Stochastic Systems with Averaging in the Scheme of Diffusion Approximation
We propose a system approach to the asymptotic analysis of stochastic systems in the scheme of series with averaging and diffusion approximation. Stochastic systems are defined by Markov processes with locally independent increments in a Euclidean space with random switchings that are described by j...
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| Date: | 2005 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509810000134144 |
|---|---|
| author | Korolyuk, V. S. Королюк, В. С. |
| author_facet | Korolyuk, V. S. Королюк, В. С. |
| author_sort | Korolyuk, V. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:02:00Z |
| description | We propose a system approach to the asymptotic analysis of stochastic systems in the scheme of series with averaging and diffusion approximation. Stochastic systems are defined by Markov processes with locally independent increments in a Euclidean space with random switchings that are described by jump Markov and semi-Markov processes. We use the asymptotic analysis of Markov and semi-Markov random evolutions. We construct the diffusion approximation using the asymptotic decomposition of generating operators and solutions of problems of singular perturbation for reducibly inverse operators. |
| first_indexed | 2026-03-24T02:47:01Z |
| format | Article |
| fulltext |
UDK 519.21
V. S. Korolgk (In-t matematyky NAN Ukra]ny, Ky]v)
STOXASTYÇNI SYSTEMY Z USEREDNENNQM
U SXEMI DYFUZIJNO} APROKSYMACI}
We suggest a system approach in the asymptotic analysis of stochastic systems in the series scheme with
averaging and diffusion approximation. Stochastic systems are determined by Markov processes with
locally independent increments in the Euclidean space with random switchings that are described by
jump Markov and semi-Markov processes.
We use the asymptotic analysis of Markov and semi-Markov random evolutions. We construct the
diffusion approximation by using the asymptotic decomposition of generating operators and solutions of
problems of singular perturbation for reducible-inverse operators.
Zaproponovano systemnyj pidxid v asymptotyçnomu analizi stoxastyçnyx system u sxemi serij z
userednennqm ta dyfuzijnog aproksymaci[g. Stoxastyçni systemy zadagt\sq markovs\kymy
procesamy z lokal\no nezaleΩnymy pryrostamy v evklidovomu prostori z vypadkovymy peremy-
kannqmy, wo opysugt\sq strybkovymy markovs\kymy ta napivmarkovs\kymy procesamy.
Vykorystovu[t\sq asymptotyçnyj analiz markovs\kyx ta napivmarkovs\kyx vypadkovyx evo-
lgcij. Dyfuzijna aproksymaciq budu[t\sq z vykorystannqm asymptotyçnoho rozkladu porod-
Ωugçyx operatoriv ta rozv’qzkiv problem synhulqrnoho zburennq dlq zvidno-obernenyx opera-
toriv.
1. Markovs\ki stoxastyçni systemy. 1.1. Procesy z lokal\no nezaleΩnymy
pryrostamy (PLNP) v evklidovomu prostori Rd , d ≥ 1, zadagt\sq porodΩu-
gçym operatorom (heneratorom) u banaxovomu prostori
� Rd( ) dijsnoznaçnyx
obmeΩenyx funkcij ϕ ( )u , u ∈ Rd
, z supremum-normog ϕ : = sup
u Rd∈
ϕ ( )u
vyrazom
Lϕ ϕ ϕ ϕ( ) ( ) ( ) ( ) ( ) ( , )u C u u u u u d
Rd
= ′ + + −[ ]∫ v vΓ . (1)
Qdro intensyvnostej velyçyny strybkiv Γ( , )u dv [ pozytyvnog obmeΩenog
mirog z q u( ) : = Γ U Rd,( ) ∈
� Rd( ) . Vektor-funkciq C u( ) = C uk ( )( , k = 1, d )
vyznaça[ determinovanu skladovu ρ( )t , t ≥ 0, evolgcijno] systemy, wo [ roz-
v’qzkom evolgcijnoho rivnqnnq
d t
dt
C t
ρ ρ( )
( )= ( ), ρ( )0 = ∈u Rd
.
Perßyj dodanok u vyrazi (1) oznaça[ skalqrnyj dobutok
C u u C u
u
uk
d
k
k
( ) ( ) : ( )
( )
′ = ∂
∂=
∑ϕ ϕ
1
.
Stoxastyçnyj proces η( )t , t ≥ 0, wo vyznaça[t\sq heneratorom (1) na malo-
mu promiΩku çasu ∆, moΩna uqvlqty u vyhlqdi sumy [1]
∆ ∆ ∆ ∆η η η ρ ν( ) : ( ) ( ) ( ) ( )t t t t t= + − = + ,
de ν( )t , t ≥ 0, [ markovs\kym strybkovym procesom z heneratorom
ΓΓϕ ϕ ϕ( ) ( ) ( ) ( , )u u u u d
Rd
= + −[ ]∫ v vΓ .
© V. S. KOROLGK, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9 1235
1236 V. S. KOROLGK
ZauvaΩennq 1 [1]. Markovs\kyj proces ν( )t , t ≥ 0, vidihra[ rol\ matema-
tyçno] modeli stoxastyçnyx system (SS) u bahat\ox prykladnyx haluzqx teori]
vypadkovyx procesiv [2, 3].
Pryklad 1. Systema postaçannq enerhi] [4].
Systema sklada[t\sq z n odnakovyx pryladiv, qki pracggt\ nezaleΩno
odyn vid odnoho. KoΩnyj prylad moΩe buty v odnomu z dvox staniv: 1 —
potrebu[ enerhi], 0 — vidnovlg[ svo] funkci]. Ças perebuvannq v koΩnomu sta-
ni?— pokaznykova vypadkova velyçyna z intensyvnistg µ u stani 1 ta λ u stani
0. Markovs\kyj proces ν( )t , t ≥ 0, vyznaça[ çyslo pryladiv, wo znaxodqt\sq u
stani 1 v moment çasu t ≥ 0. Oçevydno, wo proces ν( )t , t ≥ 0, [ procesom na-
rodΩennq ta zahybeli [5] z intensyvnostqmy perexodiv λn k( ) = ( n – k ) λ z k -ho v
( k + 1 ) -j stan i µn k( ) = k µ z k -ho v ( k – 1 ) -j stan. E = {0, 1, … , n} — mnoΩyna
staniv procesu ν( )t , t ≥ 0.
Rozhlqnemo proces νε( )t , t ≥ 0, u sxemi serij z malym parametrom seri] ε =
= 1
n
, n → ∞ . SS postaçannq enerhi] u sxemi userednennq rozhlqda[t\sq pry
takomu normuvanni:
ν ε ν
ε
ε
ε( )t t=
, ε = 1
n
.
Markovs\kyj proces νε( )t , t ≥ 0, nabuva[ znaçen\ u fazovomu prostori staniv
Eε = {ε k : 0 ≤ k ≤ n} ta zada[t\sq heneratorom
Γεϕ ε ϕ ε λ ϕ ε µ γ ϕ( ) ( ) ( ) ( ) ( ) ( ) ( )u u u u u u u= + + − −[ ]−1 , u E∈ ε .
Tut, za oznaçennqm,
λ λ( ) : ( )u u= −1 , µ µ( ) :u u= ,
γ λ µ λ µ λ( ) : ( ) ( ) ( )u u u u= + = + − .
Vykorystovugçy formulu Tejlora dlq dviçi neperervno dyferencijovno] test-
funkci] ϕ ( )u , perekonu[mos\, wo henerator ma[ asymptotyçne zobraΩennq
Γε εϕ ϕ θ ϕ( ) ( ) ( ) ( )u C u u u= ′ + ,
de
C u u u u( ) ( ) ( ) ( )= − = − +λ µ λ λ µ ,
a druhyj dodanok [ znextugçym çlenom:
θ ϕε ( ) ( )u u → 0 , ε → 0, ϕ ( )u C∈ ( )2
R .
Magçy asymptotyçne zobraΩennq heneratora, moΩna zrobyty vysnovok (dyv.
[6], teorema 2.1.1), wo ma[ misce sxema userednennq
εν
ε
ρε
t t
⇒ ( ), ε → 0.
Hranyçnyj proces ρ( )t , t ≥ 0, vyznaça[t\sq rozv’qzkom evolgcijnoho rivnqnnq
d t
dt
C t
ρ ρ( )
( )= ( ), ρ ρ( )0 0= .
Vraxovugçy vyraz ßvydkosti C u( ), moΩna zapysaty rozv’qzok rivnqnnq u
qvnomu vyhlqdi i perekonatysq v tomu, wo
ρ ρ λ
λ µ
( )t → =
+
, t → ∞.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
STOXASTYÇNI SYSTEMY Z USEREDNENNQM U SXEMI … 1237
Tut konstanta [ rozv’qzkom rivnqnnq C( )ρ = 0.
Z uraxuvannqm povedinky SS postaçannq enerhi] u sxemi userednennq istotno
vynyka[ problema vyvçennq flgktuacij procesu vidnosno toçky rivnovahy ρ :
εν
ε
ρε
t
− ,
abo vidnosno useredneno] evolgci]
εν
ε
ρε
t t
− ( ) .
Zrozumilo, wo taki flgktuaci] z nezaleΩnym normuvannqm moΩna opysaty dy-
fuzijnymy procesamy. Same taka problema i rozhlqda[t\sq v nastupnyx pid-
punktax.
1.2. Sxema userednennq. SS u sxemi serij z userednennqm zada[t\sq normo-
vanym PLNP νε( )t : = εν
εε
t
, t ≥ 0, ε > 0, z malym parametrom seri] ε → 0,
heneratorom
ΓΓε
εϕ ε ϕ ε ϕ( ) ( ) ( ) ( , )u u u u d
Rd
= + −[ ]− ∫1 v vΓ .
Osnovnog umovog u sxemi userednennq [ asymptotyçne zobraΩennq seredn\oho
znaçennq velyçyny strybkiv
C u u d C u u
Rd
ε ε
εθ( ) : ( , ) ( ) ( )= = +∫ v vΓΓ
zi znextugçym çlenom
θε( )u → 0 , ε → 0.
Dodatkovog umovog [ isnuvannq hlobal\noho rozv’qzku evolgcijnoho riv-
nqnnq
d t
dt
C t
ρ ρ( )
( )= ( ), ρ( )0 = u . (2)
Todi ma[ misce slabka zbiΩnist\ (dyv., napryklad, [3])
εν
ε
ρε
t t
⇒ ( ), ε → 0,
pry umovi zbiΩnosti poçatkovyx danyx
ενε( )0 ⇒ u, ε → 0.
1.3. SS iz rivnovahog. Rivnovahog SS u sxemi userednennq [ stalyj roz-
v’qzok ρ rivnqnnq C( )ρ = 0.
SS iz rivnovahog u sxemi dyfuzijno] aproksymaci] zada[t\sq normovanym ta
centrovanym procesom
ζ εν
ε
ε ρε ε( )t t=
− −
2
1 , t ≥ 0. (3)
Tut νε( )t , t ≥ 0, [ markovs\kym procesom z heneratorom
ΓΓε εϕ ϕ ϕ( ) ( ) ( ) ( , )u u u u d
Rd
= + −[ ]∫ v vΓ . (4)
Qdro intensyvnostej ma[ vyhlqd
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1238 V. S. KOROLGK
Γ Γ Γε ε( , ) ( , ) ( , )u d u d u dv v v= + 1 . (5)
Oçevydno, wo proces (3) [ takoΩ markovs\kym. NevaΩko perekonatysq v to-
mu, wo joho henerator ma[ vyhlqd
Lζ
ε
εϕ ε ϕ ε ϕ ρ ε( ) ( ) ( ) ( , )u u u u d
Rd
= + −[ ] +− ∫2 v vΓ . (6)
Formula (6) [ klgçovog v asymptotyçnomu analizi SS iz rivnovahog u sxemi
dyfuzijno] aproksymaci].
Osnovnog umovog [ asymptotyçne zobraΩennq perßyx dvox momentiv vely-
çyny strybkiv:
b u u d b b u u
Rd
ε ε
ερ ρ ε ρ ε ρ εθ ρ( , ) : ( , ) ( ) ( , ) ( , )= + = + +∫ v vΓ 1 , (7)
B u u d B u
Rd
ε ε
ερ ρ ε ρ θ ρ( , ) : ( , ) ( ) ( , )= + = +∫ vv v* Γ 2 (8)
zi znextugçymy çlenamy
θ ρε
i u( , ) → 0 , ε → 0, i = 1, 2.
Teorema 1 [3]. Pry umovi balansu
b C( ) ( )ρ ρ= (9)
ta zbiΩnosti poçatkovyx danyx
ζ ζε( )0 0⇒ , ε → 0,
ma[ misce slabka zbiΩnist\
εν
ε
ε ρ ζε
t t2
1 0
− ⇒− ( ) , ε → 0. (10)
Hranyçnyj dyfuzijnyj proces ζ0( )t , t ≥ 0, vyznaça[t\sq heneratorom
L0 1
2
ϕ ρ ϕ ρ ϕ( ) ( , ) ( ) ( ) ( )u b u u B u= ′ + ′′[ ]Tr .
Tut, za oznaçennqm,
Tr B u b
u
u uk r
d
kr
k r
( ) ( ) : ( )
( )
,
ρ ϕ ρ ϕ′′[ ] = ∂
∂ ∂=
∑
1
2
.
ZauvaΩennq 2. Hranyçnyj proces ζ0( )t , t ≥ 0, [ dijsno dyfuzijnym, qkwo
kovariacijna matrycq dyfuzi] B( )ρ : = bkr( )ρ[ ] ≠ 0.
Vysnovok 1. Pry dodatkovij umovi isnuvannq obmeΩeno] poxidno] ′C ( )ρ
hranyçnyj proces ζ0( )t , t ≥ 0, [ procesom Ornßtejna – Ulenbeka [5] z para-
metrom zsuvu
b u b ub( , ) ( ) ( )ρ ρ ρ= + ′1 ,
b d
Rd
1 1( ) ( , )ρ ρ= ∫ v vΓ .
1.3.1. Alhorytm dyfuzijno] aproksymaci] (ADA) v teoremi 1 budu[t\sq
z??vykorystannqm asymptotyçnoho zobraΩennq heneratora (6) markovs\koho pro-
cesu (3) u formi
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
STOXASTYÇNI SYSTEMY Z USEREDNENNQM U SXEMI … 1239
L L Lζ
ε εϕ ϕ θ ρ ϕ( ) ( ) ( ) ( )u u u= +0
zi znextugçym çlenom
θ ρ ϕε
L( ) ( )u → 0 , ε → 0, ϕ ∈ ( )C Rd3
.
Todi slabka zbiΩnist\ (10) zabezpeçu[t\sq teoremog Skoroxoda [6].
1.4. SS z evolgcijnym userednennqm. Sxema dyfuzijno] aproksymaci] SS
z evolgcijnym userednennqm rozhlqda[t\sq dlq normovanoho ta centrovanoho
procesu
ζ εν
ε
ε ρε
ε( ) ( )t t t=
− −
2
1
, t ≥ 0.
Markovs\kyj proces νε( )t , t ≥ 0, zada[t\sq heneratorom (4), (5). Evolgciq
ρ( )t , t ≥ 0, vyznaça[t\sq rozv’qzkom rivnqnnq (2). Krim toho, peredbaça[t\sq,
wo ma[ misce slabka zbiΩnist\ (sxema userednennq) [3], tobto
εν
ε
ρε
t t
⇒ ( ), ε → 0.
Osnovna umova taka sama, qk i v poperednij sxemi dyfuzijno] aproksymaci] z
rivnovahog, tobto magt\ misce asymptotyçni zobraΩennq perßyx dvox momentiv
velyçyn strybkiv (7), (8).
Teorema 2. Pry umovi balansu (9) ta zbiΩnosti poçatkovyx znaçen\ ma[
misce slabka zbiΩnist\
εν
ε
ε ρ ζε
t t t2
1 0
− ⇒− ( ) ( ), ε → 0.
Hranyçnyj dyfuzijnyj proces ζ0( )t , t ≥ 0, vyznaça[t\sq heneratorom
Lt u b u t u B t u0 1
2
ϕ ρ ϕ ρ ϕ( ) , ( ) ( ) ( ) ( )= ( ) ′ + ( ) ′′[ ]Tr . (11)
ZauvaΩennq 3. ZobraΩennq (11) heneratora hranyçnoho dyfuzijnoho pro-
cesu ζ0( )t , t ≥ 0, oznaça[, wo dvokomponentnyj markovs\kyj proces ζ0( )t ,
ρ( )t , t ≥ 0, [ odnoridnym u çasi i vyznaça[t\sq rozv’qzkom systemy stoxastyçnyx
rivnqn\
d t b t t dt t dw tζ ζ ρ σ ρ0 0( ) ( ), ( ) ( ) ( )= ( ) + ( ) , (12)
d t C t dtρ ρ( ) ( )= ( ) .
Kovariacijna matrycq σ ρ( ) vyznaça[t\sq spivvidnoßennqm
σ ρ σ ρ ρ( ) ( ) ( )* = B .
ZauvaΩennq 4. Pry dodatkovij umovi isnuvannq obmeΩeno] poxidno] ′C ( )ρ
koefici[nt zsuvu ma[ vyhlqd
b u b uC( , ) ( ) ( )ρ ρ ρ= + ′1 .
1.4.1. ADA v teoremi 2 budu[t\sq z vykorystannqm asymptotyçnoho zobra-
Ωennq heneratora dvokomponentnoho markovs\koho procesu ζ0( )t , ρ( )t , t ≥ 0 :
Lζ
ε ϕ ρ( , )u =
ε ϕ ε ρ ϕ ρ ρ εε
− ∫ + −[ ] +2
2R
u u u d( , ) ( , ) ( , )v vΓ –
– ε ρ ϕ ρ− ′1C uu( ) ( , ) + C u( ) ( , )ρ ϕ ρρ′ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1240 V. S. KOROLGK
a same, na test-funkciqx ϕ ρ( , )u ∈ C Rd3 2, ( × Rd ) ma[ misce asymptotyçne zo-
braΩennq
L Lζ
ε
ρ ρ
εϕ ρ ϕ ρ θ ϕ ρ( , ) ( , ) ( , )u u u= +0
zi znextugçym çlenom θ ϕρ
ε → 0, ε → 0, ϕ ∈ C Rd3( ). Tut henerator dvokom-
ponentnoho markovs\koho procesu ζ0( )t , ρ( )t , t ≥ 0, ma[ vyhlqd
Lρ ρϕ ρ ρ ϕ ρ ρ ϕ ρ ρ ϕ ρ0 1
2
( , ) ( , ) ( , ) ( ) ( , ) ( ) ( , )u b u u B u C uu u= ′ + ′′[ ] + ′Tr .
OtΩe, dvokomponentnyj markovs\kyj proces ζ0( )t , ρ( )t , t ≥ 0, vyznaça[t\-
sq rozv’qzkom systemy (12), a henerator hranyçnoho dyfuzijnoho procesu ζ0( )t ,
t ≥ 0, zada[t\sq vyrazom (11). Slabka zbiΩnist\ u teoremi 2 zabezpeçu[t\sq
teoremog Skoroxoda [6].
1.5. SS u sxemi fazovoho userednennq. SS u sxemi serij z malym paramet-
rom seri] ε → 0 zada[t\sq rozv’qzkom evolgcijnoho rivnqnnq v evklidovomu
prostori Rd
, d ≥ 1:
dU t
dt
C U t tε
ε
ε κ
ε
( ) ( );=
2 . (13)
Vektor-funkciq ßvydkostej ma[ vyhlqd
C u x C u x C u xε ε( ; ) ( ; ) ( ; )= + 1 . (14)
Markovs\kyj proces peremykan\ κ ( t ) , t ≥ 0, u standartnomu fazovomu
prostori ( E, E ) zada[t\sq heneratorom
Q x q x P x dy y x
Rd
ϕ ϕ ϕ( ) ( ) ( , ) ( ) ( )= −[ ]∫ . (15)
Sxema fazovoho userednennq oznaça[, wo dlq SS
dU t
dt
C U t tε
ε ε κ
ε
( )
( );=
(16)
magt\ misce umovy slabko] zbiΩnosti
U t V tε( ) ( )⇒ , ε → 0. (17)
Hranyçna evolgciq V t( ), t ≥ 0, vyznaça[t\sq rozv’qzkom userednenoho evo-
lgcijnoho rivnqnnq
dV t
dt
C V t
( )
( )= ( ) ,
C dx C x
E
( ) ( ) ( ; )v v= ∫ π .
Tut π( )B , B ∈ E , — stacionarnyj rozpodil rivnomirnoho erhodyçnoho markov-
s\koho procesu κ( )t , t ≥ 0.
Sxema dyfuzijno] aproksymaci] SS (13), (14) rozhlqda[t\sq dlq centrovano-
ho ta normovanoho procesu
ζ εε ε( ) ( ) ( )t U t V t= −[ ]−1
, t ≥ 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
STOXASTYÇNI SYSTEMY Z USEREDNENNQM U SXEMI … 1241
Osnovnog umovog [ asymptotyçne zobraΩennq funkci] ßvydkostej
C u x C x uC x u x( ; ) ( ; ) ( ; ) ( , ; )v v v vv+ = + ′ +ε ε εθε
0 , (18)
C u x C x u x1 1 1( ; ) ( ; ) ( , ; )v v v+ = +ε θε
zi znextugçymy çlenamy
θε
i x( ; )v → 0 , ε → 0, i = 0, 1.
Teorema 3 [9]. V umovax fazovoho userednennq (16), (17) ma[ misce slabka
zbiΩnist\
ε ζε− −[ ] ⇒1 0U t V t t( ) ( ) ( ) , ε → 0.
Hranyçnyj dyfuzijnyj proces ζ0( )t , t ≥ 0, vyznaça[t\sq heneratorom
Lt u b u V t u C V t u0
0 0
1
2
ϕ ϕ ϕ( ) , ( ) ( ) ( ) ( )= ( ) ′ + ( ) ′′[ ]Tr .
Tut
b u C uC b( , ) ( ) ( ) ( )v v v v= + ′ +1 0 , (19)
b dx b x
E
0 0( ) ( ) ( ; )v v= ∫ π , b x C x R C x0 0( ; ) ˆ( ; ) ˆ ( ; )v v vv= ′ ,
C dx C x
E
1 1( ) ( ) ( ; )v v= ∫ π ,
ˆ( ; ) : ( ; ) ( )C x C x Cv v v= − , (20)
C dx C x
E
0 0( ) ( ) ( ; )v v= ∫ π , C x C x R C x0 0( ; ) : ˆ( ; ) ˆ ( ; )*v v v= .
Vysnovok 2. Dvokomponentnyj markovs\kyj proces ζ0( )t , V t( ), t ≥ 0, vy-
znaça[t\sq heneratorom
L0
0
1
2
ϕ ϕ ϕ ϕ( , ) ( , ) ( , ) ( ) ( , ) ( ) ( , )u b u u C u C uu uv v v v v v vv= ′ + ′′[ ] + ′Tr . (21)
ZauvaΩennq 2 ta 3 do teoremy 1 takoΩ magt\ misce.
Tut kovariacijna matrycq σ( )v vyznaça[t\sq spivvidnoßennqm
σ σ( ) ( ) ( )*v v v= C0 .
1.5.1. ADA v teoremi 3 budu[t\sq z vykorystannqm asymptotyçnoho roz-
kladu porodΩugçoho operatora trykomponentnoho markovs\koho procesu
ζε( )t , V t( ), κε
t : = κ
ε
t
2
, t ≥ 0:
L C Cε
εϕ ε ε ϕ( , , ) ( ) ( , , )u x Q x u xv v= + +[ ]− −2 1
.
Tut Q [ heneratorom (15) markovs\koho procesu κ( )t , t ≥ 0 ; operatory
C Cε εϕ ε ϕ( ) ( ) ( ; ) ( ) ( )x u u x C u= + −[ ] ′v v
ta
Cϕ ϕ( ) ( ) ( )v v v= ′C .
Vraxovugçy osnovnu umovu (18), ma[mo asymptotyçne zobraΩennq
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1242 V. S. KOROLGK
Lε εε ε θ= + + +− −2 1
1 2Q Q x Q x xl( ) ( ) ( )
na test-funkciqx ϕ( , )u v ∈ C Rd3 2, ( × Rd ) , de
Q x u C x u1( ) ( ) ˆ( ; ) ( )ϕ ϕ= ′v ,
ˆ( ; ) : ( ; ) ( )C x C x Cv v v= − ,
Q x u b u x u C u C uu u2 0
1
2
( ) ( , ) ( , ; ) ( , ) ( ) ( , ) ( ) ( , )ϕ ϕ ϕ ϕv v v v v v vv= ′ + ′ + ′′[ ]Tr ,
b u x C x uC x( , ; ) ( ; ) ( ; )v v vv= + ′1 ;
znextugçyj çlen
θ ϕε
l x( ) → 0, ε → 0, ϕ( , )u v ∈ C Rd3 2, ( × Rd ) .
Teper vykorystovu[t\sq rozv’qzok problemy synhulqrnoho zburennq [3] dlq zri-
zanoho operatora
L0
2 1
1 2
ε ε ε: ( ) ( )= + +− −Q Q x Q x .
Hranyçnyj operator L0
dvokomponentnoho procesu ζ0( )t , V t( ), t ≥ 0, vy-
znaça[t\sq formulog [3]
L0
2 1 0 1Π Π Π Π Π= +Q x Q x R Q x( ) ( ) ( ) . (22)
Tut Π — proektor u banaxovomu prostori B ( )E , wo vyznaça[t\sq stacionar-
nym rozpodilom π( )B , B ∈ E , markovs\koho procesu κ( )t , t ≥ 0, z heneratorom
(15):
Πϕ ϕ( ) ˆ ( )x x= 1 , ˆ : ( ) ( )ϕ π ϕ= ∫
E
dx x , 1( )x ≡ 1, x ∈ E.
Potencial\nyj operator (potencial) R0 [ zvidno-obernenym operatorom do
heneratora Q i vyznaça[t\sq rivnistg (dyv. [3])
Q R R Q I0 0= = −Π .
Obçyslennq za formulog (22) dagt\ vyraz (21) heneratora L0
z uraxuvannqm
(19), (20).
Slabka zbiΩnist\ v teoremi 3 ob©runtovu[t\sq na pidstavi asymptotyçnoho
zobraΩennq (dyv. [3])
L Lε ε εϕ ϕ θ ϕ( , ) ( , ) ( ) ( , )u x u x ulv v v, = +0
na zburenyx test-funkciqx
ϕ ϕ εϕ ε ϕε( , ) ( , ) ( , ) ( , )u x u u x u xv v v v, , ,= + +1
2
2
zi znextugçym çlenom
θ ϕε
l x( ) → 0 , ε → 0, ϕ ∈ C Rd3 2, ( × Rd ) ,
ta zastosuvannq model\no] hranyçno] teoremy [3].
1.6. SS u sxemi fazovoho ukrupnennq. SS zada[t\sq rozv’qzkom evolgcij-
noho rivnqnnq v R
d
, d ≥ 1:
dU t
dt
C U t tε
ε
εκ
ε
( )
( );=
2 . (23)
Na vidminu vid pp.?1.5 markovs\kyj proces peremykan\ κε( )t , t ≥ 0, u standart-
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STOXASTYÇNI SYSTEMY Z USEREDNENNQM U SXEMI … 1243
nomu fazovomu prostori staniv ( E, E ) zaleΩyt\ vid maloho parametra seri]
ε → 0 (ε > 0) i zada[t\sq heneratorom
Q x q x P x dy y x
E
ε εϕ ϕ ϕ( ) ( ) ( , ) ( ) ( )= −[ ]∫ .
Stoxastyçne qdro ma[ vyhlqd
P x B P x B P x Bε ε( , ) ( , ) ( , )= + 2
1 , x E∈ , B ∈E .
Stoxastyçne qdro P x B( , ) uzhodΩene z rozweplennqm fazovoho prostoru sta-
niv
E E
k
N
k=
=1
∪ , E Ek k∩ ′ = ∅ , k k≠ ′ ,
takym çynom:
P x E x
x E
x E
k k
k
k
( , ) ( ) :
, ,
, .
= =
∈
∉
1
1
0
Osnovnog umovog fazovoho ukrupnennq [ rivnomirna erhodyçnist\ opornoho
markovs\koho procesu κ0( )t , t ≥ 0, z heneratorom
Q x q x P x dy y x
E
ϕ ϕ ϕ( ) ( ) ( , ) ( ) ( )= −[ ]∫
ta stacionarnymy rozpodilamy
π ρk k kdx q x q dx( ) ( ) ( )= , 1 ≤ k ≤ N,
ρ ρk
E
kB dx P x B
k
( ) ( ) ( , )= ∫ , ρk kE( ) = 1.
Vidomo [3], wo pry dodatkovij umovi
p dx P x E Ek
E
k k
k
= >∫ ρ ( ) ( , \ )1 0 , 1 ≤ k ≤ N,
ma[ misce slabka zbiΩnist\
ˆ ( ) : ˆ ( )κ κ
ε
κε t t t=
⇒v 2
2 , ε → 0. (24)
Tut v( )x = k, x ∈ Ek , — funkciq ukrupnennq. Hranyçnyj markovs\kyj proces
ˆ ( )κ t , t ≥ 0, vyznaça[t\sq porodΩugçog matryceg na ukrupnenomu prostori sta-
niv Ê = {1, 2, … , N}
ˆ ˆ ; , ˆQ q k r Ekr= ∈[ ],
ˆ ˆ ˆq q pkr k kr= , k ≠ r, q̂ q pk k r= , (25)
q̂
p
pkr
kr
k
= , p dx P x Ekr
E
k r
k
= ∫ ρ ( ) ( , )1 .
Sxema fazovoho userednennq dlq SS (22) oznaça[, wo ma[ misce slabka zbiΩ-
nist\
U t V tε( ) ˆ( )⇒ , ε → 0.
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1244 V. S. KOROLGK
Hranyçna systema
ˆ( )V t , t ≥ 0, vyznaça[t\sq rozv’qzkom evolgcijnoho rivnqnnq
z markovs\kymy peremykannqmy
dV t
dt
C V t t
ˆ( ) ˆ ˆ( ); ˆ ( )= ( )κ .
Userednena ßvydkist\ vyznaça[t\sq spivvidnoßennqm
ˆ( ; ) ( ) ( ; )C u k dx C u x
E
k
k
= ∫ π , 1 ≤ k ≤ N.
Flgktuaci] SS u sxemi fazovoho userednennq rozhlqdagt\sq dlq stoxastyçno]
systemy
dV t
dt
C V t tε
ε
ε εκ
ε
( ) ( );=
2 ,
C u x C u x C u xε ε( ; ) ( ; ) ( ; )= + 1
z pryskorenym peremykannqm. Funkciq ßvydkostej C u x( ; ) ma[ poxidnu po u,
rivnomirno obmeΩenu po x. Vidxylennq vyznaçagt\sq rozv’qzkom stoxastyçno]
evolgci]
dV t
dt
C V t t
ˆ ( ) ˆ ˆ ( ); ˆ ( )
ε
ε εκ= ( ).
Flgktuaci] zadagt\sq vyrazom
ζ εε ε ε( ) ( ) ˆ ( )t U t V t= −[ ]−1
, t ≥ 0.
Teorema 4 [10]. V umovax fazovoho ukrupnennq (24), (25) ma[ misce slabka
zbiΩnist\
ε ζε ε− −[ ] ⇒2 0U t V t t( ) ˆ ( ) ( ), ε → 0.
Hranyçnyj dyfuzijnyj proces ζ0( )t , t ≥ 0, vyznaça[t\sq heneratorom
Lt u b u V t t u B V t t u0
0
1
2
ϕ κ ϕ κ ϕ( ) , ˆ( ); ˆ ( ) ( ) ˆ( ); ˆ ( ) ( )= ( ) ′ + ( ) ′′[ ]Tr 0
. (26)
Tut
b u k C k uC k C k0
1
0( , ; ) ( ; ) ( ; ) ( ; )v v v vv= + ′ + ,
C k dx C x R C k
E
k
k
0
0( ; ) ( ) ˜( ; ) ˜ ( ; )v v vv= ′∫ π ,
B u k dx C x R C x
E
k
k
0
0( , ; ) ( ) ˜( ; ) ˜( ; )v v v= ∫ π ,
˜( ; ) : ( ; ) ˆ( ; )C x C x C kv v v= − .
1.6.1. Alhorytm dyfuzijno] aproksymaci]. V teoremi 4 hranyçnyj ope-
rator (26) budu[t\sq z vykorystannqm asymptotyçnoho rozkladu porodΩu-
gçoho?operatora trykomponentnoho markovs\koho procesu ζε( )t , V tε( ) , κε
t : =
: = κ
ε
ε t
2
, t ≥ 0:
L C Cε
εϕ ε ε ϕ( , , ) ( ) ˆ ( , , )u x Q x Q u xv v= + + +[ ]− −2 1
1 .
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STOXASTYÇNI SYSTEMY Z USEREDNENNQM U SXEMI … 1245
Tut Q [ heneratorom opornoho rivnomirnoho erhodyçnoho markovs\koho pro-
cesu κ0( )t , t ≥ 0:
Cε εϕ ε ϕ( ) ( ) ( ; ) ˜( ; ) ( )x u C u x C k u= + −[ ] ′v v ,
˜ ( ) ˜( ; ) ( )Cϕ ϕv v v= ′C k ,
Q x q x P x dy y
Ek
1 1ϕ ϕ( ) ( ) ( , ) ( )= ∫ .
Z uraxuvannqm umov teoremy 4 budu[t\sq rozv’qzok problemy synhulqrnoho
zburennq dlq zrizanoho operatora
L C0
2 1
2 1
ε ε ε= + + +− −Q x Q x Q˜ ( ) ( ) ,
wo da[ moΩlyvist\ obçyslyty hranyçnyj operator L0
trykomponentnoho pro-
cesu ζ0( )t , V t( ), ˆ ( )κ t , t ≥ 0, u vyhlqdi
L
0 ϕ ( , , )u kv = b u k u ku
0( , ) ( , , )v; v′ϕ + 1
2
Tr 0B u k u ku( , ; ) ( , , )v v′′[ ]ϕ +
+
ˆ ( ; ) ( , ; )C v vvk u k′ϕ +
ˆ ( , ; )Q u kϕ v .
Slabka zbiΩnist\ u teoremi 4 ob©runtovu[t\sq za takog Ω sxemog, wo i v teo-
remi 3.
2. Napivmarkovs\ki stoxastyçni systemy. 2.1. Sxema userednennq. SS u
sxemi serij z malym parametrom seri] ε → 0 (ε > 0) zada[t\sq rozv’qzkom evo-
lgcijnoho rivnqnnq v evklidovomu prostori Rd , d ≥ 1:
dU t
dt
C U t tε
ε κ
ε
( )
( );=
. (27)
Vektor-funkciq C u x( ; ) = C u xk ( ; )( , k = 1, d ) , u ∈ Rd , x ∈ E, zadovol\nq[ umo-
vy isnuvannq hlobal\noho rozv’qzku system
dU t
dt
C U t xx
x
( )
( );= ( ), U u Rx
d( )0 = ∈ , x ∈ E.
Proces, wo peremyka[ ßvydkosti κ( )t , t ≥ 0, [ napivmarkovs\kym u stan-
dartnomu fazovomu prostori staniv ( E, E ) i zada[t\sq napivmarkovs\kym qd-
rom?[3]
Q x B t P x B G tx( , , ) ( , ) ( )= , x ∈ E, B ∈ E , t ≥ 0,
qke vyznaça[ jmovirnosti perexodu procesu markovs\koho vidnovlennq (PMV)
κn , τn, n ≥ 0:
Q x B t( , , ) = P +κ θ κn n nB t x1 1∈ ≤ ={ }+, =
= P P+κ κ θ κn n n nB x t x1 1∈ ={ } ≤ ={ }+ .
Stoxastyçne qdro
P x B B xn n( , ) = P +κ κ1 ∈ ={ }
zada[ perexidni jmovirnosti vkladenoho lancgha Markova (VLM) κn = κ τ( )n ,
n ≥ 0; funkci] rozpodilu
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1246 V. S. KOROLGK
G t t x tx n n x( ) = P P+θ κ θ1 ≤ ={ } = ≤{ } , x ∈ E,
zadagt\ rozpodily çasiv perebuvannq θx u stanax x ∈ E. Momenty markov-
s\koho vidnovlennq zadovol\nqgt\ spivvidnoßennq
τ τ θn n n+ += +1 1, n ≥ 0, τ0 0= .
Sxema fazovoho ukrupnennq SS (27) oznaça[ slabku zbiΩnist\ v umovax
rivnomirno] erhodyçnosti napivmarkovs\koho procesu κ( )t , t ≥ 0:
U t V tε( ) ( )⇒ , ε → 0. (28)
Hranyçna determinovana evolgciq V t( ), t ≥ 0, vyznaça[t\sq, qk i v sxemi use-
rednennq z markovs\kymy peremykannqmy (dyv. p.?1.5), rozv’qzkom userednenoho
evolgcijnoho rivnqnnq
dV t
dt
C V t
( )
( )= ( ), (29)
C dx C x
E
( ) ( ) ( ; )v v= ∫ π .
Tut π( )B , B ∈ E , — stacionarnyj rozpodil napivmarkovs\koho procesu κ( )t ,
t ≥ 0, wo zadovol\nq[ spivvidnoßennq
π ρ
( )
( ) ( )
dx
dx m x
m
= ,
m x G t dtx x( ) ( )= =
∞
∫Eθ
0
, G t G tx x( ) : ( )= −1 ,
m dx m x
E
= ∫ ρ( ) ( ) ,
ρ( )B , B ∈ E , [ stacionarnym rozpodilom VLM κn , n ≥ 0, wo vyznaça[t\sq
spivvidnoßennqm
ρ ρ( ) ( ) ( , )B dx P x B
E
= ∫ , ρ( )E = 1.
2.2. Dyfuzijna aproksymaciq z rivnovahog. SS iz napivmarkovs\kymy pe-
remykannqmy u sxemi dyfuzijno] aproksymaci] z rivnovahog rozhlqda[t\sq dlq
centrovanoho ta normovanoho procesu
ζ ε ρε ε( ) ( )t U t= −[ ]−1
, t ≥ 0.
Tut SS U tε( ), t ≥ 0, vyznaça[t\sq rozv’qzkom evolgcijnoho rivnqnnq
dU t
dt
C U t tε
ε ε κ
ε
( )
( );=
2
iz pryskorenym peremykannqm O ε2( ).
Vektor-funkciq ßvydkostej
C u x C u x C u xε ε( ; ) ( ; ) ( ; )= +−1
1 .
RivnovaΩna stala ρ vyznaça[t\sq [dynym rozv’qzkom rivnqnnq
C( )ρ = 0 ,
v qkomu C u( ), u ∈ Rd
, [ userednenog ßvydkistg dlq systemy (29).
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STOXASTYÇNI SYSTEMY Z USEREDNENNQM U SXEMI … 1247
Osnovnog umovog u sxemi dyfuzijno] aproksymaci] z rivnovahog [ asympto-
tyçne zobraΩennq ßvydkostej (por. z (18))
C u x C x uC x u x( ; ) ( ; ) ( ; ) ( , ; )ρ ε ρ ε ρ εθ ρρ
ε+ = + ′ + 0 ,
C u x C u x1 1 1( ; ) ( ) ( , ; )ρ ε ρ θ ρε+ = +
zi znextugçymy çlenamy
θ ρε
1 0( , ; )u x → , ε → 0, i = 0, 1.
Teorema 5 [9]. V umovax erhodyçnoho fazovoho userednennq (24) ta pry do-
datkovyx umovax dyfuzijno] aproksymaci] SS iz napivmarkovs\kymy peremykan-
nqmy ma[ misce slabka zbiΩnist\
ε ρ ζε− −[ ] ⇒1 0U t t( ) ( ), ε → 0.
Hranyçnyj dyfuzijnyj proces ζ0( )t , t ≥ 0, vyznaça[t\sq heneratorom
L0 1
2
ϕ ρ ϕ ρ ϕ( ) ( , ) ( ) ( ) ( )u b u u B u= ′ + ′′[ ]Tr ,
parametr zsuvu
b u C uC b( , ) ( ) ( ) ( )ρ ρ ρ ρ= + ′ +1 0 ,
b dx b x
E
0 0( ) ( ) ( ; )ρ π ρ= ∫ , b x C x R C x0 0( ; ) ( ; ) ( ; )ρ ρ ρρ= ′ ,
C dx C x
E
1 1( ) ( ) ( ; )ρ π ρ= ∫ ,
kovariacijna matrycq
B B B( ) ( ) ( )ρ ρ ρ= +0 1 ,
B dx B xi
E
i( ) ( ) ( ; )ρ π ρ= ∫ , i = 0, 1,
B x C x R C x0 0( ; ) ( ; ) ( ; )ρ ρ ρ= ,
B x x C x C x1( ; ) ( ) ( ; ) ( ; )*ρ µ ρ ρ= ,
µ( ) :
( ) ( )
( )
x
m x m x
m x
= −2
22
,
m x t G dtx x2
2
0
2( ) : ( )= =
∞
∫Eθ .
Vysnovok 3. Hranyçnyj dyfuzijnyj proces ζ0( )t , t ≥ 0, [ procesom Orn-
ßtejna – Ulenbeka, parametry qkoho zaleΩat\ vid toçky rivnovahy ρ.
ZauvaΩennq 5. U vypadku napivmarkovs\kyx peremykan\ z’qvlqgt\sq do-
datkovi çleny v parametrax hranyçnoho procesu. Zokrema, kovariacijna matrycq
ma[ dodanok B1( )ρ , wo zaleΩyt\ vid xarakterystyky rozpodiliv µ( )x : =
: =
m x m x
m x
2
22( ) ( )
( )
−
. Vidomo, wo µ( )x = 0 dlq pokaznykovoho rozpodilu. Vodno-
ças dlq rozpodiliv z µ( )x < 0 matrycq B1( )ρ ne [ dodatno vyznaçenog. Pytan-
nq pro te, çy bude kovariacijna matrycq B( )ρ = B0( )ρ + B1( )ρ dodatno vyznaçe-
nog, zalyßa[t\sq vidkrytym.
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1248 V. S. KOROLGK
2.2.1. ADA v teoremi 5. Vykorystovu[t\sq asymptotyçnyj analiz rozßyre-
noho kompensugçoho operatora PMV
ζ ζ τε ε ε
n n= ( ), κ κ τn n= ( ) , τ ε τε
n n= 2
, n ≥ 0. (30)
Oznaçennq 1 [7]. Rozßyrenyj kompensugçyj operator PMV (30) zada[t\-
sq na test-funkciqx ϕ( , )u x , u ∈ Rd
, u ∈ E, spivvidnoßennqm
Lε ε εϕ ε ϕ ζ κ ϕ ζ κ( , ) ( ) , ( , ) ,u x q x u x u xn n n n= ( ) − = =[ ]−2
1 1E + + , (31)
de, za oznaçennqm, q x( ) : = 1
m x( )
, m x( ) : = Eθx =
0
∞
∫ G t dtx( ) .
OtΩe, Lε ϕ( , )u x = 0, qkwo q x( ) = 0. Neskinçennist\ vyklgça[t\sq umovog
m x( ) ≥ m > 0, x ∈ E.
Vysnovok 4. Rozßyrenyj kompensugçyj operator (31) ma[ analityçne zo-
braΩennq
L G Pε
εϕ ε ϕ( , ) ( ) ( ) ( , )u x q x x I u x= −[ ]−2
, (32)
G Cε ε
ε ρ( ) ( ) ( , )x G dt xx t
=
∞
∫
0
2 ,
sim’q napivhrup Ct xε ρ( , ) , t ≥ 0, x ∈ E, vyznaça[t\sq heneratoramy
Cε ερ ϕ ρ ε ϕ( , ) ( ) ( ; ) ( )x u C u x u= + ′ ,
operator perexidnyx imovirnostej
P ( )ϕ ϕx P x dy y
E
= ∫ ( , ) ( ), x ∈ E.
ZauvaΩennq 6. Rozßyrenyj kompensugçyj operator (32) na test-funkciqx
ϕ( , )u ⋅ ∈ C Rd1( ) ma[ zobraΩennq
L Q Qε εε ε ρ= +− −2 1
1( , )x . (33)
Tut
Qϕ ϕ ϕ( ) ( ) ( , ) ( ) ( )x q x P x dy y x
E
= −[ ]∫
[ heneratorom „suprovodΩugçoho” markovs\koho procesu κ0( )t , t ≥ 0, z inten-
syvnostqmy pokaznykovyx rozpodiliv q x( ) = 1
m x( )
.
Operator zburennq ma[ vyhlqd
Q C G( )
1
1
0( , ) ( ) ( , ) ( , ) ( )ρ ϕ ρ ρ ϕε εx u x x Q u= ,
Q x q x P x dy y
E
0 ϕ ϕ( ) : ( ) ( , ) ( )= ∫ ,
G C( )
ε ε
ερ ρ1
0
2( , ) ( ) ( , )x G t x dtx t
=
∞
∫ .
Rozklad (33) budu[t\sq z vykorystannqm alhebra]çno] totoΩnosti
gp p g p− = − + −1 1 1( )
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STOXASTYÇNI SYSTEMY Z USEREDNENNQM U SXEMI … 1249
ta rivnqnnq dlq napivhrup
d dtt tC CC= .
Zaverßu[t\sq peretvorennq intehruvannqm çastynamy. Pry c\omu vykorystovu-
[t\sq umova Kramera dlq funkcij rozpodilu
sup ( )
x E
ht
xe G dt H
∈
∞
∫ ≤ < + ∞
0
.
Analohiçni peretvorennq pryvodqt\ do takoho asymptotyçnoho rozkladu
rozßyrenoho kompensugçoho operatora (33) na test-funkciqx ϕ( , )u ⋅ ∈ C Rd3( ):
L Q Q P Q Pε εε ε ρ ρ θ ρ= + + +− −2 1
1 2( , ) ( , ) ( , )x x xe . (34)
Tut
Q1( , ) ( ) ( ; ) ( )ρ ϕ ρ ϕx u C x u= ′ ,
Q2 0( , ) ( ) ( , ; ) ( ; ) ( ) ( ; ) ( )ρ ϕ ρ ρ ϕ ρ ϕx u b u x b x u B x u= +[ ] ′ + ′′ ,
znextugçyj çlen
θ ρ ϕε
e x u( , ) ( ) → 0 , ε → 0, ϕ( )u ∈ C Rd3( ).
Teper moΩna zastosuvaty rozv’qzok problemy synhulqrnoho zburennq do
zrizanoho operatora
L Q Q Q0
2 1
1 2
ε ε ε ρ ρ= + +− − ( , ) ( , )x P x P .
Zhidno z lemog 3.2 roboty [3] hranyçnyj operator L0
u rozkladi
L L L0 1
2
2
0ε ε ε εϕ ϕ εϕ ε ϕ ϕ θ ϕ( , ) ( ) ( , ) ( , ) ( ) ( )u x u u x u x u ul= + +[ ] = + (35)
zi znextugçym çlenom θε
l vyznaça[t\sq formulog
L Q Q P Q P0
2 1 0 1Π Π Π Π Π= +( , ) ( , ) ( , )ρ ρ ρx x R x , (36)
de Π — proektor u banaxovomu prostori B( )X , wo vyznaça[t\sq stacionarnym
rozpodilom π( )B , B ∈ E , napivmarkovs\koho procesu κ( )t , t ≥ 0, abo, wo te Ω
same, suprovodΩugçoho markovs\koho procesu κ0( )t , t ≥ 0.
Obçyslennq za formulog (34) pryvodqt\ do vyrazu dlq hranyçnoho opera-
tora L0
, danoho v teoremi 5.
Ob©runtuvannq slabko] zbiΩnosti v teoremi 5 znaçno skladniße, niΩ dlq SS
z markovs\kymy peremykannqmy. Razom z tym zalyßa[t\sq standartna sxema z
vykorystannqm asymptotyçnoho zobraΩennq (34), (35) ta martynhal\no] xarak-
teryzaci] rozßyrenoho PMV (30). Vykorystovu[t\sq takoΩ kryterij kompakt-
nosti Struka – Varadana (dyv. [8]).
2.3. ADA z evolgcijnym userednennqm. SS z evolgcijnym userednennqm
rozhlqda[t\sq dlq centrovanoho ta normovanoho procesu
ζ εε ε( ) ( ) ( )t U t V t= −[ ]−1 , t ≥ 0,
z rozv’qzkom U tε( ), t ≥ 0, evolgcijnoho rivnqnnq
dU t
dt
C U t tε
ε ε κ
ε
( ) ( );=
2
z pryskorenym peremykannqm O( )ε2
. Userednena evolgciq V t( ), t ≥ 0, vyzna-
ça[t\sq rozv’qzkom rivnqnnq
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1250 V. S. KOROLGK
dV t
dt
C V t( ) ( )= ( ).
Osnovna umova u sxemi dyfuzijno] aproksymaci] z evolgcijnym userednennqm
[ takog Ω, qk i v pp. 2.2, a same, magt\ misce asymptotyçni rozklady
C u x C x uC x u xv v v vv+( ) = + ′ +ε ε εθε; ( ; ) ( ; ) ( , ; )0 ,
C u x C x u x1 1 1v v v+( ) = +ε θε; ( ; ) ( , ; )
zi znextugçymy çlenamy
θε
i u x( , ; )v → 0 , ε → 0.
Teorema 6 [13]. V umovax erhodyçnoho fazovoho userednennq (28), (29) ta
pry dodatkovyx umovax dyfuzijno] aproksymaci] SS iz napivmarkovs\kymy pere-
mykannqmy ma[ misce slabka zbiΩnist\
ζ ε ζε ε( ) ( ) ( ) ( )t U t V t t= −[ ] ⇒−1 0
, ε → 0. (37)
Hranyçnyj proces ζ0( )t , t ≥ 0, vyznaça[t\sq heneratorom
Lt u b u V t u B V t u0 1
2
ϕ ϕ ϕ( ) , ( ) ( ) ( ) ( )= ( ) ′ + ( ) ′′[ ]Tr ,
parametr zsuvu
b u C uC b( , ) ( ) ( ) ( )v v v v= + ′ +1 0 ,
b dx b x
E
0 0( ) ( ) ( )v v;= ∫ π , b x C x R C x0 0( ) ˜( ) ˜ ( )v; v; v;v= ′ ,
˜( ) ( ) ( )C x C x Cv; v; v= − ,
kovariacijna matrycq
B B B( ) ( ) ( )v v v= +0 1 ,
B dx B xi
E
i( ) ( ) ( )v v;= ∫ π , i = 0, 1,
B x C x R C x0 0( ; ) ˜( ; ) ˜( ; )v v v= ,
B x x C x C x1( ; ) ( ) ( ; ) ( ; )*v v v= µ ,
µ( ) :
( ) ( )
( )
x
m x m x
m x
= −2
22
.
Vysnovok 5. Dvokomponentnyj proces ζ0( )t , V t( ), t ≥ 0, vyznaça[t\sq he-
neratorom
L0 1
2
ϕ ϕ ϕ ϕ( , ) ( , ) ( , ) ( ) ( , ) ( ) ( , )u b u u B u C uu uv v v v v v vv= ′ + ′′[ ] + ′Tr .
OtΩe, hranyçnyj proces ζ0( )t , t ≥ 0, vyznaça[t\sq rozv’qzkom SS
d t b t V t dt V t dw tζ ζ σ0 0( ) ( ), ( ) ( ) ( )= ( ) + ( ) ,
dV t C V t dt( ) ( )= ( ) .
ZauvaΩennq 5 do teoremy 5 zalyßa[t\sq spravedlyvym i dlq teoremy 6.
2.3.1. ADA v teoremi 6. Qk i v pp.?2.2, vykorystovu[t\sq asymptotyçnyj
analiz rozßyrenoho kompensugçoho operatora PMV
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
STOXASTYÇNI SYSTEMY Z USEREDNENNQM U SXEMI … 1251
ζ ζ τε ε ε
n n= ( ), V Vn n
ε ετ= ( ), κ κ τn n= ( ), τ ε τε
n n= 2 , n ≥ 0.
TverdΩennq 1. Rozßyrenyj kompensugçyj operator zada[t\sq na test-
funkciqx ϕ( , , )u xv , u, v ∈ Rd , x ∈ E, spivvidnoßennqmy
L G Pε
εϕ ε ϕ( , , ) ( ) ( , ) ( , , )u x q x x I u xv v v= −[ ]−2
, (38)
de
G C C Cε ε
ε ε( , ) ( ) ( , )v vx G dt xx t t t=
∞
∫
0
2 .
Sim’q napivhrup Ct xε( , )v , t ≥ 0, v ∈ Rd , x ∈ E, vyznaça[t\sq heneratoramy
C
ε
εϕ ε ϕ( , ) ( ) ( ; ) ( )v vx u C u x u= + ′ .
Napivhrupa Ct , t ≥ 0, vyznaça[t\sq heneratorom Cϕ( )v = C( ) ( )v v′ϕ , a napiv-
hrupa Ct
ε , t ≥ 0, — heneratorom
Cε ϕ ε ϕ( ) ( ) ( ) ( )v v vu C= − ′−1
.
TverdΩennq 1 spravdΩu[t\sq obçyslennqm umovnoho spodivannq
E ϕ ζ κ ζ κε ε ε ε
n n n n n nV u V x+ + +( ) = = =[ ]1 1 1, , , ,v
z uraxuvannqm spivvidnoßen\
ζ ζ ζ θε ε ε ε
n n n+ += + ( )1 1 , θ ε θε
n n+ +=1
2
1: ,
a takoΩ (dyv. (33), (34))
U V un n n
ε ε εεζ ε= + = +v .
OtΩe, ma[mo
E ϕ ζ κ ε κε ε ε ε
n n n n n nV u u V x+ + +( ) = + = =[ ]1 1 1, , , ,v v =
=
0
2 2
∞
∫ G dt x u xx t t t( ) ( , ) ( ) ( , , )C C Cε
ε
ε
ε ϕv v v .
Analohiçno zauvaΩenng 6 otrymu[mo takyj vysnovok.
Vysnovok 6. Rozßyrenyj kompensugçyj operator (38) na test-funkciqx
ϕ( , , )u v ⋅ ∈ Rd( × Rd ) ma[ vyhlqd
L Q Qε εε ε= +− −2 2
1( , )v x . (39)
Zburggçyj operator
Q C C C1
0
2
ε
ε
ε εϕ( , ) ( ) ( ) ( , ) ( )v v vx u G dt x P Ix t t= −[ ]
∞
∫ .
Dali vykorystovu[t\sq alhebra]çna totoΩnist\
abc − 1 = ( a – 1 ) + ( b – 1 ) + ( c – 1 ) + ( a – 1 ) ( b – 1 ) +
+ ( a – 1 ) ( c – 1 ) + ( b – 1 ) (c – 1 ) + ( a – 1 ) ( b – 1 ) ( c – 1 ),
a takoΩ rivnqnnq dlq napivhrup ta intehruvannq çastynamy. V rezul\tati de-
tal\nyx obçyslen\ pryjdemo do asymptotyçnoho rozkladu rozßyrenoho kom-
pensugçoho operatora (39) u takij formi (dyv. (34)).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1252 V. S. KOROLGK
Vysnovok 7. Rozßyrenyj kompensugçyj operator (38) na test-funkciqx
ϕ( , , )u v ⋅ ∈ C Rd3 2, ( × Rd ) ma[ asymptotyçne zobraΩennq
L Q Q Qε εε ε θ= + + +− −2 1
1 2( , ) ( , ) ( , )v v vx P x P xl .
Tut
Q1( , ) ( ) ˜( ; ) ( )v vx u C x uϕ ϕ= ′ ,
Q2 0( , ) ( ) ( , ; ) ( ; ) ( ) ( ; ) ( )v v v vx u b u x b x u B x uϕ ϕ ϕ= +[ ] ′ + ′′ ,
znextugçyj çlen
θ ϕε
l x u( , ) ( , )v v → 0, ε → 0, ϕ ∈ ×( )C R Rd d3 2,
.
Zaverßu[t\sq pobudova hranyçnoho operatora L0
v teoremi 6 obçyslennqm
rozv’qzku problemy synhulqrnoho zburennq dlq zrizanoho operatora
L Q Q Q0
2 1
1 2
ε ε ε= + +− − ( , ) ( , )v vx P x P
z vykorystannqm formuly (36).
Ob©runtuvannq slabko] zbiΩnosti v teoremi 6 realizu[t\sq za takog Ω sxe-
mog, wo i v teoremi 5.
1. Hyxman Y. Y., Skoroxod A. V. Vvedenye v teoryg sluçajn¥x processov. – M.: Nauka, 1977. –
568 s.
2. Anysymov V. V., Lebedev E. A. Stoxastyçni lancghy obsluhovuvannq. Markovs\ki modeli:
Navç. pos. – Ky]v: Lybid\, 1992.
3. Korolyuk V. S., Korolyuk V. V. Stochastic models of systems. – Dordrecht: Kluwer, 1999.
4. Feller V. Vvedenye v teoryg veroqtnostej y ee pryloΩenyq. – M.: Myr, 1984.
5. Spravoçnyk po teoryy veroqtnostej y matematyçeskoj statystyke / Pod red. V. S. Korolg-
ka. – Kyev: Nauk. dumka, 1978.
6. Skoroxod A. V. Asymptotyçeskye metod¥ teoryy stoxastyçeskyx dyfferencyal\n¥x urav-
nenyj. – Kyev: Nauk. dumka, 1987. –?328 s.
7. Svyrydenko M. N. Martynhal\n¥j podxod v predel\n¥x teoremax dlq polumarkovskyx
processov // Teoryq veroqtnostej y ee prymenenyq. – 1986. – S. 540 – 545.
8. Stroock D. W., Varadhan S. R. S. Multidimensional diffusion processes. – Springer, 1979.
9. Korolyuk V. S., Limnios N. Diffusion approximation for evolutionary systems with equilibrium in
asymptotic split phase space // Theory Probab. and Math. Statist. – 2004. – 69.
10. Korolyuk V. S., Limnios N. Average and diffusion approximation for evolutionary systems in
asymptotic split phase space // Ann. Appl. Probab. – 2004. – 14(1). – P. 489 – 516.
11. Korolyuk V. S., Swishchuk A. V. Evolution of systems in random media. – CRC Press, 1995.
12. Skorokhod A. V., Hoppensteadt F. C., Salehi Habib. Random perturbation methods with
applications in science and engineering. – Springer, 2002. – 490 p.
13. Korolyuk V. S., Limnios N. Diffusion approximation with equilibrium of evolutionary systems
switched by semi-Markov processes // Ukr. mat. Ωurn. – 2005. – 57, # 9. – S. 1253 – 1260.
OderΩano 17.06.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
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| id | umjimathkievua-article-3681 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:47:01Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/39/717ed8146563787eabbbfa52f44a7539.pdf |
| spelling | umjimathkievua-article-36812020-03-18T20:02:00Z Stochastic Systems with Averaging in the Scheme of Diffusion Approximation Стохастичні системи з усередненням у схемі дифузійної апроксимації Korolyuk, V. S. Королюк, В. С. We propose a system approach to the asymptotic analysis of stochastic systems in the scheme of series with averaging and diffusion approximation. Stochastic systems are defined by Markov processes with locally independent increments in a Euclidean space with random switchings that are described by jump Markov and semi-Markov processes. We use the asymptotic analysis of Markov and semi-Markov random evolutions. We construct the diffusion approximation using the asymptotic decomposition of generating operators and solutions of problems of singular perturbation for reducibly inverse operators. Запропоновано системний підхід в асимптотичному аналізі стохастичних систем у схемі серій з усередненням та дифузійною апроксимацією. Стохастичні системи задаються марковськими процесами з локально незалежними приростами в евклідовому просторі з випадковими перемиканнями, що описуються стрибковими марковськими та напівмарковськими процесами. Використовується асимптотичний аналіз марковських та напівмарковських випадкових еволюцій. Дифузійна апроксимація будується з використанням асимптотичного розкладу породжуючих операторів та розв'язків проблем сингулярного збурення для звідно-обернених операторів. Institute of Mathematics, NAS of Ukraine 2005-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3681 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 9 (2005); 1235–1252 Український математичний журнал; Том 57 № 9 (2005); 1235–1252 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3681/4088 https://umj.imath.kiev.ua/index.php/umj/article/view/3681/4089 Copyright (c) 2005 Korolyuk V. S. |
| spellingShingle | Korolyuk, V. S. Королюк, В. С. Stochastic Systems with Averaging in the Scheme of Diffusion Approximation |
| title | Stochastic Systems with Averaging in the Scheme of Diffusion Approximation |
| title_alt | Стохастичні системи з усередненням у схемі дифузійної апроксимації |
| title_full | Stochastic Systems with Averaging in the Scheme of Diffusion Approximation |
| title_fullStr | Stochastic Systems with Averaging in the Scheme of Diffusion Approximation |
| title_full_unstemmed | Stochastic Systems with Averaging in the Scheme of Diffusion Approximation |
| title_short | Stochastic Systems with Averaging in the Scheme of Diffusion Approximation |
| title_sort | stochastic systems with averaging in the scheme of diffusion approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3681 |
| work_keys_str_mv | AT korolyukvs stochasticsystemswithaveragingintheschemeofdiffusionapproximation AT korolûkvs stochasticsystemswithaveragingintheschemeofdiffusionapproximation AT korolyukvs stohastičnísistemizuserednennâmushemídifuzíjnoíaproksimacíí AT korolûkvs stohastičnísistemizuserednennâmushemídifuzíjnoíaproksimacíí |