ 66 12
2
10 2 11
ε εε ε ε εY t B t dt B t Y t dW t( ) ( ) ( ) ( ) ( )+ .
Zapys¥vaq poslednee uravnenye v yntehral\nom vyde y berq matematyçeskoe
oΩydanye, poluçaem
M MY t Yε ε
12 12 0( ) ( )= +
+
0
12 11
2
10 212 12 66
t
Y s A s ds Y s U s ds Y s B s ds∫ − + +
ε ε εε ε ε ε ε εM M M( ) ( ) ( ) ( ) ( ) ( ) . (16)
Uslovyq teorem¥ pozvolqgt utverΩdat\, çto funkcyq
( )( ) ( ) ( )M MY s Y s A sε ε ε
12 1 12− = γ ( )s ≥ c > 0
opredelena, tohda yz (16) ymeem
M MY t Y s ds
t
ε ε ε
γ12 12
0
0 12( ) ( ) exp ( )= −
∫ +
+
0
11
2
10 212 12 66
t
s
t
d Y s U s ds Y s B s ds∫ ∫−
+
exp ( ) ( ) ( ) ( ) ( )
ε
γ τ τ
ε εε ε ε εM M . (17)
Pust\ 0 < < +∞λ , 0 < < +∞α . Yspol\zuq neravenstvo vyda
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PONYÛENYE PORQDKA SYSTEMÁ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1593
ξη λ ξ η
λ
≤ +p
p q
qp q
, 1 1 1
p q
+ = ,
y ocenky yz uslovyq teorem¥, yz (17) poluçaem
MY t K C
c
C
cε ε λ εα
12
6
12
12
2
6
11
2
( ) ≤ + + +
+
0
12 11
2 6 5
1212 12 66
t
c t s c
Y s ds∫ − −{ } +
exp
( )
( )/
/
ε ε
λ α
ε εM .
Polahaq v poslednem neravenstve λ = 1, α ε6 5/ = , naxodym
MY t K C T
cε ε λ
12
6
12
12( ) ≤ + +
11
2
2
6
C
cε
+
+
0
2 1212 12 1 66
t
c t s
C Y s ds∫ − −{ } +[ ]exp ( ) ( )
ε ε εM . (18)
Yz (18) v sylu neravenstva Hronuolla ymeem
MY t
K C T
c
C
c
C
c
c t s
ds
t
ε ε ε ε ε
12
6
12 2
6
2
0
11
2
1 66 12 12
( ) exp exp
( )≤ + +
+ − −{ }
∫ ≤
≤
K C T
c
C
c
C
cε ε6
12 2
6
211
2
1 66+ +
+
exp =
Q12
6ε
.
Takym obrazom, (14) ymeet mesto. Dalee, v sylu toho çto
X t X Y s dsp p
t
ε ε ε
− −= + ∫2 2
0
0( ) ( ) ( ) ,
ymeet mesto ocenka
M sup ( )
0
2 12
≤ ≤
−
t T
pX tε ≤ 211
12
12
6
K T Q+
ε
= ˆ ( )Q ε . (19)
Yspol\zuq promeΩutoçnug ocenku (19), poluçaem
M X tp
ε
−2 12
( ) ≤ ˆ ( )Q ε ,
M X tp
ε
−3 12
( ) ≤ 2 211 11 12K T Q+ ˆ ( )ε ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)
M X ti
ε( )
12
≤ 2 1 2 2 211 11 12 11 12 3 11 12 2K T T T Qp i p i( ( ) ) ( ) ˆ( )+ + … + +− − − − ε ,
i = 0, … , p – 3 .
Dalee na osnove (20) ustanovym unyversal\nug ocenku.
Dejstvytel\no, pust\ snaçala 2 111 12T ≤ , tohda
sup ( )
,0 0 2
12
≤ ≤ ≤ ≤ −t T i p
iX tM ε ≤ 211Kp Q+ ˆ ( )ε . (21)
V sluçae 2 111 12T > yz (20) ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1594 B. V. BONDAREV, E. E. KOVTUN
M X ti
ε( )
12
≤ 2 2 1
2 1
211
11 12 1
11 12
11 12 2K
T
T
T Q
p
p( ) ( ) ˆ( )
−
−−
−
+ ε , i = 0, … , p – 2 . (22)
Yz (21) y (22) sleduet ocenka
M X ti
ε( )
12
≤
Q̂12
6ε
, i = 0, … , p – 2 . (23)
Pry v¥polnenyy (14) y (23) spravedlyva ocenka
M ε ∂
∂ε
ε
ε2
0
2
3
2
4
1
t
p
Y s d
ds A s
A
x
s ds∫
−
( )
( )
( ) ≤ ε2 2Q̃ . (24)
Dejstvytel\no, v sylu toho çto
d
ds A s
A
x
s A
A
x s
A
A
x
A
sp p p
1 33
2
4
2
2 2ε
ε
ε
ε
ε
ε ε∂
∂
∂
∂ ∂
∂
∂
∂
∂( )
( )
−
−
− −
=
−
+
+
∂
∂ ∂
∂
∂
∂
∂
∂
∂ ∂
∂
∂
∂
∂
ε
ε
ε ε
ε
ε
ε
ε ε
ε
2
2 2
1
1
3 2
2 2
13 3
A
x x
A
A
x
A
x
X s
A
x x
A
A
x
A
x
X s
p p j
p
p j p j
j
− − =
−
− −
+−
+ −
∑( ) ( ) +
+
∂
∂ ∂
∂
∂
∂
∂
ε ε ε
ε
2
2 2 2 2
3
A
x x
A
x
A
x
Y s
p p p p− − − −
−
( ) ,
ymeem
M ε ∂
∂ε
ε
ε2
0
2
3
2
4
1
t
p
Y s d
ds A s
A
x
s ds∫
−
( )
( )
( ) ≤ ε ε
8 3
8
16
4 8
0
123T C
c
C Y s ds
t
+
∫ M ( ) +
+ ε ε
8
8
16
4 8
0
12 2 33C
c
C Y s ds
t
+
∫ (M ( )) /
+
+ ε ε ε
8 3
8
16
4 8 3
1
2
0
12 2 3 12 1 32 3( ) ( ) [ ( )]) )/ /p C
c
C T Y s X s ds
j
p t
j− +
=
−
∑ ∫ ( (M M .
Yz posledneho neravenstva s uçetom (14), (23) sleduet ocenka (24). Tohda
M ψ ε4( ) ≤ 93 12
1 3
4
4
12
2 3
12
4 4
12
1 3
4
+ + + +K Q
c
C K Q
c
T C Q
c
/ / /( )
+
+ T p C
c
Q Q T
C Q
c
T C
c
Q T Q
4 4 4
4 12 12
1 3 4
8
12
1 3
12
5
2
2
8
12 12
1 3 44
3
( ˆ ) ˜/
/
/+ + + +
=
�
Q. (25)
S uçetom (25) yz (8) sleduet ocenka
M X tp
ε
−2 4
( ) ≤
5
16
363 4
4
4
12
12
2
4
4K T C
c
C
c
T
C
c
Q+ +
+ +
�
= Q . (26)
V sylu toho çto
M X tp
ε
−2 4
( ) ≤ Q ,
M X tp
ε
−3 4
( ) ≤ 8 8 4K T Q+ ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (27)
M X ti
ε( )
4
≤ 8 1 8 8 84 4 3 4 2K T T T Qp i p i( ( ) ( ) ) ( )+ + … + +− − − − ,
i = 0, … , p – 3,
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PONYÛENYE PORQDKA SYSTEMÁ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1595
na osnove (27) poluçym unyversal\nug ocenku.
Dejstvytel\no, pust\ snaçala 8 14T ≤ , tohda
sup ( )
,0 0 2
4
≤ ≤ ≤ ≤ −t T i p
iX tM ε ≤ 8Kp Q+ . (28)
V sluçae 8 14T > yz (27) ymeem
M X ti
ε( )
4
≤ 8 8 1
8 1
8
4
4
4K
T
T
T Q
p i
p i( ) ( )
−
−−
−
+ ≤
≤ 8 8 1
8 1
8
4 2
4
4 2K
T
T
T Q
p
p( ) ( )
−
−−
−
+ , i = 0, … , p – 2 . (29)
Yz (28) y (29) sleduet ocenka (15).
LemmaN2Ndokazana.
Opyraqs\ na (14), (15), netrudno ubedyt\sq v tom, çto spravedlyva ocenka
M − + + −
− −
− −
ε ε ε ∂
∂
∂
∂ε ε ε ε
ε
ε
ε
ε
ε
εA t Y t A Y
A t
A
x
t Y t
A
A
x
Y
p p
1 1 2
3
2
2
3
2
20 0 1 1
0
0 0( ) ( ) ( ) ( )
( )
( ) ( )
( )
( ) ( ) –
– ε ∂
∂
∂
∂
∂
∂ε
ε
ε ε
ε
ε
ε ε
ε
ε
ε
0
2
1
2
2
2
1
2
1 1 1
t
i
p
i
i
A
A
x
X s Y s
A
A
x
X s Y s
A
A
s
Y s ds∫ ∑+ +
=
−
−
( ) ( ) ( ) ( ) ( ) –
– ε ∂
∂
ε ∂
∂ε
ε
ε ε
ε
ε
ε ε
0
3
2 0
3
2
2
1 1
t
p
t
pA s
A
x
Y s U s ds
A
A
x
B s Y s dW s∫ ∫
− −
−
( )
( ) ( ) ( ) ( ) ( ) ≤ εQ̂ . (30)
Yz (24) sleduet, çto ymeet mesto ocenka
M ε ∂
∂ε
ε
ε2
0
2
3
2
2
1
t
p
Y s d
ds A s
A
x
s ds∫
−
( )
( )
( ) ≤ εQ̃ . (31)
Otmetym, çto yz (9) s uçetom ocenok (30), (31) sleduet predstavlenye
X t X A s U s dsp p
t
ε ε ε ε
− − −= + ∫2 2
0
10( ) ( ) ( ) ( ) +
+
0
3
2
2
0
11
t
p
t
A s
A
x
s B s ds A s B s dW s∫ ∫
−
−+ +
ε
ε
ε ε ε
∂
∂
ψ ε
( )
( ) ( ) ( ) ( ) ( ) ( ) , (32)
pryçem
M Q Qψ ε ε2 0( ) ˆ ˜( )≤ + → , ε → 0. (33)
„Formal\no predel\noe” pry ε → 0 uravnenye dlq (32) ymeet vyd
X t X A s U s dsp p
t
0
2
0
2
0
0
1
00− − −= + ∫( ) ( ) ( ) ( ) +
+
0 0
3
0
2
0
2
0
0
1
0
1
t
p
t
A s
A
x
s B s ds A s B s dW s∫ ∫
−
−+
( )
( ) ( ) ( ) ( ) ( )
∂
∂
. (34)
V uslovyqx teorem¥ uravnenye (34) ymeet syl\noe reßenye.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1596 B. V. BONDAREV, E. E. KOVTUN
Ocenym „blyzost\” reßenyj uravnenyj (24) y (26) v metryke ρ( , )X Y =
= ( )sup ( ) ( ) /
0
2 1 2
≤ ≤
−
t T
X t Y tM , podhonqq raznost\ reßenyj v πtoj metryke pod so-
otvetstvugwee neravenstvo Hronuolla. Poskol\ku sxodymost\ koπffycyen-
tov uravnenyq (32) ymeetsq lyß\ na ohranyçenn¥x mnoΩestvax, neobxodymo
ymet\ ocenky veroqtnostej v¥xoda za rastuwyj uroven\ velyçyn
sup ( )
0
0
≤ ≤t T
iX t , i = 0, … , p – 2 .
DokaΩem sledugwee vspomohatel\noe utverΩdenye.
Lemma/3. Pust\ vektor Z t X t X t X tp
0 0 0
1
0
2( ) ( ), ( ), , ( ){ }= … − . V uslovyqx teo-
rem¥ spravedlyva ocenka
M sup ( )
0
0
≤ ≤t T
Z t ≤ pQ
�
. (35)
Dokazatel\stvo. Rassmotrym systemu
X t X A s U s dsp p
t
0
2
0
2
0
0
1
00− − −= + ∫( ) ( ) ( ) ( ) +
+
0
0
3 0
2
0
2
0
0
1
0
t
p
t
A s
A
x
s B s ds A s B s dW s∫ ∫−
−
−+( ) ( ) ( ) ( ) ( ) ( )
∂
∂
,
(36)
X t X X s dsi i
t
i
0 0
0
0
10( ) ( ) ( )= + ∫ + , i = 0, … , p – 3 .
Yz pervoho uravnenyq (36) v sylu toho, çto
M sup ( ) ( ) ( )
0 0
0
1
0
2
≤ ≤
−∫
t T
t
A s B s dW s ≤ 4
2
2
C T
c
,
ymeem ocenku
sup ( )
0
0
2 2
≤ ≤
−
t T
pX t ≤ 3 6( )K L+ ,
hde
L = T C
c
T C
c
C T
c
2 2
2
2 6
6
2
24+ + .
S uçetom poslednej ocenky netrudno ubedyt\sq v v¥polnenyy neravenstva
M sup ( )
0
0
2
≤ ≤t T
iX t ≤ 3 1 3 3 3 36 2 2 2 2 2K T T T Lp i p i( )( ) ( )+ + … + +− − − − ,
i = 0, … , p – 2 .
Opqt\ rassmotrym dva sluçaq. Pust\ snaçala 0 3 12< ≤T , tohda
sup sup ( )
0 2 0
0
2
≤ ≤ − ≤ ≤i p t T
iX tM ≤ 3 36 K p L+ . (37)
V sluçae 1 3 2< T ymeet mesto ocenka
sup sup ( )
0 2 0
0
2
≤ ≤ − ≤ ≤i p t T
iX tM ≤ 3 3 1
3 1
3 36
2 1
2
2 2K
T
T
T L
p
p( ) ( )
−
−−
−
+ . (38)
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PONYÛENYE PORQDKA SYSTEMÁ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1597
Yz (37), (38) sleduet, çto unyversal\noj ocenkoj budet velyçyna
sup sup ( )
0 2 0
0
2
≤ ≤ − ≤ ≤i p t T
iX tM ≤ 3
3 1
3 1
3 3 16
2
2
2K
T
T
p T L
p
p( )
( )
−
−
+
+ +[ ] =
�
Q.
Yz posledneho sleduet, çto spravedlyva ocenka
M sup ( )
0
0
≤ ≤t T
Z t = M sup ( )
/
0 0
2
0
2
1 2
≤ ≤ =
−
∑
t T i
p
iX t ≤ M
i
p
t T
iX t
=
−
≤ ≤
∑
0
2
0
0
2
1 2
sup ( )
/
≤
≤
i
p
t T
iX t
=
−
≤ ≤
∑
0
2
0
0
2
1 2
M sup ( )
/
≤ p X t
i p t T
isup sup ( )
/
0 2 0
0
2
1 2
≤ ≤ − ≤ ≤
M ≤ pQ
�
,
t. e. ocenka (35) ymeet mesto.
LemmaN3Ndokazana.
Lemma/4. Esly v¥brat\ R( ) ( )ε δ ε= → +∞−1 , ε → 0, to v uslovyqx teo-
rem¥ na traektoryqx vektora Z t0( ) spravedlyv¥ ocenky
M
0
3
2
2
0 0
3
0
2
0
2
2
1 1
t
p
t
pA s
A
x
s B s ds
A s
A
x
s B s ds∫ ∫
− −
−
ε
ε
ε
∂
∂
∂
∂( )
( ) ( )
( )
( ) ( ) ≤
≤ C1 0δ ε( ) → , ε → 0,
(39)
M
−( )∫ − −
0
1
0 0 0
1
0 0 0
t
A s Z s U s Z s A s Z s U s Z s dsε ε( ) ( ) ( ) ( ), ( ) , ( ) , ( ) , ( ) +
+
0
1
0 0 0
1
0 0 0
2t
A s Z s B s Z s A s Z s B s Z s dW s∫ − −−( )
ε ε( ) ( ) ( ) ( ), ( ) , ( ) , ( ) , ( ) ( ) ≤
≤ C2 0δ ε( ) → , ε → 0.
Dokazatel\stvo. Netrudno ubedyt\sq v tom, çto na traektoryqx vektora
Z t0( ) v¥polnqetsq neravenstvo
M
0
3
2
2
0 0
3
0
2
0
2
2
1 1
t
p
t
pA s
A
x
s B s ds
A s
A
x
s B s ds∫ ∫
− −
−
ε
ε
ε
∂
∂
∂
∂( )
( ) ( )
( )
( ) ( ) ≤
≤ 6 4 2
6
0
0
2C T
c
B s B s
t T
M sup ( ) ( )
≤ ≤
−ε +
+ 3 274 2
6
0 2
0
2
2 10 2
12
0
0
2C T
c
A
x
t
A
x
t
C T
c
A s A s
t T p p t T
M Msup ( ) ( ) sup ( ) ( )
≤ ≤ − − ≤ ≤
− + −∂
∂
∂
∂
ε
ε . (40)
Pust\ χ δ εsup ( ) ( )
0
0
1
≤ ≤
−≤
t T
Z t — yndykator sob¥tyq sup ( ) ( )
0
0
1
≤ ≤
−≤
t T
Z t δ ε , toh-
da yz (40) s uçetom (35) ymeem ocenku
M
0
3
2
2
0 0
3
0
2
0
2
2
1 1
t
p
t
pA s
A
x
s B s ds
A s
A
x
s B s ds∫ ∫
− −
−
ε
ε
ε
∂
∂
∂
∂( )
( ) ( )
( )
( ) ( ) ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1598 B. V. BONDAREV, E. E. KOVTUN
≤ 9 27 2
4 2
6
10 2
12
2
0
0
1C T
c
C T
c
C Z t
t T
+
>
≤ ≤
−M χ δ εsup ( ) ( ) +
+
9 274 2
6
10 2
12
C T
c
C T
c
+
δ ε( ) ≤ C1 0δ ε( ) → , ε → 0.
Analohyçno
M
−( )∫ − −
0
1
0 0 0
1
0 0 0
t
A s Z s U s Z s A s Z s U s Z s dsε ε( ) ( ) ( ) ( ), ( ) , ( ) , ( ) , ( ) +
+
0
1
0 0 0
1
0 0 0
2t
A s Z s B s Z s A s Z s B s Z s dW s∫ − −−( )
ε ε( ) ( ) ( ) ( ), ( ) , ( ) , ( ) , ( ) ( ) ≤
≤ 2
0
1
0 0 0
1
0 0 0
2
T A s Z s U s Z s A s Z s U s Z s ds
T
−( )∫ − −
ε ε( ) ( ) ( ) ( ), ( ) , ( ) , ( ) , ( ) +
+
0
1
0 0 0
1
0 0 0
2
T
A s Z s B s Z s A s Z s B s Z s ds∫ − −−( )
ε ε( ) ( ) ( ) ( ), ( ) , ( ) , ( ) , ( ) ≤
≤ 4 2
2
0
0 0 0
2T
c
U s Z s U s Z s
t T
sup , ( ) , ( )( ) ( )
≤ ≤
−ε +
+ 4 12
4
0
0 0 0
2C T T
c
A s Z s A s Z s
t T
( ) sup , ( ) , ( )( ) ( )+ −
≤ ≤
ε +
+ 2
2
0
0 0 0
2T
c
B s Z s B s Z s
t T
sup , ( ) , ( )( ) ( )
≤ ≤
−ε ≤ C2 0δ ε( ) → , ε → 0.
LemmaN4 dokazana.
Perejdem k dokazatel\stvu teorem¥. V¥çytaq yz (32) ravenstvo (34), s uçe-
tom (39) y (40) poluçaem ocenku
M X t X tp p
ε
− −−2
0
2 2
( ) ( ) ≤ 5 0 02
0
2 2
M X Xp p
ε
− −−( ) ( ) +
+ 5
0
1
0
1
0 0 0
2
T A s Z s U s Z s A s Z s U s Z s ds
t
∫ − −−M ε ε ε ε( ) ( ) ( ) ( ), ( ) , ( ) , ( ) , ( ) +
+ 5 1
0
3
2
2T
A s Z s
A
x
s Z s B s Z s
t
p
∫
−
M
ε ε
ε
ε ε ε
∂
∂( )
( ) ( )
, ( )
, ( ) , ( ) –
– 1
0
3
0
0
2
0 0
2
0
2
A s Z s
A
x
s Z s B s Z s ds
p( )
( ) ( )
, ( )
, ( ) , ( )
∂
∂ −
+
+ 5
0
1
0
1
0 0 0
2
M
t
A s Z s B s Z s A s Z s B s Z s dW s ds∫ − −−[ ]ε ε ε ε( ) ( ) ( ) ( ), ( ) , ( ) , ( ) , ( ) ( ) + 5 2Mψ ε( ) ≤
≤ 10
0
1 1
0 0
2
T A s Z s U s Z s A s Z s U s Z s ds
t
∫ − −−M ε ε ε ε ε ε( ) ( ) ( ) ( ), ( ) , ( ) , ( ) , ( ) +
+ 10 1
0
3
2
2T
A s Z s
A
x
s Z s B s Z s
t
p
∫
−
M
ε ε
ε
ε ε ε
∂
∂( )
( ) ( )
, ( )
, ( ) , ( ) –
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
PONYÛENYE PORQDKA SYSTEMÁ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1599
– 1
3
0
0
2
0
2
0
2
A s Z s
A
x
s Z s B s Z s ds
pε
ε
∂
∂( )
( ) ( )
, ( )
, ( ) , ( )
−
+
+ 10
0
1 1
0 0
2
M
t
A s Z s B s Z s A s Z s B s Z s dW s∫ − −−[ ]ε ε ε ε ε ε( ) ( ) ( ) ( ), ( ) , ( ) , ( ) , ( ) ( ) +
+ 5 10 5 0 02
1 2
2
0
2 2
Mψ ε δ ε ε( ) ( ) ( ) ( )( )+ + + −− −T C C X Xp pM ≤
≤ 10 81 1 10 1 1
12
12
3
3
2 2
2
2
0
0
2
T
C
c
C
c
T C
c
C
c
Z s Z s ds
t
+
+ + +
−∫( ) ( ) ( )M ε +
+ 5 10 5 0 02
1 2
2
0
2 2
Mψ ε δ ε ε( ) ( ) ( ) ( )( )+ + + −− −T C C X Xp pM ,
t. e. dlq lgboho 0 < ≤t T
sup ( ) ( )
0
2
0
2 2
≤ ≤
− −−
τ
ε τ τ
t
p pX XM ≤
≤ 5 10 52
1 2 0
0
0
2
M Mψ ε δ ε ε ε( ) ( ) ( ) ( ) ˜ ( ) ( )+ + + + −∫T C C r D Z s Z s ds
t
. (41)
Dalee, pust\ 0 < ≤t T , tohda netrudno ubedyt\sq v tom, çto spravedlyv¥
ocenky
sup ( ) ( )[ ]
0
0
2
≤ ≤
−
τ
ε τ τ
t
i iX XM ≤
≤ 2 1 2 2 2 20
2 2 2 2 2 2 2
0
2
0
2 2r T T T T X Xp i p i
t
p p( ) sup ( ) ( )( ) ( ) ( ) [ ]ε τ τ
τ
ε+ + +{ } + −− − − −
≤ ≤
− −M ,
(42)
i = 0, … , p – 2 .
Ustanovym unyversal\nug ocenku. Pust\ snaçala 0 2 12< ≤T , tohda, ys-
pol\zuq (42), ymeem
sup ( ) ( ) ( ) sup ( ) ( )[ ] [ ]
0
0
2
0
0
2
0
2 22
≤ ≤ ≤ ≤
− −− ≤ + −
τ
ε
τ
ετ τ ε τ τ
t
i i
t
p pX X r p X XM M ,
(43)
i = 0, … , p – 2 .
Pry 1 2 2< T na osnovanyy (34), (42) ymeem
sup ( ) ( ) ( ) ( ) ( ) sup ( ) ( )[ ] [ ]
0
0
2
0
2
2
2
0
2
0
2 22 2 1
2 1
2
≤ ≤ ≤ ≤
− −− ≤ −
−
+ −
τ
ε
τ
ετ τ ε τ τ
t
i i
p
p
t
p pX X r
T
T
T X XM M ,
(44)
i = 0, … , p – 2 .
Yz (43), (44) sleduet unyversal\naq ocenka
sup ( ) ( ) ( ) ˆ sup ( ) ( )
,
[ ] [ ]
0 0 2
0
2
0
0
2
0
2 2
≤ ≤ ≤ ≤ − ≤ ≤
− −− ≤ + −
τ
ε
τ
ετ τ ε τ τ
t i p
i i
t
p pX X r D D X XM M .
(45)
Yz (45), v çastnosty, sleduet takΩe ocenka
M M[ ] [ ]( ) ( ) ( ) ˆ sup ( ) ( )Z t Z t r pD pD X X
t
p p
ε
τ
εε τ τ− ≤ + −
≤ ≤
− −
0
2
0
0
2
0
2 2 . (46)
Podstavlqq v (46) ocenku (41), poluçaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1600 B. V. BONDAREV, E. E. KOVTUN
M M[ ]( ) ( ) ˆ ˜ ( ) ( )Z t Z t pDD Z s Z s ds
t
ε ε− ≤ −∫0
2
0
0
2
+
+ r pD pD T C C r0
2
1 2 05 10 5( ) ˆ ( ) ( ) ( ) ( )ε ψ ε δ ε ε+ + + +{ }M .
Yspol\zuq lemmu Hronuolla, yz posledneho neravenstva ymeem
sup ( ) ( )[ ]
0
0
2
≤ ≤
−
t T
Z t Z tM ε ≤
≤ p r D D Q Q T C C pDDT0 1 25 5 10( )( ) ˆ ( ( ˆ ˜ ) ( ) ( )) exp ˆ ˜ε ε δ ε+ + + + +{ } { } .
TeoremaNdokazana.
PryloΩenye
Q K C T
c
C
c
C
c12
12 2 211
2
1 66= + +
+
exp [ ] ,
ˆ ( ) ( )[ ] [ ]Q K
T
T
Kp T K T Q
p
p
12
11
11 12 1
11 12
11 11 12 2 11 12
122 2 1
2 1
2 2 1 2= −
−
+ + + +
−
− ,
˜ ˆ( )/ / /Q p
C
c
C T Q Q Q Q2 4
8
16
4 8 4
12 12
2 3
12
2 3
12
1 33= +
+ + ,
�
Q K
T
T
p T
T C
c
T C
c
C T
c
p
p= −
−
+
+ +[ ] + +
3 3 1
3 1
3 3 1 46
2
2
2
2 2
2
2 6
6
2
2
( ) ( ) ,
�
Q
K Q
c
C K Q
c
T C Q
c
T p C
c
Q Q=
+ + + + +93 12
1 3
4
4
12
2 3
12
4 4
12
1 3
4
4 4 4
4 12 12
1 3
/ / /
/( ) ( )ˆ +
+ T
C Q
c
T C
c
Q T Q4
8
12
1 3
12
5
2
2
8
12 12
1 3 44
3
/
/ ˜+ + +
,
Q K T C
c
C
c
T
C
c
Q= + +
+ +
5
16
363 4
4
4
12
12
2
4
4
�
,
Q K
T
T
p T Q
p
p= −
−
+
+ +[ ]−8
8 1
8 1
8 14
4
4
4 2( )
( ) ,
ˆ ( ) ( ) [ ]Q
c
c Q K C Q K c p T C Q Q Q T C Q TC Q= + + + + + + +[ ]92
6
4 2 2 2 2 2 2 4 4 ,
C
C T
c
C T
c
pQ
C T
c
C T
c1
6 2
6
12 2
12
1 2
4 2
6
10 2
12
18 54 9 27= +
+ +[ ] /�
,
C
T T
c
C T T
c
pQ
C T T
c
C T T
c2 2
2
4
1 2
2
2
4
4
2 2 1 4 1 2 1 8 1= + + +
+ + + +( ) ( ) [ ˜ ] ( ) ( )/ ,
˜ ( )D T
C
c
C
c
T C
c
C
c
= +
+ + +
10 81 1 10 1 1
12
12
3
3
2 2
2
2
,
D
T
T
p
p
= −
−
+
2 2 1
2 1
2
2
( ) , ˆ ( )D T p= +[ ]2 12 ,
D p D D Q Q T C C pDDT= + + + + +{ } { }( ) ˆ ( ˆ ˜ ) ( ) exp ˆ ˜5 5 10 1 2 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
PONYÛENYE PORQDKA SYSTEMÁ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1601
V¥vod¥. V stat\e rassmatryvagtsq uravnenyq, opys¥vagwye dvyΩenye ças-
tyc v b¥stroperemenn¥x sluçajn¥x polqx. V rqde sluçaev neobxodymo znat\
skorost\ sblyΩenyq πtyx reßenyj (naprymer, dlq ocenky blyzosty srednyx
πnerhetyçeskyx xarakterystyk). V dannoj rabote pry uslovyqx, bolee Ωestkyx,
çem v [5], ustanovlena ocenka skorosty sblyΩenyq reßenyq uravnenyq, soder-
Ωaweho p-g proyzvodnug po vremeny, s reßenyem stoxastyçeskoho dyfferen-
cyal\noho uravnenyq, soderΩaweho ( )p − 1 -g proyzvodnug po vremeny v
metryke ρ( , ) sup ( ) ( )( ) /X Y X t Y t
t T
= −
≤ ≤0
2 1 2M . Esly poqvytsq neobxodymost\
pryvesty qvn¥j vyd konstant¥ D, to πto netrudno sdelat\ s pomow\g cepoçky
postoqnn¥x, zapysann¥x v pryloΩenyy.
1. Krylov M. M., Boholgbov M. M. Pro rivnqnnq Fokkera – Planka // Vçen. zap. AN USSR. –
1939. – # 4. – S. 5 – 158.
2. Yl\yn A. M., Xas\mynskyj R. Z. Ob uravnenyqx brounovskoho dvyΩenyq // Teoryq veroqtnos-
tej y ee prymenenyq. – 1964. – 9, v¥p. 4. – S. 466 – 491.
3. Panovskyj P., Fylyps M. Klassyçeskaq πlektrodynamyka. – M.: Fyzmathyz, 1963. – 432 s.
4. Dubko V. A. PonyΩenye porqdka system¥ stoxastyçeskyx dyfferencyal\n¥x uravnenyj s
mal¥m parametrom pry starßej proyzvodnoj // Teoryq sluçajn¥x processov. – 1980. –
V¥p. 8. – S. 35 – 41.
5. Skoroxod A. V. Ob usrednenyy stoxastyçeskyx uravnenyj matematyçeskoj fyzyky //
Problem¥ asymptotyçeskoj teoryy nelynejn¥x kolebanyj. – Kyev: Nauk. dumka, 1977. –
S.N196 – 208.
6. Kovtun E. E. Ocenka skorosty sxodymosty v stoxastyçeskyx systemax s mal¥m parametrom
pry starßej proyzvodnoj // Tr. Yn-ta prykl. matematyky y mexanyky NAN Ukrayn¥. – 2005.
– V¥p. 10. – S.N88 – 97.
7. Hyxman Y. Y., Skoroxod A. V. Stoxastyçeskye dyfferencyal\n¥e uravnenyq y yx prylo-
Ωenyq. – Kyev: Nauk. dumka, 1982. – 612 s.
Poluçeno 17.03.2005,
posle dorabotky — 23.08.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
|
| id | umjimathkievua-article-3557 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:44:45Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/80/42cdc5b7ee71f4d1ba0407b57adf5e80.pdf |
| spelling | umjimathkievua-article-35572020-03-18T19:57:46Z Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence Понижение порядка системы стохастических дифференциальных уравнений с малым параметром при старшей производной. Оценка скорости сходимости Bondarev, B. V. Kovtun, E. E. Бондарев, Б. В. Ковтун, Е. Е. Бондарев, Б. В. Ковтун, Е. Е. In the metric $\rho(X, Y) = (\sup\limits_{0 \leq t \leq T} M|X(t) - Y(t)|^2)^{1/2} $ for an ordinary stochastic differential equation of order $p \geq 2$ with small parameter of the higher derivative, we establish an estimate of the rate of convergence of its solution to a solution of stochastic equation of order $p - 1$. У метриці $\rho(X, Y) = (\sup\limits_{0 \leq t \leq T} M|X(t) - Y(t)|^2)^{1/2} $ встановлено оцінку швидкості збіжності розв'язку звичайного стохастичного диференціального рівняння порядку $p \geq 2$ з малим параметром при старшій похідній до розв'язку стохастичного рівняння порядку $p - 1$. Institute of Mathematics, NAS of Ukraine 2006-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3557 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 12 (2006); 1587–1601 Український математичний журнал; Том 58 № 12 (2006); 1587–1601 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3557/3848 https://umj.imath.kiev.ua/index.php/umj/article/view/3557/3849 Copyright (c) 2006 Bondarev B. V.; Kovtun E. E. |
| spellingShingle | Bondarev, B. V. Kovtun, E. E. Бондарев, Б. В. Ковтун, Е. Е. Бондарев, Б. В. Ковтун, Е. Е. Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence |
| title | Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence |
| title_alt | Понижение порядка системы стохастических дифференциальных уравнений с малым параметром при старшей производной. Оценка скорости сходимости |
| title_full | Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence |
| title_fullStr | Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence |
| title_full_unstemmed | Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence |
| title_short | Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence |
| title_sort | order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. estimate for the rate of convergence |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3557 |
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