Two-limit problems for almost semicontinuous processes defined on a Markov chain
We consider almost upper-semicontinuous processes defined on a finite Markov chain. The distributions of functionals associated with the exit of these processes from a finite interval are studied. We also consider some modifications of these processes.
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| Datum: | 2007 |
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509399424958464 |
|---|---|
| author | Karnaukh, E. V. Карнаух, Є. В. |
| author_facet | Karnaukh, E. V. Карнаух, Є. В. |
| author_sort | Karnaukh, E. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:20Z |
| description | We consider almost upper-semicontinuous processes defined on a finite Markov chain. The distributions of functionals associated with the exit of these processes from a finite interval are studied. We also consider some modifications of these processes. |
| first_indexed | 2026-03-24T02:40:29Z |
| format | Article |
| fulltext |
UDK 519.21
{. V. Karnaux (Ky]v. nac. un-t im. T. Íevçenka)
DVOHRANYÇNI ZADAÇI
DLQ MAJÛE NAPIVNEPERERVNYX PROCESIV,
ZADANYX NA LANCGHU MARKOVA
Almost upper semicontinuous processes defined on a finite Markov chain are considered. Distributions
of functionals associated with the exit of these processes from a finite interval are investigated. Some
modification of such processes is also considered.
Rassmatryvagtsq poçty poluneprer¥vn¥e sverxu process¥, zadann¥e na koneçnoj cepy Marko-
va. Yzuçagtsq raspredelenyq funkcyonalov, svqzann¥x s v¥xodom πtyx processov yz ohrany-
çennoho yntervala. Rassmatryvaetsq takΩe modyfykacyq dann¥x processov.
Zadaçi, pov’qzani z vyxodom iz intervalu, dlq procesu z nezaleΩnymy pryrosta-
my rozhlqdalys\ u bahat\ox robotax (dyv., napryklad, [1, 2]). Analohiçni prob-
lemy doslidΩuvalys\ dlq procesiv na skinçennomu lancghu Markova (LM) za
umovy napivneperervnosti [3, 4]. Dlq blukan\ na zliçennomu LM dvohranyçnu
zadaçu bulo rozhlqnuto v [5]. U danij statti rozhlqdagt\sq rozpodily deqkyx
funkcionaliv, pov’qzanyx z vyxodom z obmeΩenoho intervalu, dlq procesu z ne-
zaleΩnymy pryrostamy na skinçennomu LM, za umovy, wo cej proces peretyna[
dodatnyj riven\ lyße pokaznykovo rozpodilenymy strybkamy (majΩe napivnepe-
rervnyj proces [6]).
Rozpodily perestrybkovyx funkcionaliv, wo opysugt\sq intehral\nymy riv-
nqnnqmy na pivosi, vyznaçagt\sq proekcijno-faktoryzacijnym metodom iz za-
stosuvannqm (zamist\ kanoniçno]) neskinçenno podil\no] faktoryzaci]. V danij
roboti doslidΩuvani funkcionaly opysugt\sq intehral\nym rivnqnnqm na in-
tervali, qke prodovΩu[t\sq na pivprqmu. Pry rozv’qzanni prodovΩenoho riv-
nqnnq vykorystovu[t\sq metod, rozvynenyj M. H. Krejnom v [7, 8] iz zastosuvan-
nqm imovirnisnyx faktoryzacijnyx totoΩnostej.
Rozhlqnemo dvovymirnyj proces Markova
Z ( t ) = { ξ ( t ) , x ( t ) } , t ≥ 0,
de x ( t ) — skinçennyj nezvidnyj neperiodyçnyj LM z mnoΩynog staniv E ′ =
= { 1, … , m } ta matryceg perexidnyx imovirnostej
P ( t ) = etQ , t ≥ 0, Q = N ( P – I ) ,
N = δ νkr k k r
m
, =1, νk — parametry pokaznykovo rozpodilenyx vypadkovyx vely-
çyn ζk (ças perebuvannq x ( t ) v stani k ), P = pkr — matrycq perexidnyx imo-
virnostej vkladenoho lancgha; π = ( π1 , … , πm ) — stacionarnyj rozpodil, ξ ( t )
— odnoridnyj proces z umovno nezaleΩnymy pryrostamy pry fiksovanyx zna-
çennqx x ( t ) (dyv. [3, c. 13]).
Evolgciq procesu Z ( t ) opysu[t\sq matryçnog xarakterystyçnog funkci[g
(x. f.):
Φt ( α ) = E[ / ]( ), ( ) ( )e x t r x ki tαξ = =0 = Eei tαξ( ) = etΨ( )α , Ψ ( 0 ) = Q.
Dali budemo rozhlqdaty procesy, qki magt\ kumulqntu
Ψ ( α ) = ΛF C C i I I e dK x Qi x
0
1
0
00 1( ) ( )( ) ( )− −( ) + − +−
−∞
∫α α , (1)
de
© {. V. KARNAUX, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4 555
556 {. V. KARNAUX
dK x0( ) = NdF x dx( ) ( )+ Π ; F ( x ) = P{ / }; ( ) ( )χ ζkr x x r x k< = =1 0 ;
χkr — strybky ξ ( t ) v momenty perexodu x ( t ) zi stanu k v r ;
Π( )dx = ΛdF x0( ); F0 ( x ) = δkr kF x0( ) ;
F xk
0( ) — funkci] rozpodilu strybkiv ξ ( t ) , qkwo x ( t ) = k ; Λ = δ λkr k ; λk —
parametry pokaznykovo rozpodilenyx vypadkovyx velyçyn
′ζk (ças miΩ dvoma
susidnymy strybkamy ξ ( t ) , qkwo x ( t ) = k ); C = δkr kc ; ck — parametry po-
kaznykovo rozpodilenyx dodatnyx strybkiv ξ ( t ) , qkwo x ( t ) = k . Proces Z ( t )
z takog kumulqntog [ majΩe napivneperervnym zverxu procesom i vyznaçenyj v
[6, c. 43].
Qkwo çerez θs poznaçyty pokaznykovo rozpodilenu vypadkovu velyçynu z
parametrom s > 0 ( { } ),P θs
stt e t> = ≥− 0 , ne zaleΩnu vid Z ( t ) , to x. f. ξ ( θs )
zapysu[mo tak:
Φ ( s, α ) = Eei sαξ θ( ) = s e dtst
t
0
∞
−∫ Φ ( )α = s sI( ( ))− −Ψ α 1,
Ps = s e P t dtst
0
∞
−∫ ( ) = Φ ( s, 0 ) = s sI Q( )− −1.
Poznaçymo momenty perßoho dosqhnennq dodatnoho (vid’[mnoho) rivnq çerez
τ+( )x = inf : ( ){ }t t x> >0 ξ , x > 0
( τ−( )x = inf : ( ){ }t t x> <0 ξ , x < 0 ) ,
ta moment perßoho vyxodu z intervalu ( x – T, x ) , 0 < x < T, T > 0, çerez
τ ( x, T ) = inf : ( ) ( , ){ }t t x T x> ∉ −0 ξ .
Vvedemo podi]
A x+( ) = { }: ( ( , ))ω ξ τ x T x≥ , A x−( ) = { }: ( ( , ))ω ξ τ x T x T≤ − .
Todi dlq x > 0 moΩemo zapysaty
τ ( x, T ) �
τ τ ω
τ τ ω
+ +
+
− −
−
= ∈
= − ∈
( , ) ( ), ( ),
( , ) ( ), ( ).
x T x A x
x T x T A x
Perestrybky v moment vyxodu z intervalu vyznaçymo tak:
γ T x− ( ) = x T x T− − −ξ τ( )( , ) , γ T x+ ( ) = ξ τ( )( , )+ −x T x .
U perßij çastyni statti znajdemo utoçnennq dlq heneratrys:
B s xT ( , ) = E e x T x x x T r x ks x T− ≥ = =[ ]τ ξ τ τ( , ), ( , ) , ( , ) ( )( ) ( ) / 0 =
= E e A xs x T−
+
+[ ]τ ( , ), ( ) ,
B s xT ( , ) = E e x T x T x x T r x ks x T− ≤ − = =[ ]τ ξ τ τ( , ), ( , ) , ( , ) ( )( ) ( ) / 0 =
= E e A xs x T−
−
−[ ]τ ( , ), ( ) ,
B ( s, x, T ) = Ee s x T− τ( , ) , V ( s, α, x, T ) = E e x Ti
s
sαξ θ τ θ( ), ( , ) >[ ],
V s x T±( , , , )α = E e A xi x s x TTαγ τ± ±−
±[ ]( ) ( , ), ( ) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
DVOHRANYÇNI ZADAÇI DLQ MAJÛE NAPIVNEPERERVNYX PROCESIV … 557
V s x T±( , , , )α = E e A xi x T s x Tαξ τ τ( ( , )) ( , ), ( )
± ±−
±[ ].
Poznaçymo mnoΩynu obmeΩenyx absolgtno intehrovnyx na intervali I ⊂
⊂ ( – ∞, ∞ ) funkcij i, vidpovidno, mnoΩynu ]x intehral\nyx peretvoren\ tak:
Lm I( ) = G x G x G x dx k r mkr kr
I
( ) ( ) : ( ) ; , ,= < ∞ =
∫ 1 ,
� m I0 ( ) = g g g C e G x dx k r mkr kr kr
i x
kr
I
0 0 0 1( ) ( ) : ( ) ( ) ; , ,α α α α= = + =
∫ .
Vvedemo operacig proektuvannq na � m
0 (( , ))−∞ ∞ :
C g I+[ ]( )α = e G x dxi x
I
α ( )∫ , C g I+[ ]( )α 0 = C e G x dxi x
I
+ ∫ α ( ) ,
C g+[ ]−( )α = C g+[ ] −∞( ) ( , )α 0 , C g+[ ]+( )α = C g+[ ] ∞( ) ( , )α 0 .
Slid zaznaçyty, wo V s x T x T xm( , , , ) (( , ))α ∈ −�0 , V s x T xm
+ ∈ ∞( , , , ) ([ , ))α �0 , V ( s,
α, x, T ) ∈ � m x T0 (( , ])−∞ − .
Dali, vvedemo ekstremumy dlq ξ ( t ) i vidpovidni funkci] rozpodiliv:
ξ±( )t = sup (inf) ( )
0≤ ≤u t
uξ , ξ± = sup (inf) ( )
0≤ ≤∞u
uξ ,
ξ( )t = ξ ξ( ) ( )t t− + ,
∨
ξ( )t = ξ ξ( ) ( )t t− − ,
P s x+( , ) = P{ }( )ξ θ+ <s x , x > 0, P s x−( , ) = P{ }( )ξ θs x< , x < 0,
p s+( ) = P{ }( )ξ θ+ =s 0 , q s+( ) = P p ss − +( ),
p s+
∗( ) = p s Ps+
−( ) 1, R s+
∗( ) = C p s+
∗( ).
Lema$1 [3, c. 49]. Dlq dvovymirnoho procesu Markova Z ( t ) = { }( ), ( )ξ t x t ma[
misce faktoryzacijna totoΩnist\
Φ ( s, α ) = Eei sαξ θ( ) =
Φ Φ
Φ Φ
+
− −
−
− +
( , ) ( , ),
( , ) ( , ),
s P s
s P s
s
s
α α
α α
1
1
(2)
de
Φ± ( s, α ) = Eei sαξ θ± ( ) , Φ−( , )s α = Eei sαξ θ( ) , Φ+( , )s α =
∨
Eei sαξ θ( ) .
Teorema$1. Dlq procesu Z ( t ) z kumulqntog (1) B s xT ( , ) vyznaça[t\sq
spivvidnoßennqm
sB s xT ( , ) = s I p s e p s dP s y e C sR s x
x T
C x y T( ( )) ( ) ( , ) ( )( ) ( )− −+
∗ −
+
∗
−∞
−
− − −+
∗
∫ 0 –
– ( ( )) ( ) ( , ) ( )( ) ( )I p s e R s dP s y e dzC s
x
R s z
x z T
C x y z T− +
∗ −
+
∗
−∞
− −
− − − −∫ ∫+
∗
0
0 , 0 < x < T,
(3)
C sT
0 ( ) = ΛF I C e B s z dz
T
Cz T
0
0
0( ) ( , )+
∫ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
558 {. V. KARNAUX
Dovedennq. Zi stoxastyçnyx spivvidnoßen\ dlq τkr x T+ ( , ), de nyΩni indeksy
vidpovidno poznaçagt\ poçatkovyj stan ta stan lancgha x ( t ) v moment vyxodu z
( , )x T x− ( ( ) , ( ( , )) )x k x x T r0 = =τ :
τkr x T+ ( , ) =
′ ′ < >
′ + − ′ < − < <
+ − ′ > − < <
+
+
ζ ζ ζ ξ
ζ τ ξ ζ ζ ξ
ζ τ χ ζ ζ χ
k k k k
k kr k k k k
k jr kj k k kj
x
x T x T x
x T x T x
, , ,
( , ), , ,
( , ), , ,
(4)
vyvodqt\sq rivnqnnq
B s xkr
T ( , ) = λ λ ν
k
s y
x
k
x T
x
k kr
Te dy dF z dF z B s x zk k
0
0 0
∞
− + +
∞
−
∫ ∫ ∫+ −
( ) ( ) ( ) ( , ) +
+ ν λ ν
k
j
m
s y
x T
x
kj jr
Te dy dF z B s x zk k
=
∞
− + +
−
∑ ∫ ∫ −
1 0
( ) ( ) ( , ) , 0 < x < T,
qki moΩna zapysaty v matryçnij formi
( ) ( , )sI N B s xT+ +Λ = ΛF x dK z B s x z
x T
x
T
0 0( ) ( ) ( , )+ −
−
∫ , 0 < x < T,
(5)
B s xT ( , ) = 0, x ≥ T, B s xT ( , ) = I, x < 0.
Pislq zaminy B s xT ( , ) = I B s xT− ( , ) z (5) dlq B s xT ( , ) , 0 < x < T, otryma[mo
rivnqnnq
( ) ( , )sI N B s xT+ +Λ = ( ) ( ) ( , )sI Q dK z B s x zT− + −
−∞
∞
∫ 0 , 0 < x < T,
qke pislq prodovΩennq na pivvis\ x > 0 bude maty vyhlqd
( ) ( , )sI N B s xT+ +Λ = ( ) ( ) ( , ) ( ) { }sI Q dK z B s x z e C s I x TT Cx T− + − + >
−∞
∞
−∫ 0 0 . (6)
Poznaçymo Cε ( x ) = e I xx− >ε { }0 i rozhlqnemo zamist\ (6) rivnqnnq dlq
Y T s xε( , , ), x > 0, ε > 0:
( ) ( , , )sI N Y T s x+ +Λ ε = ( ) ( )sI Q C x− ε +
+
−∞
∞
−∫ − + >dK z Y T s x z e C s I x TCx T
0 0( ) ( , , ) ( ) { }ε . (7)
Zastosovugçy intehral\ne peretvorennq po x do (7), otrymu[mo rivnqnnq
( ( )) ˜ ( , , )sI Y T s− Ψ α αε = ( ) ( ) { }sI Q e e dz e e C s I z T dzi z z i z Cz T− + >
∞
−
∞
−∫ ∫
0 0
0
α ε α –
– ˜ ( ) ˜ ( , , )K Y T s0 α αε[ ]− ,
(8)
˜ ( )K0 α =
0
0
∞
∫ e dK zi zα ( ), ˜ ( , , )Y T sε α =
0
∞
∫ e Y T s z dzi zα
ε( , , ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
DVOHRANYÇNI ZADAÇI DLQ MAJÛE NAPIVNEPERERVNYX PROCESIV … 559
Vykorystovugçy (2), pislq proektuvannq []+ z (8) oderΩu[mo spivvidnoßennq
sY T s˜ ( , , )ε α = Φ Φ+
− −
∞
−−
∫( , ) ( , ) ( )s P s sI Q e e dzs
i z zα α α ε1
0
+
+
0
0
∞
−
+
∫ >
e e C s I z T dzi z Cz Tα ( ) { } ,
pislq obernennq qkoho ma[mo
sY T s xε( , , ) =
0
1
0x
s
x y zdP s z P dP s y sI Q e∫ ∫+
−
−∞
− − − −−( , ) ( , )( ) ( )ε +
+
0
1
0
0
x
s
x z T
C x y z TdP s z P dP s y e C s∫ ∫+
−
−∞
− −
− − − −( , ) ( , ) ( )
min{ , }
( ) . (9)
Oskil\ky Y T s x B s xT
ε( , , ) ( , )→ pry ε → 0, 0 < x < T, to z (9) vyplyva[
s B s xT ( , ) =
0 0
1
0
x x
s
x z T
C x y z TdP s z sI Q dP s z P dP s y e C s∫ ∫ ∫+ +
−
−∞
− −
− − − −− +( , )( ) ( , ) ( , ) ( )( ) .
Vraxovugçy, wo [6, c. 45]
P{ }( )ξ θ+ >s x = ( )( ) ( )I p s e PR s x
s− +
∗ − +
∗
, x > 0, (10)
otrymu[mo (3).
Dali navedemo bez dovedennq analoh totoΩnostej Peçers\koho (dyv. [2,
c. 108]).
Lema$2. Dlq Z ( t ) magt\ misce nastupni totoΩnosti:
V ( s, α, x, T ) = Φ( , ) ( , , , ) ( , , , )s I V s x T V s x Tα α α− −( )+ − , (11)
V ( s, α, x, T ) = Φ Φ+
− −
+ − ∞
−[ ]( , ) ( , )( ( , , , ))
[ , )
s P s I V s x Ts x T
α α α1 , (12)
V ( s, α, x, T ) = Φ Φ−
− +
− −∞
−[ ]( , ) ( , )( ( , , , ))
( , ]
s P s I V s x Ts x
α α α1 . (13)
Poznaçymo spil\nyj rozpodil { }( ), ( ), ( )ξ θ ξ θ ξ θs s s
+ −
tak:
H T x ys( , , ) = P ξ θ ξ θ ξ θ θ( ) , ( ) , ( ) , ( ) ( )/s s s sy x x T x r x k< < > − = ={ }+ − 0 =
= P ξ θ τ θ( ) , ( , )s sy x T< >{ } .
Teorema$2. Dlq procesu Z ( t ) z kumulqntog (1) spil\ni rozpodily { ( , ),τ+ x T
γ T x+ ( )} ta { }( , ), ( ( , ))τ ξ τ+ +x T x T vyznaçagt\sq spivvidnoßennqmy
V s x T+( , , , )α = B s x C C i IT ( , ) ( )− −α 1, 0 < x < T,
(14)
V s x T+( , , , )α = e V s x Ti xα α+( , , , ).
X. f. ξ θ( )s do momentu vyxodu z intervalu ma[ vyhlqd
V s x T( , , , )α = Φ Φ+
− − −
− ∞
− −( )[ ]( , ) ( , ) ( , ) ( )
[ , )
s P s I e B s x C C i Is
i x T
x T
α α αα1 1 , (15)
vidpovidnyj rozpodil ma[ wil\nist\ ( , )x T y x y− < < ≠ 0
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
560 {. V. KARNAUX
hs ( T, x, y ) = ∂
∂y
H T x ys( , , ) =
= p s P s y I y I p s e R s dP s z
x T
y
R s y z
+
∗ −
+
∗
−
− −
+
∗ −′ < + − ∫ +
∗
( )( ( , )) { } ( ( )) ( ) ( , )
min( ; )
( )( )0
0
–
–
p s dP s z B s x Ce I p s e
y x
T C y x z
y x T
R s
+
∗
−∞
−
− − − −
+
∗
− −
−∫ ∫− − +
∗
( ) ( , ) ( , ) ( ( ))( )
( )
( )
0
v ×
×
−∞
− −
+
∗ − − − − −∫
y x
T C y x zR s dP s z B s x Ce d
v
v v( ) ( , ) ( , ) ( ) (16)
z atomom v nuli
P ξ θ τ θ( ) , ( , )s sx T= >{ }0 =
= s sI N x r x k Ikr+ − = = ={ } −( )( )−Λ P χ ζ0 01
1, ( ) ( )/ . (17)
Imovirnist\ nevyxodu z intervalu ( , )x T x− vyznaça[t\sq spivvidnoßennqm
P τ θ( , )x T s>{ } =
x T
x
sd H T x y
−
∫ ( , , ). (18)
Dlq heneratrys τ ( x, T ) ta τ
–
( x, T ) ma[mo
B ( s, x, T ) = I x T Ps s− >{ } −P τ θ( , ) 1, 0 < x < T,
(19)
BT ( s, x ) = B ( s, x, T ) – B
T
( s, x ) , 0 < x < T.
Dovedennq. Zi stoxastyçnyx spivvidnoßen\ (4) dlq τ+( , )x T ta
γ T krx+ ( ) �
ξ ζ ζ ξ
γ ξ ζ ζ ξ
γ χ ζ ζ χ
k k k k
T k kr k k k
T kj jr k k kj
x
x x T x
x x T x
, , ,
( ) , , ,
( ) , , ,
′ < >
− ′ < − < <
− ′ > − < <
+
+
dlq γ T x+ ( ) vyvodymo rivnqnnq
( ) ( , , , )sI N V s x T+ + +Λ α = ΛF e C C i ICx
0
10( ) ( )− −− α + dK z V s x z T
x T
x
0( ) ( , , , )+
−
−∫ α .
Vraxovugçy (5), otrymu[mo perßu formulu z (14). Druha vyplyva[ z oznaçennq
γ T x+ ( ). Spivvidnoßennq (15) vyplyva[ z (12). Pislq obernennq po α z (15) otry-
ma[mo (16) ta (17).
Vykorystovugçy (16), (17) ta analoh formul Bratijçuka [1, c. 187], moΩemo
oderΩaty matryçnyj analoh dlq heneratrys spil\nyx rozpodiliv { ( , ),τ+ x T
ξ τ( )}( , )+ x T ta { ( )}( , ), ( , )τ ξ τ− −x T x T .
Teorema$3. Dlq Z ( t ) magt\ misce nastupni spivvidnoßennq:
s e x T zs x TE − ++
>[ ]τ ξ τ( , ), ( ( , )) =
x T
x
sd H T x y K z y
−
∫ −( , , ) ( )0 , z > x , (20)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
DVOHRANYÇNI ZADAÇI DLQ MAJÛE NAPIVNEPERERVNYX PROCESIV … 561
s e x T zs x TE − −−
<[ ]τ ξ τ( , ), ( ( , )) =
x T
x
sd H T x y K z y
−
∫ −( , , ) ( )0 , z < x – T , (21)
K x0( ) =
x
dK y
∞
∫ 0( ) , x > 0, K x0( ) =
−∞
∫
x
dK y0( ), x < 0.
Dovedennq. Zhidno z [4, c. 469]
Ei
s x Te f x x T x x T f x i− −[ ] −τ ξ τ τ( , ) ( )( ( , )), ( ( , )) ( , ) = Ei
st
x T
e g x t x t dt− −∫ ( ( ), ( ))
( , )
ξ
τ
0
, (22)
de f — obmeΩena funkciq, g = Af – sf, A — henerator napivhrupy, qkyj vyzna-
ça[t\sq kumulqntog Ψ ( α ) . Dlq pravo] çastyny rivnqnnq ma[mo
Ei
x T
ste g x t x t dt
0
τ
ξ
( , )
( ( ), ( ))∫ − − =
j
m
st
ie g x t j x T t x t j dt
=
∞
−∑ ∫ − > =[ ]
1 0
E ( ( ), ), ( , ) , ( )ξ τ =
=
j
m
x T
x
s ijs g x y j d H T x y
=
−
−
∑ ∫ −
1
1 ( , ) ( ( , , )) . (23)
Qkwo prypustyty, wo f ( x, i ) = I x z ir{ }≥ − δ , z > 0, i, r ∈ E ′ , to
Ei
s x Te f x x T x x T f x i− −[ ] −τ ξ τ τ( , ) ( )( ( , )), ( ( , )) ( , ) =
= Ei
s x T
Te x z x x T r− + ≥ =[ ]τ γ τ( , ), ( ) , ( ( , )) , (24)
g ( x, j ) =
−∞
∞
∫ − ≤ −I x y z dK yjr{ } ( )0 = K x zjr
0 ( )+ . (25)
Pidstavlqgçy (24) ta (25) v (22) ta beruçy do uvahy (23), otrymu[mo
Ei
s x T
Te x z x x T r− + ≥ =[ ]τ γ τ( , ), ( ) , ( ( , )) = s K x y z d H T x y
j
m
x T
x
jr
s ij
−
= −
∑ ∫ − +1
1
0 ( ) ( ( , , )) .
(26)
Vykorystovugçy oznaçennq γ T x+ ( ), z (26) oderΩu[mo (20). Analohiçno znaxody-
mo formulu (21).
Rozhlqnemo povedinku funkcij B s xT ( , ) ta H T x ys( , , ) pry s → 0. Pozna-
çymo M ( y ) = lim ( ) ( , )s s p s P s y→
−
+
∗ −
0
1 ; isnuvannq ci[] funkci] vyplyva[ z takyx
mirkuvan\. Zhidno z [6, c. 46] dlq heneratrysy ξ θ( )s ma[mo
lim ( ) ( )
s
rs p s e s
→
−
+
∗
0
1 E ξ θ = – ( )( ) ( ) ( )p C rI C rI ir+
∗ − −− − −0 1 1Ψ .
Todi moΩemo vyznaçyty M ( y ) qk funkcig, dlq qko]
−∞
∫
0
e dM yry ( ) = – ( )( ) ( ) ( )p C rI C rI ir+
∗ − −− − −0 1 1Ψ .
Krim toho, z [6, c. 41] vyplyva[
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
562 {. V. KARNAUX
p+
∗( )0 = I x r x k− < ∞ = ={ }( )+ +P τ τ( ) , ( ( )) ( )/0 0 0 .
Naslidok$1. Dlq procesu Z ( t ) ma[mo
lim ( , , )
s
ss h T x y
→
−
0
1 = ′ < + − +
∗
−
− −∫ +
∗
M y I y I p e CdM z
x T
y
R y z( ) { } ( ( )) ( )
min( ; )
( )( )0 0
0
0 –
–
( ( )) ( ) ( )
( )
( ) ( )
I p e C dM z B x Ce d
y x T
R
y x
T C y x z
− +
∗
− −
−
−∞
− −
∫ ∫+
∗ − − − −
0
0
0 v
v
v
v –
–
−∞
−
∫
− − −
y x
TdM z B x Ce
C y x z
( ) ( )
( )
, (27)
B xT ( ) = lim ( , )
s
TB s x
→0
= ( ( )) ( ) ( )( ) ( )I p e dM y e CR x
x T
C x y T− −+
∗ −
−∞
−
− −+
∗
∫0 00
0 –
– ( ( )) ( ) ( )( ) ( )I p e C dM y e dzC
x
R z
x z T
C x y z T− +
∗ −
−∞
− −
− − −∫ ∫+
∗
0 0
0
0
0 , (28)
CT
0 0( ) = ΛF I C e B z dzCz T
T
0
0
0( ) ( )+
∫ .
Prypustymo dali, wo χkr ≡ 0. Qkwo x ( t ) = k, k = 1, … , m, to poklademo
ξ ( t ) =
n t
n
k
n t
n
k
k k≤ ′ ≤
∑ ∑′ −
ν ν
ξ ξ
( ) ( )
,
de ′νk t( ) ta νk t( ) — procesy Puassona z intensyvnostqmy λk
1
ta λk
2
vidpovid-
no,
′ξn
k
ta ξn
k
— nezaleΩni dodatni vypadkovi velyçyny;
′ξn
k
pokaznykovo roz-
podileni z parametramy ck
, ξn
k
magt\ dovil\nyj rozpodil z obmeΩenym
matematyçnym spodivannqm mk
. Zrozumilo, wo v c\omu vypadku proces Z ( t ) =
= { ξ ( t ) , x ( t ) } [ majΩe napivneperervnym zverxu z kumulqntog
Ψ ( α ) = Λ ΠF C C i I I e I dx Qi x
0
1
0
0( ) ( ) ( ) ( )( )− − + − +−
−∞
∫α α ,
de
Λ = δ λ λkr k k( )1 2+ , F0 0( ) = δ λ λ λkr k k k
1 1 2/ ( )+ , Π( )dx = ΛF dF x0 0
10( ) ( ),
F0 0( ) = I F− 0 0( ), F x0
1( ) = δ ξkr n
k xP{ }− < , x < 0.
Rozhlqnemo proces ηB u t, ( ), qkyj vyznaça[t\sq stoxastyçnymy spivvidnoßen-
nqmy
ηB u
kr t, ( ) =
u t t T
B t T T T
t T t T x T j T
kr
B B
jr
j
+ <
∈ < ∞
− > = < ∞
−
ξ
η
ξ
( ), ,
, ( , ), ,
( ), , ( ) , ,
,
1
1 2 1
2 2 2 1
1
(29)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
DVOHRANYÇNI ZADAÇI DLQ MAJÛE NAPIVNEPERERVNYX PROCESIV … 563
de verxni indeksy kr oznaçagt\, wo x ( t ) = r, x ( 0 ) = k ; T1 � τ+( )v , v = B – u
ta T2 � T1 + ζ∗
, ζ∗ — moment perßoho vid’[mnoho strybka, wo ne zaleΩyt\ vid
T1
.
Proces ηB u t, ( ) nazyva[t\sq procesom ryzyku v markovs\komu seredovywi zi
stoxastyçnog funkci[g premij ta obmeΩenym rezervom (dyv. [9 – 11]). Rozhlq-
nemo takoΩ dyvidendnyj proces Y tB u, ( ) � u t tB u+ −ξ η( ) ( ), (dyv. [9, c. 169]).
Teorema$4. Rozpodil η θB u s, ( ) vyznaça[t\sq x. f.
ΦB u s, ( , )α = Ee
i B u sαη θ, ( )
=
=
e i I p s e I p s e R s R s i I si B i R sCα αα α α( ) ( ) ( )( ) ( ) ( ) ( ) ( , )( )− − −( ) −+
∗ −
+
∗ −
+
∗
+
∗ − −+
∗v v 1Φ +
+
( )( ) ( ( ) ) ( ) ˜ ( , )( )I p s e sI F Q se F sR s i B
B− + − +( )+
∗ − −+
∗ v Λ Λ Φ0
1
00 0α α , (30)
˜ ( , )ΦB s α = dF z sB B z0
1
0
( ) ( , ),
−∞
+∫ Φ α = e F sI F Q p si Bα ( ) ( )( ) ( ( ) ) ( )Λ Λ0
1
0
10 0−
+
∗ −+ − ×
× ( ) ( ) ( ) ( ( ) )( ) ( )sI Q dz I p s e s sI F QR s z+ − − + −(−
−∞
+
∗ −∫ +
∗
Λ Π Λ1
0
0
10 +
+ e C i I I p s e C p s R s i I si z R s zα α α α( ) ( ) ( )( ) ( ) ( ) ( , )( )− − −( ) − )+
∗
+
∗
+
∗ − −+
∗ 1Φ . (31)
Qkwo m1
0 = π λ λk k k k kk
m
c m( / )1 2
1
−=∑ > 0, to dlq ηB u, = lim ( ),s B u s→0η θ
ΦB u, ( )α = Ee
i B uαη , = e I p e p Q dzi B Rα ( )( ) (( ) ( )( )− −+
∗ −
+
∗ −
−∞
+
∗
∫0 0 1
0
v Λ Π ×
× − + − − + −( )( )−
+
∗ − − −+
∗
e I p e F Q C C i Ii z R zα α α αΨ Λ Ψ1 0
0
1 1 10 0( ) ( ) ( ( ) ) ( )( ) ( )( ) , (32)
P{ },ηB u B= = ( )( ) ( )( )I p e p QR− −+
∗ −
+
∗ −+
∗
0 0 1v Λ ×
×
−∞
+
∗ −∫ − −+
∗
0
0
0
10 0Π Λ( ) ( ) ( ( ) )( ) ( )dz I p e F QR z . (33)
Dlq dyvidendnoho procesu YB u s, ( )θ ma[mo
Ee
YB u s−µ θ, ( )
=
I I p s e I R s PR s
s− − +( )+
∗ −
+
∗ −+
∗
( ) ( )( ) ( )( ) vµ µ 1 , (34)
P{ }, ( )YB u sθ = 0 = P{ }( )ξ θ+ <s v = P I p s e Ps
R s
s− − +
∗ − +
∗
( )( ) ( )v . (35)
Dovedennq. Z (29) vyplyva[, wo x. f. ηB u t, ( ) zadovol\nq[ intehral\ne spiv-
vidnoßennq, qke pislq peretvorennq Laplasa – Karsona nabyra[ vyhlqdu
ΦB u s, ( , )α =
e ei u i
s
sα αξ θ ξ θE ( ), ( )+ <[ ]v +
+
e e T I e P e T e si B sT s
s
sT s
B
α ζ ζ αE E E E− − − −<[ ] − + < ∞[ ]∗ ∗1 1
1 1, , ˜ ( , )( )v Φ . (36)
Zhidno z [3, c. 50; 6, c. 43] ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
564 {. V. KARNAUX
E e TsT− < ∞[ ]1
1, = ( )( ) ( )I p s e R s− +
∗ − +
∗ v ,
Ee s− ∗ζ = ( ( ) ) ( )sI F Q F+ − −Λ Λ0
1
00 0 , (37)
E ei
s
sαξ θ ξ θ( ), ( )+ <[ ]v =
E e P si
s s
sαξ θ ξ θ α
+ + − −<[ ]( ), ( ) ( , )v 1Φ =
=
( ) ( ) ( )( ) ( ) ( ) ( ) ( , )( ( ))C i I p s I p s e R s R s i I si I R s− − −( ) −+
∗
+
∗ −
+
∗
+
∗ − −+
∗
α α αα v 1Φ .
Pidstavlqgçy poperedni formuly v (36), otrymu[mo (30) ta
˜ ( , )ΦB s α = e I dF z I p s ei B R s zα − −
−∞
+
∗∫ +
∗
0
0
1( ) ( )( ) ( ) ×
× ( ( ) ) ( ) ( )( )sI F Q F e dF z i Ii z C+ −
−
−
−
−∞
∫Λ Λ0
1
0
1 0
0
10 0 α α –
–
−∞
+
∗
+
∗
+
∗ − −∫ −
−+
∗
0
0
1 1dF z I p s e C p s R s i I sR s z( ) ( ) ( ) ( ) ( , )( ) ( )( ) α αΦ –
–
−∞
+
∗ −∫ − + −
+
∗
0
0
1
0
10dF z I p s e s sI F QR s z( ) ( ) ( ( ) )( ) ( ) Λ . (38)
Zi stoxastyçnyx spivvidnoßen\ dlq τ+( )x (dyv. [3, c. 62]) vyplyva[ rivnqnnq
dlq p s+
∗ ( ):
( ) ( )( )sI Q I p s+ − − +
∗Λ = Λ ΛF F dF z I p s eC p s z
0 0
0
0
10 0( ) ( ) ( ) ( )( ) ( )+ −
−∞
+
∗∫ +
∗
. (39)
Pidstavlqgçy (39) v (38), oderΩu[mo (31). ZauvaΩymo, wo dlq m1
0 > 0 ξ+ ma[
vyrodΩenyj rozpodil, tomu perßyj dodanok formuly (36) prqmu[ do 0 pry
s → 0. Oskil\ky ( )I e s− − ∗E ζ = s sI F Q Ps( ( ) )+ − − −Λ 0
1 10 , to druhyj dodanok v
(36) takoΩ prqmu[ do 0. Wodo tret\oho zaznaçymo, wo zhidno z teoremogT3 [6]
ta perßog formulog v (2) ma[mo
lim ( )( )
s
s p s
→
+
∗ −
0
1 = p+
∗ ,
lim ( ) ( ) ( , )( )
s
s p s R s i I s
→
−
+
∗
+
∗ − −−
0
1 1α αΦ = lim ( ( ))( )
s
C i I sI
→
− −− −
0
1 1α αΨ =
= – ( ) ( )C i I− − −α α1 1Ψ .
Takym çynom, z (36) pislq perexodu do hranyci pry s → 0 vyplyva[ (32). Spiv-
vidnoßennq (34), (35) vyplyvagt\ iz zobraΩennq Y tB u
kr
, ( ) �
max , ( )0 ξkr t+ −( )v .
1. Bratyjçuk N. S., Husak D. V. Hranyçn¥e zadaçy dlq processov s nezavysym¥my pryrawe-
nyqmy. – Kyev: Nauk. dumka, 1990. – 264 s.
2. Peçerskyj E. A. Nekotor¥e toΩdestva, svqzann¥e s v¥xodom sluçajnoho bluΩdanyq yz ot-
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DVOHRANYÇNI ZADAÇI DLQ MAJÛE NAPIVNEPERERVNYX PROCESIV … 565
rezka y poluyntervala // Teoryq veroqtnostej y ee prymenenyq. – 1974. – 19, # 1. –
S.T104 – 119.
3. Husak D. V. Hranyçni zadaçi dlq procesiv s nezaleΩnymy pryrostamy na skinçennyx lan-
cghax Markova ta dlq napivmarkovs\kyx procesiv. – Ky]v: In-t matematyky NAN Ukra]ny,
1998. – 320 s.
4. Korolgk V. S., Íurenkov V. M. Metod rezol\vent¥ v hranyçn¥x zadaçax dlq sluçajn¥x
bluΩdanyj na cepqx Markova // Ukr. mat. Ωurn. – 1977. – 29, # 4. – S.T464 – 471.
5. Bratijçuk M. S. Vlastyvosti operatoriv, zv’qzanyx z markovs\kymy adytyvnymy procesamy.
I, II // Teoriq jmovirnostej ta mat. statystyka. – 1996. – # 55. – S.T20 – 29; 1997. – # 57. –
S.T1 – 9.
6. Gusak D. V., Karnaukh E. V. Matrix factorization identity for almost semi-continuous processes on
a Markov chain // Theory Stochast. Process. – 2005. – 11(27), # 1-2. – P. 41 – 47.
7. Krejn M. H. Yntehral\n¥e uravnenyq na poluprqmoj s qdrom, zavysqwym ot raznostej ar-
humentov // Uspexy mat. nauk. – 1958. – 13, # 5. – S.T3 – 120.
8. Hoxberh Y. C., Krejn M. H. System¥ yntehral\n¥x uravnenyj na poluprqmoj s qdramy, za-
vysqwymy ot raznostej arhumentov // Uspexy mat. nauk. – 1958. – 13, # 2. – S.T3 – 72.
9. Bühlmann H. Mathematical methods in risk theory. – Berlin: Springer, 1970. – 330 p.
10. Husak D. V. Pro modyfikaci] procesiv ryzyku // Teoriq jmovirnostej ta mat. statystyka. –
1997. – # 56. – S.T87 – 95.
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OderΩano 01.02.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
|
| id | umjimathkievua-article-3330 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:40:29Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6e/84cab0e7426cc50a6743ee9e66ba176e.pdf |
| spelling | umjimathkievua-article-33302020-03-18T19:51:20Z Two-limit problems for almost semicontinuous processes defined on a Markov chain Двограничні задачі для майже напівнеперервних процесів, заданих на ланцюгу Маркова Karnaukh, E. V. Карнаух, Є. В. We consider almost upper-semicontinuous processes defined on a finite Markov chain. The distributions of functionals associated with the exit of these processes from a finite interval are studied. We also consider some modifications of these processes. Рассматриваются почти полунепрерывные сверху процессы, заданные на конечной цепи Маркова. Изучаются распределения функционалов, связанных с выходом этих процессов из ограниченного интервала. Рассматривается также модификация данных процессов. Institute of Mathematics, NAS of Ukraine 2007-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3330 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 4 (2007); 555–565 Український математичний журнал; Том 59 № 4 (2007); 555–565 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3330/3404 https://umj.imath.kiev.ua/index.php/umj/article/view/3330/3405 Copyright (c) 2007 Karnaukh E. V. |
| spellingShingle | Karnaukh, E. V. Карнаух, Є. В. Two-limit problems for almost semicontinuous processes defined on a Markov chain |
| title | Two-limit problems for almost semicontinuous processes defined on a Markov chain |
| title_alt | Двограничні задачі для майже напівнеперервних процесів, заданих на ланцюгу Маркова |
| title_full | Two-limit problems for almost semicontinuous processes defined on a Markov chain |
| title_fullStr | Two-limit problems for almost semicontinuous processes defined on a Markov chain |
| title_full_unstemmed | Two-limit problems for almost semicontinuous processes defined on a Markov chain |
| title_short | Two-limit problems for almost semicontinuous processes defined on a Markov chain |
| title_sort | two-limit problems for almost semicontinuous processes defined on a markov chain |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3330 |
| work_keys_str_mv | AT karnaukhev twolimitproblemsforalmostsemicontinuousprocessesdefinedonamarkovchain AT karnauhêv twolimitproblemsforalmostsemicontinuousprocessesdefinedonamarkovchain AT karnaukhev dvograničnízadačídlâmajženapívneperervnihprocesívzadanihnalancûgumarkova AT karnauhêv dvograničnízadačídlâmajženapívneperervnihprocesívzadanihnalancûgumarkova |