Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations
We prove a general theorem on the convergence of solutions of stochastic differential equations. As a corollary, we obtain a result concerning the convergence of solutions of stochastic differential equations with absolutely continuous processes to a solution of an equation with Brownian motion.
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| Дата: | 2010 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2010
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2952 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508955172667392 |
|---|---|
| author | Ral’chenko, K. V. Shevchenko, H. M. Ральченко, К. В. Шевченко, Г. М. |
| author_facet | Ral’chenko, K. V. Shevchenko, H. M. Ральченко, К. В. Шевченко, Г. М. |
| author_sort | Ral’chenko, K. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:41:02Z |
| description | We prove a general theorem on the convergence of solutions of stochastic differential equations. As a corollary, we obtain a result concerning the convergence of solutions of stochastic differential equations with absolutely continuous processes to a solution of an equation with Brownian motion. |
| first_indexed | 2026-03-24T02:33:25Z |
| format | Article |
| fulltext |
UDK 519.21
K. V. Ral\çenko, H. M. Íevçenko (Ky]v. nac. un-t im. T. Íevçenka)
NABLYÛENNQ ROZV’QZKIV STOXASTYÇNYX
DYFERENCIAL|NYX RIVNQN|
IZ DROBOVYM BROUNIVS|KYM RUXOM
ROZV’QZKAMY VYPADKOVYX ZVYÇAJNYX
DYFERENCIAL|NYX RIVNQN|*
We prove the general theorem on the convergence of solutions of stochastic differential equations. As a
corollary, we obtain a result on the convergence of solutions of stochastic differential equations with
absolutely continuous processes to a solution of an equation with the fractional Brownian motion.
Dokazana obwaq teorema o sxodymosty reßenyj stoxastyçeskyx dyfferencyal\n¥x uravnenyj.
Kak sledstvye, poluçen rezul\tat o sxodymosty reßenyj stoxastyçeskyx dyfferencyal\n¥x
uravnenyj s absolgtno neprer¥vn¥my processamy k reßenyg uravnenyq brounovskym dvy-
Ωenyem.
Vstup. Brounivs\kyj rux uprodovΩ dovhoho çasu buv i zalyßa[t\sq populqr-
nog modellg vypadkovosti pry doslidΩenni procesiv u pryrodoznavstvi, na fi-
nansovyx rynkax towo. Sutt[vym obmeΩennqm u zastosuvanni brounivs\koho ru-
xu [ te, wo vin ma[ nezaleΩni pryrosty, i, takym çynom, vypadkovyj ßum, porod-
Ωuvanyj nym, [ „bilym”, tobto nekorel\ovanym. Prote bahato procesiv u pryro-
doznavstvi, komp’gternyx mereΩax, na finansovyx rynkax towo magt\ vlasty-
vist\ dovhostrokovo] zaleΩnosti, tobto korelqci] vypadkovoho ßumu u takyx
procesax spadagt\ u çasi povil\no. Dlq modelgvannq takyx procesiv vykorys-
tovu[t\sq drobovyj brounivs\kyj rux.
Stoxastyçnym dyferencial\nym rivnqnnqm iz drobovym brounivs\kym ruxom
prysvqçeno bahato statej, i odni[g z pryçyn c\oho [ te, wo intehral vidnosno
drobovoho brounivs\koho ruxu moΩna vyznaçaty riznymy sposobamy. Odyn zi
sposobiv — ce potra[ktorne vyznaçennq. Uperße joho bulo zaproponovano u
[1], de stoxastyçnyj intehral vyznaçavsq qk intehral Gnha, pizniße u [2] roz-
hlqdalasq pobudova stoxastyçnoho intehrala za dopomohog tak zvanyx „ßerßa-
vyx tra[ktorij” dlq dovil\nyx H. Dlq potra[ktornyx stoxastyçnyx dyferen-
cial\nyx rivnqn\ iz drobovym brounivs\kym ruxom isnuvannq ta [dynist\ rozv’qz-
kiv bulo dovedeno u stattqx [3, 4] u vypadku dovhostrokovo] zaleΩnosti (H >
> 1 / 2), [2, 5] u vypadku H > 1 / 4 (dyv. takoΩ [6]). Inßyj sposib — kvadratyçne
intehruvannq za dopomohog teori] prostoriv, porodΩuvanyx qdramy. Uperße ta-
ku konstrukcig intehrala bulo rozhlqnuto u [7], pizniße — [8, 9]. Stoxastyçni
dyferencial\ni rivnqnnq iz takym intehralom rozhlqdalys\ u [7]. Dlq zahal\-
no] potra[ktorno] konstrukci] stoxastyçnoho intehrala, zaproponovano] u [10],
stoxastyçni dyferencial\ni rivnqnnq vyvçalys\ u [11]. Nareßti, rivnqnnq z
intehralom Skoroxoda rozhlqdalysq u [12, 13]. Bil\ß dokladnyj ohlqd litera-
tury z ci[] tematyky moΩna znajty u [14].
U bahat\ox vypadkax analiz rivnqn\ iz drobovym brounivs\kym ruxom vyqv-
lq[t\sq dosyt\ skladnym, tomu vynyka[ potreba v nablyΩenomu rozv’qzuvanni
takyx rivnqn\. Pytannq aproksymaci] drobovoho brounivs\koho ruxu vklgça[ v
sebe pytannq modelgvannq, qke rozhlqdalosq bahat\ma avtoramy. Dlq nas, od-
nak, bil\ß cikavym [ aproksymaci] takymy procesamy, qki dozvolqgt\ prostißyj
analiz z toçky zoru baΩanyx, nasampered finansovyx, zastosuvan\. Odnym iz
najprostißyx metodiv [ metod dyskretyzaci] çasu v stoxastyçnomu dyferen-
cial\nomu rivnqnni, (dyv. [7, 15, 16]). Aproksymaci] drobovoho brounivs\koho ru-
xu za dopomohog semimartynhaliv vyvçalysq u [17].
*
Pidtrymano prohramog „Marie Curie Actions”, hrant # PIRSES-GA-2008-230804.
© K. V. RAL|ÇENKO, H. M. ÍEVÇENKO, 2010
1256 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 9
NABLYÛENNQ ROZV’QZKIV STOXASTYÇNYX DYFERENCIAL|NYX RIVNQN| … 1257
U danij statti my prodovΩu[mo doslidΩennq pytan\, zapoçatkovane u [18], a
same, nablyΩennq drobovoho brounivs\koho ruxu absolgtno neperervnymy
procesamy. Na Ωal\, rezul\taty, otrymani u [18], stosugt\sq lyße zbiΩnosti
samoho drobovoho brounivs\koho ruxu ta stoxastyçnyx intehraliv za drobovym
brounivs\kym ruxom vid dosyt\ hladen\kyx procesiv, tomu ci rezul\taty ne
vda[t\sq zastosuvaty dlq dovedennq zbiΩnosti rozv’qzkiv stoxastyçnyx
dyferencial\nyx rivnqn\. U danij statti my dovodymo dlq aproksymacij bil\ß
syl\nu zbiΩnist\, niΩ u [18]. Qk naslidok, oderΩu[mo rezul\tat pro zbiΩnist\
rozv’qzkiv vidpovidnyx stoxastyçnyx dyferencial\nyx rivnqn\.
Stattg pobudovano takym çynom. U p.N1 navedeno neobxidni oznaçennq ta
formulgvannq. U p.N2 vstanovleno rezul\tat pro zbiΩnist\ aproksymacij dro-
bovoho brounivs\koho ruxu absolgtno neperervnymy procesamy. U p.N3 dovedeno
zahal\nu teoremu pro zbiΩnist\ rozv’qzkiv stoxastyçnyx dyferencial\nyx riv-
nqn\ i, qk naslidok, otrymano zbiΩnist\ dlq aproksymacij, pobudovanyx u p.N2.
1. Oznaçennq. 1.1. Elementy drobovoho çyslennq. U c\omu punkti roz-
hlqda[t\sq pobudova potra[ktornoho intehrala.
Nexaj f ∈ L a b1( , ) ta α > 0. Livo- ta pravostoronnij intehraly Rimana –
Liuvillq vid funkci] f porqdku α [ vyznaçenymy majΩe dlq vsix x ∈ ( , )a b za
dopomohog formul (oznaçennq 2.1 [19])
I f x x y f y dya
a
x
+
−= −∫α α
α
( ) :
( )
( ) ( )
1 1
Γ
,
I f x y x f y dyb
x
b
−
−
−=
−
−∫α
α
α
α
( ) :
( )
( )
( ) ( )
1 1
Γ
vidpovidno, de ( )− −1 α = e i− πα
, Γ( )α = r e drrα− −∞
∫ 1
0
— hamma-funkciq Ejlera.
Obrazy prostoru L a bp( , ) pid di[g operatoriv Ia+
α
, Ib−
α
poznaçagt\sq vid-
povidno I Lp
α
α
+ ( ) ta I Lb
p
−
α ( ) .
Dlq 0 < α < 1 vid funkci] f : a b,[ ] → R, livo- ta pravostoronnq poxidni Ri-
mana – Liuvillq vyznaçagt\sq takym çynom:
D f x
d
dx
x y f y dya
a
x
+
−=
−
−∫α α
α
( ) :
( )
( ) ( )
1
1Γ
,
D f x
d
dx
y x f y dyb
x
b
−
+
−=
−
−
−∫α
α
α
α
( ) :
( )
( )
( ) ( )
1
1
1
Γ
.
U vypadku, koly hranyci f a( )+ i g b( )− isnugt\ i skinçenni, poznaçymo
f xa+ ( ) = f x f a xa b( ) ( ) ( )( , )− +( )• ,
g xb− ( ) = g x g b xa b( ) ( ) ( )( , )− −( )• .
Oznaçennq 1. Prypustymo, wo dlq funkcij f, g isnugt\ hranyci f a( )+ ,
g a( )+ , g b( )− , a takoΩ fa+ ∈ I La
p
+
α ( ) , gb− ∈ I Lb
q
−
−1 α ( ) dlq deqkyx 1 / p +
+ 1 / q ≤ 1, 0 ≤ α ≤ 1. Todi intehral Gnha, abo uzahal\nenyj intehral Stil\-
t\[sa vid funkci] f za funkci[g g zada[t\sq rivnistg
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 9
1258 K. V. RAL|ÇENKO, H. M. ÍEVÇENKO
f x dg x D f x D g x
a
b
a a b
a
b
b( ) ( ) : ( ) ( ) (= −∫ ∫ + + −
−
−1 1α α α )) dx N+
+ f a g b g a( ) ( ) ( )+ − − +( ) . (1)
Ce vyznaçennq dozvolq[ intehruvaty funkci], wo zadovol\nqgt\ umovu Hel\-
dera. Nahada[mo, wo C a bλ ,[ ] — prostir funkcij, qki zadovol\nqgt\ umovu
Hel\dera z pokaznykom λ :
f x f y( ) ( )− ≤ C x y− λ , x, y ∈ a b,[ ] .
TverdΩennq 1 (teorema 4.2.1 [1]). Nexaj x ∈ C a bλ ,[ ] , g ∈ C a bµ ,[ ] z λ +
+ µ > 1. Todi umovy oznaçennqN1 vykonugt\sq z bud\-qkym α ∈ (1 – µ, λ ) t a
p = q = ∞. Bil\ß toho, vyznaçenyj za (1) uzahal\nenyj intehral Stil\t\[sa
f x dg x
a
b
( ) ( )∫ zbiha[t\sq z intehralom Rimana – Stil\t\[sa:
R S− = −(∫ ∑ ∗
+f x dg x f x g x g x
a
b
i
i
i i( ) ( ) : lim ( ) ( ) ( )
π
1 )) ,
de π = a{ = x0 ≤ x0
∗ ≤ x1 ≤ … ≤ xn−1 ≤ xn−
∗
1 ≤ xn = b} , a takoΩ π =
= maxi i ix x+ −1 .
1.2. Drobovyj brounivs\kyj rux. Nexaj ( , , )Ω F P — povnyj imovirnisnyj
prostir.
Oznaçennq 2. Drobovym brounivs\kym ruxom (DBR) z parametrom Xgrsta
H ∈ (0, 1) nazyva[t\sq centrovanyj haussivs\kyj proces BH = B tt
H , ≥{ }0 z i
stacionarnymy pryrostamy ta kovariacijnog funkci[g
E B Bt
H
s
H( ) =
1
2
2 2 2t s t sH H H+ − −( ) .
NevaΩko baçyty, wo pryrosty drobovoho brounivs\koho ruxu zadovol\nqgt\
rivnist\
E B Bt
H
s
H( )2 = t s H− 2 ,
zvidky, zavdqky tomu, wo BH
[ haussivs\kym procesom, vyplyva[, wo vin ma[
neperervnu modyfikacig zhidno z teoremog Kolmohorova. Bil\ß toho (dyv., na-
pryklad, [14], hl. 1.16), joho tra[ktori] majΩe napevno naleΩat\ do C Tβ 0,[ ]
dlq vsix T > 0, β ∈ (0, H).
Qk vidomo [20], DBR B tt
H , ≥{ }0 pry H ∈ (1 / 2, 1) moΩna podaty u vyhlqdi
Bt
H = s dY t Y s Y dss t
t
s
t
α α αα:= −∫ ∫ −
0
1
0
,
(2)
Yt = C s u s du dWH
s
tt
s
− −−
∫∫ α α( ) 1
0
,
de W tt , ≥{ }0 — vinerivs\kyj proces, α = H – 1 / 2, a stala
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #9
NABLYÛENNQ ROZV’QZKIV STOXASTYÇNYX DYFERENCIAL|NYX RIVNQN| … 1259
CH =
2
3
2
1
2
2 2
1
H H
H H
Γ
Γ Γ
−
+
−
( )
/22
1
2
H −
.
1.3. Prostory B[sova (drobovi prostory Sobol[va). Nexaj dlq b ∈ (0, 1)
ϕβ β
f
t
t f t f t f s t s ds( ) : ( ) ( ) ( ) ( )= + − −∫ − −
0
1 ,
i W0
β = W T0 0β ,[ ] — prostir vymirnyx funkcij f : 0, T[ ] → R z normog
f t
t T
f0
0
,
,
: sup ( )β
βϕ=
∈[ ]
< ∞.
TakoΩ nexaj W1
β = W T1 0β ,[ ] — prostir funkcij f : 0, T[ ] → R z normog
f
f t f s
t s
f u f s
us t T
1
0
, : sup
( ) ( )
( )
( ) ( )
(β β=
−
−
+
−
≤ < ≤ −−
+∫ s
du
s
t
)1 β < ∞,
i W2
β = W T2 0β ,[ ] — prostir funkcij f : 0, T[ ] → R z normog
f
f s
s
ds
T
2
0
, :
( )
β β= ∫ +
f s f u
s u
du ds
sT ( ) ( )
( )
−
− +∫∫ β 1
00
< ∞.
ZauvaΩymo, wo prostory Wi
β , i = 0, 2, [ banaxovymy vidnosno vidpovidnyx norm;
⋅ 1,β [ lyße napivnormog.
Dlq bud\-qkoho 0 < ε < β ∧ (1 – β)
C T W T C Ti
β ε β β ε+ −[ ] ⊂ [ ] ⊂ [ ]0 0 0, , , , i = 0, 2, C T W Tβ ε β+ [ ] ⊂ [ ]0 02, , .
OtΩe, tra[ktori] DBR BH
dlq majΩe vsix ω ∈Ω , bud\-qkoho T > 0 i bud\-
qkoho 0 < β < H naleΩat\ W T1 0β ,[ ] .
2. Aproksymaciq drobovoho brounivs\koho ruxu absolgtno neperervny-
my procesamy. Dlq pobudovy aproksymacij drobovoho brounivs\koho ruxu sko-
rysta[mosq ide[g, zaproponovanog u [18].
ZauvaΩymo, wo v (2) my ne moΩemo pominqty porqdok intehruvannq ta za-
pysaty
Yt = C s u s dW duH
H H
ut
s
1 2 3 2
00
/ /( )− −−∫∫ ,
oskil\ky vnutrißnij intehral [ rozbiΩnym. Ideq polqha[ u tomu, wob „vidstupy-
ty” vid u u vnutrißn\omu intehrali. OtΩe, aproksymaci] drobovoho brounivs\-
koho ruxu absolgtno neperervnymy procesamy pobudu[mo tak:
Y C u s s dW du dut H s
ut
ε α α
φε
: ( )
( )
= −
− −∫∫ 1
00
, (3)
B s dYt
H
s
t
, :ε α ε= ∫
0
. (4)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 9
1260 K. V. RAL|ÇENKO, H. M. ÍEVÇENKO
Tut nabir dijsnyx vymirnyx nespadnyx funkcij φε : 0, T[ ] → 0, T[ ] , ε ∈ (0, 1),
zadovol\nq[ umovy:
i) φε ( )0 = 0, 0 < φε ( )t < t, t ∈ 0, T( ] ,
ii) φ φε ε( ) ( )t s− ≤ L t s− , t, s ∈ 0, T( ] ,
iii) fε : = sup ( ),t T t t∈[ ] −( )0 φε → 0, ε → 0 +,
iv) dlq bud\-qkoho ε ∈ (0, 1) sup
( ),t T
t
t∈[ ]0 φε
≤ K < + ∞.
U roboti [18] dovedeno, wo dlq dovil\noho β ∈ (0, 1 – H) ma[ misce zbiΩnist\
B BH H− →,
,
ε
γ1
0P , ε → 0 +. (5)
Vidomo, wo dlq zbiΩnosti intehraliv Gnha vid funkci] z C Tα 0,[ ] dostatn\o
zbiΩnosti intehratoriv u W1
γ pry γ > 1 – α (dyv., napryklad, [3] abo lemuN1).
Oskil\ky my doslidΩuvatymemo zbiΩnist\ intehraliv vid funkcij iz prostoru
Cα
, de α < H, to nam potribno posylyty rezul\tat (5).
Teorema 1. Dlq dovil\noho γ ∈ (0, H) ma[ misce zbiΩnist\
B BH H,
,
ε
γ
− →
1
0P , ε → 0 +.
Dovedennq. Za oznaçennqm
B BH H− ,
,
ε
γ1
= sup ,
0≤ < ≤s t T
s t∆ ,
de
∆s t, =
∆ ∆B B
t s
t
H
s
H, ,
( )
ε ε
γ
−
−
+
∆ ∆B B
u s
du
u
H
s
H
s
t , ,
( )
ε ε
γ
−
− +∫ 1 ,
∆ B B Bt
H
t
H
t
H, ,:ε ε= − = C s u s u du dWH
t
s
s
s
− −∫ ∫ −
−
α
φ
α
φ
α
ε ε
0
1
1( ) ( )
( ) +
+ s u s u du dW
t
t
s
t
s
− −∫ ∫ −
α
φ
α α
ε ( )
( ) 1 ,
φ−1
poznaça[ obernenu do φ funkcig.
Dovedemo, wo pry 0 < δ < 2(H – γ) ma[ misce ocinka
E ∆ ∆B Bt
H
s
H, ,ε ε−( )2 ≤ ′ − −C t s fH
H2 δ
ε
δ , t, s ∈ 0, T[ ] , (6)
de ′CH — deqka stala.
Prypustymo, wo 0 ≤ s < t ≤ T. MoΩlyvi dva vypadky.
Vypadok 1: t > s > φε ( )t > φε ( )s .
Za vlastyvistg izometri]
E ∆ ∆B Bt
H
s
H, ,ε ε−( )2 = C u u dH
s
t
s
u
2 2 1
1
− −∫ ∫ −
−
α
φ
φ
α
φ
α
ε
ε ε
( )
( ) ( )
( )v v v
2
du +
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #9
NABLYÛENNQ ROZV’QZKIV STOXASTYÇNYX DYFERENCIAL|NYX RIVNQN| … 1261
+ u u d du u u
s
t
s
t
− − − −−
−∫ ∫2 1
2
2α α α α α( ) ( )v v v v+ 11
2
s
t
t
s
d du∫∫
v vα
φε ( )
= :
: = I I I1 2 3+ + .
Ocinymo koΩen z tr\ox intehraliv. Dlq drobovoho brounivs\koho ruxu ma[mo
t s H− 2 = E B Bt
H
s
H−( )2 =
= C u u d du uH
s
t
u
t t
2 2 1
2
2
0
− − −∫ ∫ −
+α α α α( )v v v ∫∫ ∫ −
−( )v v vu d du
u
t
α α1
2
.
Zvidsy vyplyva[, wo
I t s H
3
2≤ − ,
I1 ≤ C u u d duH
s
s
t
2 2
0
1
2
− −∫ ∫ −
α α α( )v v v ≤ t s H− 2 ,
I2 ≤ C u u d duH
s
s
t
2 2
0
1
2
− −∫ ∫ −
α α α( )v v v ≤ t s H− 2 .
Takym çynom,
E ∆ ∆B Bt
H
s
H, ,ε ε−( )2 ≤ 3 2t s H− ≤ 3 2t s t tH− −( )−δ
ε
δφ ( ) ≤
≤ 3 2t s fH− −δ
ε
δ .
OtΩe, u vypadku 1 ocinku (6) dovedeno.
Vypadok 2: t > φε ( )t > s > φε ( )s .
U c\omu vypadku
E ∆ ∆B Bt
H
s
H, ,ε ε−( )2 = C u u dH
s
s
s
u
2 2 1
1
− −∫ ∫ −
−
α
φ
α
φ
α
ε
ε
( )
( )
( )v v v
2
du +
+ u u d du u
u
u
− − −−
−
∫2 1
2
2
1
α α
φ
α α
φ
ε
ε
( )
( )
v v v +
(( )
( )
( )
t
t
u
t
s
t
u d du∫ ∫∫ −
−v v vα α
φε
1
2
= :
: = J J J1 2 3+ + .
Ocinymo koΩen z tr\ox intehraliv:
J1 ≤ C
u
u
uH
s
s
s
u
2
1 2
1
1
φε
α
φ
α
φ
ε
ε−
−
−∫ ∫
−
( )
( )
( )
( )
v vv vα d du
2
≤
≤ C K u u s u duH
s
s
2 2 2 1 2α
ε
α α
φ
α φ
ε
− − − − −( )∫ ( ( ) ) ( )
( )
≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 9
1262 K. V. RAL|ÇENKO, H. M. ÍEVÇENKO
≤ C K s u duH
s
s
2 2 2 1 2α
ε
α
φ
α φ
ε
− − −( )∫ ( )
( )
≤
C K
s sH
2 2
2
2 1
2 1
α
ε
α
α α
φ
( )
( )
+
−( ) − =
= C H s s H
1
2( ) ( )−( )φε ≤ C H t s s sH
1
2( ) ( ) ( ) ( )φ φ φε ε
δ
ε
δ−( ) −( )− ≤
≤ C H t s s sH
2
2( ) ( )− −( )−δ
ε
δφ ,
J2 ≤ C u u du
s
t
3
1 2
φε
α
φε
− −( )∫ ( )
( )
≤ C H t u u u du
s
t
3
2 1( ) ( ) ( )
( )
− −( )− −∫ α δ
φ
ε
δε
φ ≤
≤ C H t u du t t
s
t
t T
3
2
0
( ) ( ) sup ( )
( )
,
− −( )−
∈[ ]∫ α δ
φ
ε
δ
ε
φ ≤ C H t s fH
4
2( ) − −δ
ε
δ ,
J3 ≤ C
t
t
t u duH
t
t
2
2
2 2
φ
α
ε
α
α
φε
( )
( )
( )
−− ∫ ≤ C H t t H
1
2( ) ( )−( )φε ≤
≤ C H t s t tH
1
2( ) ( )− −( )−δ
ε
δφ .
Takym çynom, u vypadku 2 ocinku (6) takoΩ dovedeno.
ZauvaΩymo, wo, oskil\ky ∆ Bt
H ,ε – ∆ Bs
H ,ε
ma[ normal\nyj rozpodil, z (6)
vyplyva[, wo dlq vsix p > 0, δ ∈ (0, H – γ), s, t ∈ 0, T[ ]
E ∆ ∆B Bt
H
s
H p, ,ε ε−( ) ≤ C t s fH p
H p p
,
( )− −δ
ε
δ , (7)
de CH p, — deqka stala.
Dali nam znadobyt\sq rivnomirnyj po t, s analoh ocinky (6). Z nerivnosti
Ìarsia – Rodemixa – Ramsi (dyv. [21], teorema 1.4) vyplyva[, wo dlq bud\-qkyx
p > 0, α > 1 / p vykonano nerivnist\
sup
, ,
, ,
/ , ,
t s T
t
H
s
H
p p p
B B
t s
C
∈[ ] −
−
−
≤ ′
0
1
∆ ∆ε ε
α α αξ , (8)
de ′C pα, — deqka nevypadkova stala,
ξα
ε ε
α,
, ,
p
x
H
y
H p
p
TT B B
x y
dx dy=
−
−
+∫∫
∆ ∆
1
00
1/ p
.
Pry p > 1 / H, δ ∈ (0, H – 1 / p), α ∈ (1 / p, H – δ) ma[mo
E( ),ξα p
p =
E ∆ ∆B B
x y
dx dy
x
H
y
H p
p
TT
, ,ε ε
α
−( )
− +∫∫ 1
00
≤
≤ C f x y dx dyH p
p H p
TT
,
( )
ε
δ δ α− − − −∫∫ 1
00
= C fH p
p
, ,δ ε
δ .
Todi, vraxovugçy (8) i pokladagçy dlq θ ∈ (γ, H) p = 2 / (H – θ), α = (θ + H) / 2,
δ = (H – θ) / 4, otrymu[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #9
NABLYÛENNQ ROZV’QZKIV STOXASTYÇNYX DYFERENCIAL|NYX RIVNQN| … 1263
E sup
, ,
, ,
t s T
t
H
s
H p
p
B B
t s∈[ ]
−
−0
∆ ∆ε ε
θ ≤ C fH
H p
,
( ) /
θ ε
θ1 4−
z deqkog stalog CH ,θ
1
.
Z ostann\o] ocinky vyplyva[, wo dlq dovil\nyx θ ∈ (γ, H), κ ∈ (0, 1) isnu[
stala Cκ taka, wo jmovirnist\ podi]
A s t T B Bt
H
s
H
ε
ε ε: , , , ,= ∈[ ] −{dlq vsix vykonano0 ∆ ∆ ≤
≤ C t s f H
κ
θ
ε
θ− }−( )/4
ne menßa za 1 – κ.
Na mnoΩyni Aε dlq vsix s, t T∈[ ]0,
∆s t, ≤ C f t s u s duH
s
t
κ ε
θ θ γ θ γ( )/− − − −− + −
∫4 1 =
= C f t sH
κ ε
θ θ γθ γ( )/ ( )− − −+ −( ) −4 11 ,
zvidky
B BH H,
,
ε
γ
−
1
≤ C f TH
κ ε
θ θ γθ γ( )/ ( )− − −+ −( )4 11 .
Todi dlq bud\-qkoho a > 0
lim ,
,ε
ε
γ
κ
→ +
− ≥( ) ≤
0 1
P B B aH H ,
oskil\ky dlq dostatn\o malyx C f TH
κ ε
θ θ γθ γ( )/ ( )− − −+ −( )4 11 < a.
OtΩe, pry κ → 0 + oderΩymo
lim ,
,ε
ε
γ→ +
− ≥( )0 1
P B B aH H = 0.
Teoremu dovedeno.
3. NablyΩennq rozv’qzkiv stoxastyçnyx dyferencial\nyx rivnqn\.
Rozhlqnemo stoxastyçne dyferencial\ne rivnqnnq z drobovym brounivs\kym ru-
xom Bt
H
, H ∈ (1 / 2, 1):
X X b s X ds s X dBt s
t
s
t
s
H= + +∫ ∫0
0 0
( , ) ( , )σ , t T∈[ ]0, . (9)
U statti [3] navedeno umovy isnuvannq ta [dynosti rozv’qzku takoho rivnqnnq.
Ma[ misce takyj rezul\tat.
TverdΩennq 2. Nexaj B tt , ≥{ }0 — drobovyj brounivs\kyj rux z paramet-
rom Xgrsta H ∈
1
2
1, , vyznaçenyj na povnomu jmovirnisnomu prostori
( , , )Ω F P . N e x a j X0 [ vypadkovog velyçynog, a koefici[nty b t x( , ) ,
σ( , )t x majΩe napevno zadovol\nqgt\ nastupni umovy ( )Hb , ( )Hσ z deqkym
nevypadkovym β > 1 – H ta stalymy LN , MN , qki moΩut\ zaleΩaty vid ω:
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 9
1264 K. V. RAL|ÇENKO, H. M. ÍEVÇENKO
( ) :
)
H
N
b
i lokal\na lipßycevist\
dlq koΩnoho
:
> 0 iisnu[ take woL
b t x b t y L x y x y N
N
N
>
− ≤ − ∀ ≤ ∀
0 ,
( , ) ( , ) , tt T
b t x L x
∈[ ]
≤ +(
0
10
, ,
)
( , )
ii linijne zrostannq:
)) ∀ ∈ ∀ ∈[ ]
x t TR 0, ;
( ) :
)
( , ) ( , ) ,
H
b t x b t y M x y x
σ
i lipßycevist\:
− ≤ − ∀ ∈R,, ∀ ∈ t T0, ,
)
)
ii
ii
lokal\na hel\derovist\
dl
:
qq koΩnoho isnu[ take woN M
x
t x
x
N> >
∂
∂
−
∂
∂
0 0 ,
( , )σ σ(( , ) , , ,
)
t y M x y x y N t TN≤ − ∀ ≤ ∀ ∈ 0
iii hel\derovisst\ za çasom:
σ σ σ σ( , ) ( , ) ( , ) ( ,t x t y
x
t x
x
t− +
∂
∂
−
∂
∂
yy M t s
x t T
)
, .
≤ −
∀ ∈ ∀ ∈
β
R 0
Poznaçymo
α β0
1
2
: min ,= { } .
Todi dlq dovil\noho α ∈ ( , )1 0− H α isnu[ [dynyj rozv’qzok X ∈ L0 Ω( ,
F , P ; W T0 0α ,[ ]) , qkyj [ rozv’qzkom rivnqnnq (9), do toho Ω, dlq P-majΩe vsix
ω ∈Ω vykonu[t\sq
X C T( , ) ,ω α⋅ ∈ [ ]−1 0 .
Rozhlqnemo mnoΩynu procesiv BH , ,ε ε >{ }0 , qki nablyΩagt\ proces BH
.
Nexaj X ε
— rozv’qzok rivnqnnq
Xt
ε = X0 + b s X dss
t
( , )ε
0
∫ + σ ε ε( , ) ,s X dBs
t
s
H
0
∫ , t T∈[ ]0, .
Teorema 2. Prypustymo, wo koefici[nty rivnqnnq (9) zadovol\nqgt\ umo-
vy ( )Hb , ( )Hσ i dlq deqkoho γ ∈ ( / , )1 2 H
B BH H,
,
ε
γ
− →
1
0P , ε → 0 +.
Todi
sup
,t T
t tX X
∈[ ]
− →
0
0ε P , ε → 0 +.
Rozhlqnemo na W T0 0β ,[ ] normu, ekvivalentnu do ⋅ 0,β :
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #9
NABLYÛENNQ ROZV’QZKIV STOXASTYÇNYX DYFERENCIAL|NYX RIVNQN| … 1265
f e t
t T
t
f0
0
, ,
,
: sup ( )β λ
λ βϕ=
∈[ ]
− .
Dlq dovedennq nam budut\ potribni taki lemy.
Lema 1 [3]. Prypustymo, wo vykonano umovu ( )Hσ , β ∈( , / )0 1 2 , f ∈
∈ W T0 0β ,[ ] , g ∈ W T1
1 0− [ ]β , . Todi dlq bud\-qkoho λ ≥ 1 spravedlyvi taki
tverdΩennq:
1) isnu[ intehral G f tg( , )( )( )σ : = σ ⋅ ⋅( )∫ , ( )f dg
t
0
, t T∈[ ]0, ;
2) G fg( , )( )σ ∈ C T1 0− [ ]β , � W T0 0β ,[ ] ;
3) G fg( , )
, ,
( )σ
β λ0
≤ C g f1 1
2 1
01Λ −
− +( )β
β
β λλ( ) , , , d e C1 zaleΩyt\ lyße
vid β, T i σ;
4) dlq dovil\nyx f, h ∈ W T0 0β ,[ ] takyx, wo f hT T
∗ ∗∨ ≤ R
G f G hg g( , ) ( , )
, ,
( ) ( )σ σ
β λ
−
0
≤ C g K K f hf h2
2 1
1 01λ β
β β λ
−
− + + −Λ ( ) ( ) , , ,
de K f = sup
( )
,r T
r sr f f
r s
ds∈[ ] +
−
−∫0 10 β , C2 zaleΩyt\ lyße vid β, T, R, σ.
Toçne oznaçennq Λ1−β( )g moΩna znajty v [3]. Dlq dovedennq teoremyN2
nam bude dostatn\o toho, wo dlq g ∈ W T1
1 0− [ ]β , ma[ misce ocinka
Λ1 1 1− −≤β β( ) ,g C g .
Lema 2 [3]. Nexaj β ∈ ( , / )0 1 2 , vykonano umovu ( )Hb , f ∈ W T0 0β ,[ ] . Todi
dlq bud\-qkoho λ ≥ 1 spravedlyvi taki tverdΩennq:
1) isnu[ intehral Lebeha F f tb( )( )( ) : = b s f s ds
t
, ( )( )∫0 , t T∈[ ]0, ;
2) F fb( )( ) ∈ C T1 0− [ ]β , ;
3) F fb( )
, ,
( )
0 β λ
≤ C f3
2 1
01λ β
β λ
− +( ), , , de C3 zaleΩyt\ lyße vid β , T
ta b;
4) nexaj f, h ∈ W T0 0β ,[ ] z f hT T
∗ ∗∨ ≤ R,
todi
F f F hb b( ) ( )
, ,
( ) ( )−
0 β λ
≤ C f h4
1
0λβ β λ
− − , , ,
de C4 zaleΩyt\ vid β, T, R, b.
Dovedennq teoremy 2. Poznaçymo β : = 1 – γ.
Spoçatku dovedemo, wo KX ε rivnomirno po ε obmeΩene za jmovirnistg.
Oçevydno, wo
K f ≡ sup
( ),r T
r s
r f f
r s
ds
∈[ ] +
−
−∫
0
1
0
β ≤ C f 0,β .
Z dovedennq teoremyN5.1 [3] vyplyva[, wo
X ε
β0,
≤ 2 1 0
0+( )X e Tλ ε( ) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 9
1266 K. V. RAL|ÇENKO, H. M. ÍEVÇENKO
de
λ ε0( ) ≤ 2 3 1 1
1 2 1
C C BH+( )( )−
−
Λ β
ε β
( ), /
.
Oskil\ky
Λ1−β
ε( ),BH ≤ BH ,
,
ε
β1 1−
i
B BH H,
,
ε
β
− →
−1 1
0P , ε → 0 +,
to Λ1−β
ε( ),BH
obmeΩene za jmovirnistg rivnomirno po ε. Zvidsy X ε
β0,
ob-
meΩene za jmovirnistg rivnomirno po ε, a tomu j KX ε obmeΩeni za jmovirnis-
tg rivnomirno po ε ta Xt
ε
obmeΩeni za jmovirnistg rivnomirno po ε, t.
ZauvaΩymo, wo dostatn\o dovesty zbiΩnist\
P sup , sup , sup
t
t t
t
t
t
tX X X R X R− > ≤ ≤
ε εδ → 0, ε → 0 +,
dlq vsix δ > 0, R > 0. My oderΩymo potribnyj rezul\tat, oskil\ky Xt ta Xt
ε
rivnomirno obmeΩeni za jmovirnistg.
OtΩe, prypustymo, wo Xt ta Xt
ε
obmeΩeni deqkym R > 0. Ma[mo
X = X F X G Xb BH
0 + +( ) ( , )( ) ( )σ ,
X ε = X F X G Xb BH
0 + +( ) ( , )( ) ( )
,ε σ εε
,
de F f tb( )( )( ) : = b s f s ds
t
, ( )( )∫0 , G f tg( , )( )( )σ : = σ ⋅ ⋅( )∫ , ( )f dg
t
0
, t T∈[ ]0, .
Zapyßemo
X X− ε
β λ0, ,
≤ F X F Xb b( ) ( )
, ,
( ) ( )− ε
β λ0
+
+ G X G XB BH H( , ) ( , )
, ,
( ) ( )σ σ ε
β λ
−
0
+ G XBH( , )
, ,
,
( )σ ε
β λ
ε
0
.
Vykorystovugçy lemy 1 ta 2, ocing[mo
X X− ε
β λ0, ,
≤ C B K K X XH
X Xλ β
β
ε
β λε
2 1
1 1 0
1 1−
−
+( ) + + −
, , ,
( ) +
+ C B B XH Hλ β ε
β
ε
β λ
2 1
1 1 0
1−
−
− +( ),
, , ,
.
Qkwo
Θ( , )λ ε = C B K K XH
X Xλ β
β
ε
β λε
2 1
1 1 0
1 1−
−
+( ) + + +{ }, , ,
( ) < 1 / 2,
to
X X− ε
β λ0, ,
≤ B BH H−
−
,
,
ε
β1 1
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #9
NABLYÛENNQ ROZV’QZKIV STOXASTYÇNYX DYFERENCIAL|NYX RIVNQN| … 1267
Zvidsy
P X X c− >( )ε
β λ0, ,
≤ P B B cH H− >( )−
,
,
ε
β1 1
+ P Θ( , ) /λ ε >( )1 2 .
ZauvaΩymo, wo
sup
,t T
t tX X
∈[ ]
−
0
ε ≤ e X XTλ ε
β λ
−
0, ,
,
tomu dlq dovil\noho δ > 0
P sup
,t T
t tX X
∈[ ]
− >
0
ε δ ≤ P B B eH H T− >( )−
−,
,
ε
β
λδ
1 1
+ P Θ( , ) /λ ε >( )1 2 .
Ale lehko pokazaty, wo Θ( , )λ ε → 0 za jmovirnistg pry λ → ∞ rivnomirno
poNNε.
Teoremu dovedeno.
Naslidok 1. Qkwo koefici[nty rivnqnnq (9) zadovol\nqgt\ umovy ( )Hb ,
Hσ , BH ,ε
vyznaçeno za dopomohog formul (3) – (4), i funkci] φε zadovol\-
nqgt\ umovy i) – iv), navedeni na poçatku p.N2, to ma[ misce zbiΩnist\
sup
,t T
t tX X
∈[ ]
− →
0
0ε P , ε → 0 +.
1. Zähle M. Integration with respect to fractal functions and stochastic calculus I // Probab. Theory
Relat. Fields. – 1998. – 111, # 3. – P. 333 – 372.
2. Coutin L., Qian Zh. Stochastic analysis, rough path analysis and fractional Brownian motions //
Ibid. – 2002. – 122, # 1. – P. 108 – 140.
3. Nualart D., Răscanu A. Differential equations driven by fractional Brownian motion // Collect.
Math. – 2002. – 53, #1. – P. 55 – 81.
4. Ruzmaikina A. A. Stieltjes integrals of Hölder continuous functions with applications to fractional
Brownian motion // J. Statist. Phys. – 2000. – 100, # 5 – 6. – P. 1049 – 1069.
5. Coutin L., Qian Zh. Stochastic differential equations for fractional Brownian motions // C. r. Acad.
sci., Sér. I. Math. – 2000. – 331, # 1. – P. 75 – 80.
6. Nourdin I. A simple theory for the study of SDEs driven by a fractional Brownian motion, in di-
mension one // Sémin. probab. XLI. Some papers are selected contributions of the seminars in Nan-
cy 2005 and Luminy 2006. (Lect. Notes Math., 1934.) – Berlin: Springer, 2008. – P. 181 – 197.
7. Lin S. J. Stochastic analysis of fractional Brownian motions // Stochast. Rep. – 1995. – 55, # 1 –
2. – P. 121 – 140.
8. Duncan T. E., Hu Y., Pasik-Duncan B. Stochastic calculus for fractional Brownian motion. I.
Theory // SIAM J. Control Optim. – 2000. – 38, # 2. – P. 582 – 612.
9. Alòs E., Mazet O., Nualart D. Stochastic calculus with respect to Gaussian processes // Ann. Pro-
bab. – 2001. – 29, # 2. – P. 766 – 801.
10. Russo F., Vallois P. Forward, backward and symmetric stochastic integration // Probab. Theory
Relat. Fields. – 1993. – 97, # 3. – P. 403 – 421.
11. León J. A., Tudor C. Semilinear fractional stochastic differential equations // Bol. Soc. mat. mex.,
III. – 2002. – 8, # 2. – P. 205 – 226.
12. Mißura G. S. Kvazilinijni stoxastyçni dyferencial\ni rivnqnnq z drobovo-brounivs\kog
komponentog // Teoriq imovirnostej i mat. statystyka. – 2004. – # 68. – S. 95 – 106.
13. Tindel S., Tudor C. A., Viens F. Stochastic evolution equations with fractional Brownian motion //
Probab. Theory Relat. Fields – 2003. – 127, # 2. – P. 186 – 204.
14. Mishura Yu. Stochastic calculus for fractional Brownian motion and related processes // Lect. No-
tes Math. – Berlin: Springer, 2008. – xviii + 393 p.
15. Mishura Yu., Shevchenko G. The rate of convergence for Euler approximations of solutions of sto-
chastic differential equations driven by fractional Brownian motion // Stochastics. – 2008. – 80,
# 5. – P. 489 – 511.
16. Nordin I., Neuenkirch A. Exact rate of convergence of some approximation schemes associated to
SDEs driven by a fractional Brownian motion // J. Theor. Probab. – 2007. – 20, # 4. – P. 871 – 899.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 9
1268 K. V. RAL|ÇENKO, H. M. ÍEVÇENKO
17. Thao T. H. An approximate approach to fractional analysis for finance // Nonlinear Anal., Real
World and Appl. – 2006. – 7, # 1. – P. 124 – 132.
18. Androwuk T. NablyΩennq stoxastyçnoho intehralu po drobovomu brounivs\komu ruxu
intehralamy po absolgtno neperervnym procesam // Teoriq imovirnostej i mat. statystyka. –
2005. – # 73. – S. 11 – 20.
19. Samko S. H., Kylbas A. A., Maryçev O. Y. Yntehral¥ y proyzvodn¥e drobnoho porqdka y ne-
kotor¥e yx pryloΩenyq. – Mynsk: Nauka y texnyka, 1987. – 688 s.
20. Norros I., Valkeila E., Virtamo J. An elementary approach to a Girsanov formula and other analy-
tical results on fractional Brownian motions // Bernoulli. – 1999. – 5, # 4. – P. 571 – 587.
21. Garsia A. M., Rodemich E. Monotonicity of certain functionals under rearrangement // Ann. Inst.
Fourier. – 1974. – 24, # 2. – P. 67 – 116.
OderΩano 30.09.09,
pislq doopracgvannq — 29.06.10
π
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|
| id | umjimathkievua-article-2952 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:33:25Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ce/79b683b2283fc9c931b73a751bea20ce.pdf |
| spelling | umjimathkievua-article-29522020-03-18T19:41:02Z Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations Наближення розв'язків стохастичних диференціальних рівнянь із дробовим броунівським рухом розв'язками випадкових звичайних диференціальних рівнянь Ral’chenko, K. V. Shevchenko, H. M. Ральченко, К. В. Шевченко, Г. М. We prove a general theorem on the convergence of solutions of stochastic differential equations. As a corollary, we obtain a result concerning the convergence of solutions of stochastic differential equations with absolutely continuous processes to a solution of an equation with Brownian motion. Доказана общая теорема о сходимости решений стохастических дифференциальных уравнений. Как следствие, получен результат о сходимости решений стохастических дифференциальных уравнений с абсолютно непрерывными процессами к решению уравнения с броуновским движением. Institute of Mathematics, NAS of Ukraine 2010-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2952 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 9 (2010); 1256–1268 Український математичний журнал; Том 62 № 9 (2010); 1256–1268 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2952/2654 https://umj.imath.kiev.ua/index.php/umj/article/view/2952/2655 Copyright (c) 2010 Ral’chenko K. V.; Shevchenko H. M. |
| spellingShingle | Ral’chenko, K. V. Shevchenko, H. M. Ральченко, К. В. Шевченко, Г. М. Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations |
| title | Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations |
| title_alt | Наближення розв'язків стохастичних диференціальних рівнянь із дробовим броунівським рухом розв'язками випадкових звичайних диференціальних рівнянь |
| title_full | Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations |
| title_fullStr | Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations |
| title_full_unstemmed | Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations |
| title_short | Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations |
| title_sort | approximation of solutions of stochastic differential equations with fractional brownian motion by solutions of random ordinary differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2952 |
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