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) . .!. X , X Rk. 1.1. X : D ! X ( X - - ) M | X . X Aj fMg- , A := fX ;1(M)g: fX ;1 (M)g = X ;1 ( fMg). 2 2 ,
D := fD X : X ;1 (D) 2 Ag - , . . -
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, t 2 T , , A { -
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A = fXt;1(Bt) Bt 2 Bt t 2 T g Xt 2 FjBt t 2 T , , , A F . 7
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. ? X = X (t !) $+ S $, $ ; T D Xt, $ $ X (t !) t2T & t 2 T $ F j Bt- .,. Xt. I X (t !) ! , $ # $ . J ! . .
X (t) Xt . C T R, t 2 T $ ( ", T Z). C T Rd, d > 1, . / . . ( T R), !
. & , ,
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( ( , () , , (, , .. / . < , ; ; ( . .
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XU =
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y, U y(t) 2 Xt t 2 U ( ,
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. 1.1
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6 y 2 XU |
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U
q
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-
. 1.2
$ % Bt (. . 1.2). & , CU (t Bt) = U;1t Bt, U t := U ftg t 2 U .
XU $ " - BU , ! , . . BU = fU;t1Bt t 2 U g: (1.2)
1.3 , BU $ $% & U t t 2 U . B ( 1.5) , - % BU & $ & $ , ..
U;1V BV , V {
U , $ BV 2 BV . @ 1.4. '.#. X = X (t !), . . F j Bt- .. X (t ), t 2 T , F j BT - X : D ! XT , "( ! 2 D " X ( ! ), . . X X ( ! ): ! ;! ) , (1.3) F jBt - ..
F j BT -
(1.3) ,
fX (t ) t 2 T g
2 ! CT (t Bt), t 2 T , Bt 2 Bt. ? = f! : X (t !) 2 Btg 2 F , X (t ) F j Bt-
X;1(CT (t Bt))
. < (1.2)
1.2. < , X (1.3), X (t !) = T t X(!). ) 1.1
,
U;1V (BV ) BU V U T (1.4)
. . U V 2 BU jBV ( ! XV ! XU ). < ,
(
( - ) . 2 & , $ 1.4 % $, %& $ $ $ (D F ) P . * X : D ! X F j B- .!. (D F ) P . * X B , PX PX ;1 , PX ;1 (B ) := P (X ;1 (B )) B 2 B: (1.5)
, (1.5) B. ) 1.4 , .. X (t), t 2 T , $ PX (XT BT ). L . !. X L(X) ("law" { ). * , X , 9
'' q $$ ' $$ && %% & %% X (! )q B 2B
! X ;1(B) 2F
() F P )
(X B PX )
. 1.3
. . X = fXt t 2 T g. ', L(Xt t 2 T ) = L(Yt t 2 T ) , PX = PY - BT ( Y = fYt t 2 T g). @ % , & $+ P (! : X ( !) 2 B ), B 2 BT .
,$ < ; % $ % $+ $ $ - % BT . <
CT =
J 2F (T )
T;1J BJ :
@ 1.5. T | N (T ) | -
BT = fCT g.
2 * ET :=
T.
BT =
U 2N (T )
BT U
BT U = T;1U BU
(1.6)
S B , . .
X , TU T
U 2N (T )
( - 6 - BT U . * , ET - . < , XT = T;U1 XU 2 BT U U T , A = T;1U B , B 2 BU , XT n A = T;U1 (XU n B ) 2 BT U . L An 2 ET , n 2 N. ? An 2 BT Un S1 Un 2 N (T ). / , An 2 BT M , M = Un 2 N (T ), n=1
. . BT V BT U V U T . B
, V U T T V = U V T U , ! T;1V = T;U1 U;1V (1.7)
S1
BT V = T;1V BV = T;1U (U;V1 BV ) T;U1 BU = BT U (1.4). ) , An 2 BT M ET . n=1 < , BT U BT T = BT U T (T T |
XT ). *! ET BT . / , t 2 T , U 2 N (T ) , t 2 U , T;1t Bt = BT ftg BT U ET . (1.2) BT ET . * Cn = T;1Jn Bn, Bn 2 BJn , Jn 2 F (T ), n = 1 2. < J = J1 J2. ? Cn = T;J1 (J;J1n Bn), J;J1n Bn 2 BJ , n = 1 2, (1.4). / , Cn 2 T;1J BJ , n = 1 2, C1 C2 2 T;1J BJ CT . < , XT = T;1J XJ 2 CT CT . B, CT BT (1.4). *! fCT g BT . / , T;1t Bt CT t 2 T , BT = = fT;t1Bt t 2 T g fCT g. 2 ) , 1.5
, ;% B 2 BT &$+ y (y XT ) &$ B $ $ + $ $ &$ U T ( U B , 10
T , (1.6)
BT = BT T ). L T = N0 1], Xt = R t 2 T . ? , , , C N0 1] 2= BT ,
. . , . J ,
S = = fy 2 XT : sup y(t) 6 g 2= BT 2 R. ) , T t2T
,
- . ' ! . / , F j B- , $ X (X B) $ $ &. 3 , .!. X PX . <
! 1.6. +"" Q (X B) .. X . 2 D = X , F = B, P = Q X = I (
). 2 , (XT BT ) Q, X (t !), ( PX = Q, X (t !) = !(t) !() 2 XT : (1.8)
(1.8)
XT P ! , . L
(1.1) - . * (XT BT ) Q. ? (XU BU ), U T ,
QU := QT;1U ($ % Q), . (1.4). T V
#" ! #" ! #" ! T U
(XT BT Q)
z
U V
zj
(XU BU QU = QT;1U ) (XV BV QU U;1V ) . 1.4
*! V U T , (1.5), (1.7), QT;1V = Q(T;1U U;1V ) = = (QT;U1 )U;1V . $ Q: QV = QU U;1V BV V U T: (1.9) & , + % $ $ $$+ BT , $ QU , &$ U ( . . (1.9) V U 2 F (T ), F (T ) {
T ). ',
. ',
Rk . @ 1.7 (/ ). fXt t 2 T g | - Bt = B (Xt). (XJ BJ ) QJ , J 2 F (T ). (XT BT ) ( Q, QJ = QT;1J J 2 F (T ). 11
1 (
Xt t 2 T , { B!) 2. /
. *, T | ;% $ -
&$. A + $ $ $ (Xt Bt), t 2 T , $ 1.7, % , $ $ $ $+ (. B1.2 B2.11).
$ 1.8.
X = X (t !), T
" $ PX .
1.7. ( .#. (D F P ), QJ , J 2 F (T ),
2 * 1.7
Q (XT BT ), QJ = QT;J1 J 2 F (T ). * 1.6 .!. X : D ! XT , PX = Q. ) , X (t !) (1.8). ' QJ = PX T;1J J 2 F (T ). 2 $ 1.9. fXt t 2 T g | " . (Xt B (Xt)) Qt , t 2 T . ( (D F P ) .. Xt : D ! Xt, "( F j B(Xt)- , PXt = Qt B (Xt), t 2 T . B
! 1.10. ' QJ (XJ BJ ), J 2 F (T ),
,
QI (BI ) = QJ J;I1(BI ) (1.10) " $ % BI = fy 2 XI : y (t) 2 Bt t 2 I Bt 2 Bt g, J 2 F (T ) I J . 2 + BI % ;$ + &$ QI XI (K | , ? 2 K, A1 A2 2 K A1 \ A2 2 K A2 n A1 A1 A2 6 ( ( A1 A2) (
K). 8 , BI = fQI g (1.11) T ( , BI = I;t1BtP , ! BI t2I ). < , & $ + K ( ) & ; - % fKg (., ., N?, . 3, x3]). 2 2 B
1.9. B I 2 F (T ) BI 2 BI Bt, t 2 I , Y QI (BI ) := Qt(Bt): (1.12) t2I
' , QI QI , , (1.11) BI ( (
7, . (7.28)). B , , (1.10), . . J;I1(BI ) = fy 2 XJ : y(t) 2 Bt t 2 2 I y(t) 2 Xt t 2 J n I g, Qt(Xt) = 1, t 2 T . *
1.8 . . X (t !) T (D F P ), QI = PXT;I1, I 2 F (T ). * !
Qt = Qftg = PXT;t1 = PXt t 2 T
12
. . X (t !) = T t X(!) f! : X (t !) 2 Btg = f! : X() 2 T;1t Btg Bt 2 B(Xt). & , Xt = X (t ), t 2 T , |
.!., . . I 2 F (T ) BI Bt, t 2 I ,
P
\ t2I
Y
f! : Xt 2 Btg =
t2I
P (! : Xt 2 Bt):
(1.13)
(1.13)
P (T I XQ2 BI ) = P (X 2 T;1I (BI )) = PXT;I1 (BI ) = QI (BI ), Q | PXt (Bt) = Qt(Bt). ? , (1.13) t2I t2I (1.12). 2 B , & $ $$+, $ + $ $ +$ .,. * . 1.11. * 1 2 : : : | (...) , (D F P ) ( Rk (k > 1). n X 1 Pn (B !) = n 1 B ( j (!)) B 2 B(Rk) n 2 N: (1.14) j =1 1 B () |
B , . .
(
1 x 2 B (1.15) 0 x 2= B: * . .
.!., , (1.14) n 2 N ,
B , . . T = B(Rk). 1.12. * f j j 2 Zdg { , ... Rk. * | - B(Rd), d > 1. < 1 B (x) =
$
Sn (B !) =
X j 2Zd
j (!) (nB \ Cj ) B 2 B(N0 1]d) n 2 N
(1.16)
Cj = (j ; 1 j ] = (j1 ; 1 j1] : : : (jd ; 1 jd ] |
j 2 Zd, nB = fx = ny y 2 B g. ) d = 1 " " 1 + : : : + n n. 1.13. B f j j 2 Ng ... "$ " X (1.17) Xt(!) = maxfn : j (!) 6 tg t > 0 j 6n
3; $ $ &$ ;, ! Xt (!) = 0, 1(!) > t. ( , Xt(!) R = R f1g ( ! B(R) B B f1g, B 2 B(R)). ' ( . * , t0 = 0, ,
j ,
, . ? Xt (" ") (0 t]. 13
1.14. L (D F P ) f j j 2 Ng, f j j 2 Ng , f j j j 2 Ng L( j ) = L( 1), L( j ) = L( 1), j 2 N.
B c, y0
Yt(!) = y0 + ct ;
X
Xt ( ! ) j =1
j (!) t > 0
(1.18)
Xt (!) (1.17). * (1.18) $ / { % . J , , y0 { , cP{ , j { j = ji=1 i (j 2 N). ? Yt { t. 1.15. * { ( 6 0), - B(X )
X . * Y X1 X2 : : : { ! (D F P ) , Y {
= (X ), X1 X2 : : : { ... , ( FjB(X ) { PX1 (B ) = (B )= (X ) B 2 B(X ).
1.9 ( , + Y Xk $ $ ). B(X ) D Z (B !),
Z (B !) =
X
Y (!) j =1
1 B (Xj (!))
(1.19)
Z (B !) = 0, Y (!) = 0. I (1.19) ( ( ) . / - ! ,
. ' , $&$ ;$. * X = fXt t 2 T g Y = fYt t 2 T g (D F P ) t 2 T ( (Xt Bt), ( ), " "
P (! : Xt(!) = Yt(!)) = 1 t 2 T: 2 Y, ! X, # $ X ( ! , , X Y). 3 % $ $+ $$ $ $ , . . F
P - 0 0. ? , $ : T = D = N0 1], F = B(N0 1]), P { 2 N0 1], Xt(!) = 1 ftg(!) (. (1.15)), Yt(!) 0 t 2 T ! 2 D. < , X , Y . * X Y ( ,
, t 2 T ( (Xt Bt)) - , PX = PY BT . C X Y ()
, (Xt Bt) t 2 T {
, ! ! . 14
* X = fXt t 2 T g Y = fYt t 2 T g, (D F P ) ( (Xt Bt) t 2 T , ,
P (! : Xt(!) 6= Yt(!) t 2 T ) = 0
.. 1 ! . 1 ! X Y. B
, s 2 T
f! : Xs (!) 6= Ys (!)g f! : Xt(!) 6= Yt(!) t 2 T g (- F ).
. D. 1.1. C (Xn Bn) n 2 N { + $ +$+ $ $ ( ) Qn { Bn (n 2 N), (D F P ) .!. Xn 2 FjBn
, L(Xn ) = Qn n 2 N. 1 ( ( .
@ 81.2 () ?, . N?, . 266]). (X1 B1) Q1 n 2 N xk 2 Xk , k = 1 : : : n (Xn+1 Bn+1 ) Qn (x1 : : : xn P ), Qn ( : : : P B ) 2 Fn jB (R) " B 2 Bn+1 , Fn = nk=1 Bk ( - - , ( B1 : : : Bn). Bk 2 Bk ,
W
k = 1 : : : n (n > 2)
Pn(B1 : : : Bn ) =
Z B1
Q1(dx1)
Z B2
Q2(x1P dx2) : : :
Z Bn
Qn(x1 : : : xn;1P dxn ) (1.20)
( (1.20) 0 ).
(D F P ) ( " .. Xn , L(X1 : : : Xn ) = Pn n 2 N.
D. 1.3. *
- A 2 R, (
D, , 2A - (
1.5). D. 1.4. *
G g : Rd ! R, . * , G 2 BT (Xt = R t 2 T = Rd). ' PX(G), X = fXt = (1 + f (t) ) t 2 Rdg, f : Rd ! R ( ( ) , { , N-1,2]. D. 1.5. B ,
(N0 1] B(N0 1]) P ), P { 2 ,
fXt t 2 Rg , . . , P (Xt = 0) = P (Xt = 1) = 1=2 t (
1.9). D. 1.6. B ,
(...)
. . X (t !) t 2 R ! 2 D, , 1 X ( !) R. 15
D. 1.7. * C (R) {
, R. & " ", ( , . .
DJ = fx 2 C (R) : x(t) 2 Bt t 2 J g, Bt 2 B(R) t 2 J J 2 F (R), Y QJ (DJ ) :=
t2J
Q0(Bt)
Q0 { B(R). B , QJ J 2 F (R)
C (R) , % ;$ $ $$+; (
1.9). D. 1.8. , (1.17) (1.18)? D. 1.9. * k = 1 (1.16), . . f j g { ... , d = 1. * Xn(t !) = p1n Sn (N0 t] !) t 2 N0 1]: * ! , ) { 2 RP ) { " "( ", .. X (B ) = 1 B (j ) B R: j 2Z
< , ( ) ) . 1.9 (
C N0 1]
/ DN0 1]) 2,3 N?]. D. 1.10. * k = 1 (1.14). * Fn(x !) = Pn ((;1 x] !) x 2 R: * Fn(x !), # $ . , j N0 1], j { , . . P ( j = 1) = p, P ( j = 0) = 1 ; p, 0 < p < 1? ,
M B(Rk) - (.-.) Q, B(Rk), " > 0 S1(") : : : SN(") 2 B(Rk), N = N ("), , B 2 M Si(") Sj(")
Si(") B Sj(") Q(Sj(") n Si(")) < ". @ 81.11 (. N?, . 421]). M { , .- . Q. Pn (B ! ) { $, # (1.14), P1 = Q.
sup jPn (B !) ; Q(B )j ! 0 n ! 1
e
B 2M
(1.21)
e
! 2 D, D n D D0 P (D0 ) = 0. 1 , (D F P ) , (1.21) " 1. D. 1.12. B , M = f(;1 x] x 2 Rkg, . .
(;1 x1] (;1 xk ], (x1 : : : xk ) 2 Rk, .-. -
( ) Q. ? $ $ ! { / $ .. n ! 1 ! Fn(x !) := Pn ((;1 x] !) F1 (x). 16
D. 1.13. B , Q = P 2 Rk M {
, (1.21) 1
(
, ( . 1.12, k > 2). D. 1.14. * ,
, ( . 1.13, , Q 2 ( k > 2). ) ! , ! ,
, ( , ., ., N?], N?]. * A B(N0 1]d), d > 1. B F : A ! R kF kA = sup jF (A)j. A2A * F(A) = jA()j, A 2 A, > 0 A() = fx 2 Rd : (x A) < g, (x A) = inf y2A (x y), { j j { 2 Rd. *,
A B(N0 1]d) ,
kF kA ! 0 ! 0 + :
(1.22)
D. 1.15. B , (1.22) , A {
Na1 b1] Nad bd] N0 1]d. * ( . @ 81.16 ( % +< , . N?]). fXj j 2 Nd g { , ( ... c. A B (N0 1]d) " (1.22).
kn;dSn () ; j jckA ! 0 . . n ! 1 P Sn (A) = j 2nA Xj , nA = fx = ny y 2 Ag, n 2 N, . .
n;d { "( .
(1.16), < ,
, . N?], N?]. $ (1.17) . * , 1 { , . .
k, k 2 Z, > 0 { , " " ( 1 { , k = 0 1 2 : : : ). . < # $ " u(t) t > 0, (0 t], . . u(t) = EXt . D. 1.17. B , u(t) < 1 t > 0. ) ( $ $ $ . @ 81.18 (E , , ., ., N?, . 46]). 3 $ $
u(t + s) ; u(t) ! s=c s > 0, c = E 1
t ! 1 " # 1=1 = 0). 3 $ - 1 . < . N?], N?].
( # s,
17
D. 1.19. < 6 , Z (B ), (1.19), B 2 B(X )
(Z+ A), Z+ = f0 1 : : : g = f0g N, A { -
Z+ ( , fZ (B ) B 2 B(X )g {
,
B(X )). $+ { - (X B), .. X = 1m=1 Xm, Xm 2 B 0 < (Xm) < 1, m 2 N. ! $ ; (1.19) $+ ; % . ' (D F P ) Y (1) X1(1) X2(1) : : : Y (m) X1(m) X2(m) : : :
, Y (m) *
m = (Xm) L(Xj(m)) = ()= (Xm ) Bm = B \ Xm m j 2 N: *
Zm (D !) = Z (B !) =
X
Y (m) (!) j =1
1 X m=1
1 D (Xj(m)(!)) D 2 Bm ! 2 D:
Zm (B \ Xm !) B 2 B ! 2 D:
(1.23)
< , Z (B !) . C
, ) ? (. . 1.1),
+ 1.15 & $ $+ + $ $ (X B). D. 1.20. B , (B ) = 1 B 2 B, Z (B ) = 1 .., (B ) < 1, Z (B ) < 1 .. @ 81.21 (. N?, . 24]). , .. $ (1.23), :
1. 3 " n 2 N " "( Z (B1 ) : : : Z (Bn ) . 2. 3 B (B ).
2 B
Z (B )
-
-
B. 8 , ; $ = 1 , , ; % $ $$+; . Z (B ) , " ",
B 2 B. ! ( 3.
.
Z ( !) " $
B1 : : : Bn 2 B
-
D. 1.22. N. N?, . 370]] * B 2 B , 0 < (B ) < 1, { - (X B). * B = nq=1Bq , Bq 2PB, Bj \ Bq = q 6= j (q j = 1 : : : n). ? k1 : : : kn 2 Z+ k = nq=1 kq 18
P (Z (B1) = k1 : : : Z (Bn) = knjZ (B ) = k) =
k!
(B1) k
1
(Bn ) kn
(B ) : (1.24) ? , B = B(Rd), d > 1, $ (1.24) , " $ ", < &$ B ( 6) $ & & $ $+ $ , ;$ k , B . = k ! k ! (B ) 1 n
/ , , , ( (., ., N?, ?] ). ? ,
. D. 1.23. * X = fXt t 2 T g Y = fYt t 2 T g, (D F P ),
(X B) t 2 T , , PX = PY BT , X Y ! . D. 1.24. * (
) X = fXt t 2 T g
D L 2= BT , f! : X ( !) 2 Dg 2 F f! : X ( !) 2 Lg 2= F . D. 1.25. * ! X = fXt t 2 T g Y = fYt t 2 T g,
D 2= BT , A = f! : X ( !) 2 Dg 2 F B = f! : Y ( !) 2 Dg 2 F , P (A) 6= P (B ). 8 1.24 1.25 , & $+ $-
$ &$, ; - % , ,$ &, $ $$+ $ &$ % $+ % fX (tj ) tj 2 T j = 1 : : : ng n 2 N. D. 1.26. * ! , -
. * % < -
$$ $, $ ( &) % $ , $ % $ "<" $ .
) , , , ,
! . < ( . < $, $ $ & t 2 T ;$ Xt t 2 T , &, " $" (D F P ) ( (
!). ? , & $ $+ X = fXt t 2 T g $$ $ $ (D^ F P ). S
, ( ( - ) (D^ F P ) = (D F P ) (D0 F 0 P 0). ! X D D0
Y (!e) = X (!) !e = (! !0) 2 D D0:
(1.25)
? , , PY = PX BT . 19
L
N?] . 120 { 130. 2 , ( !
N?]. *, $ $ % $ . * (Xt Bt)t2T (Yt Et)t2T {
ht : Xt ! Yt, ht 2 BtjEt t 2 T . / ( 2. D. 1.27. B V T (V T )
hV : XV ! YV ,
hV (x) = y x = (xt t 2 V ) y = (ht(xt) t 2 V )
(1.26)
* , hV 2 BV jEV V T (EV { - Q YV = t2V Yt). D. 1.28. * QJ {
(XJ BJ ) J 2 F (T ). B , J = QJ h;J 1 { (YJ EJ ) J 2 F (T ), hJ (1.26). D. 1.29. * (Xt Bt)t2T (Yt Et)t2T {
# ( (Xt Bt) (Yt Et) t 2 T ), .. t 2 T ht : Xt ! Yt, , ht 2 BtjEt h;t 1 2 Et jBt. *, (YT ET ) (
J = QJ h;J 1 J 2 F (T ). ? Q = hT QJ J 2 F (T ) ( hT = (h;T 1 );1). D. 1.30. * (Yt Et) (Zt At) t 2 T , .. Yt Zt Et At t 2 T ( EV AV V T ). * (YJ EJ ) J , J 2 F (T ). B ,
PJ (A) = J (A \ YJ ) A 2 AJ
(1.27)
(ZJ AJ ) J 2 F (T ). * ! , , PJ = J EJ J 2 F (T ). D. 1.31. * (ZT AT ) (
P , ( PJ , ; 1 ; 1 J 2 F (T ), ( (1.27). B , PhT ( .. P (hT ) ) (XT BT ) QJ (XJ BJ ), J 2 F (T ). & , $ ,
.. Ys = Xg;1(s) s 2 S , g : T ! S { . * , , , , fXt t 2 U g fXt t 2 V g U V T
U V .
(T ) (T ) ( , (x y ) = 0, x = y ). ', Xt = R EXt2 < 1 t 2 T , " " ( )
(s t) = (E(Xt ; Xs )2)1=2:
(1.28)
!
, L.2. B , <.C. 2 B. L !, 20
. A + $$ $, $% $$+ -
, $ + $ & % . * ,
* 1, . ? ( N?], N?]. < , ( . * $ ,
..
(., ., N?], N?]).
(. N?], N?] N?], N?]).
21
2. . . - " . , " . - . CT . . " $ . $ . / Qt1 ::: tn t1 : : : tn 2 T . 0 (Rn B(Rn)). / . $ 1 1. . 2 ( ).
B
, . 2.1. Xt, t 2 T , | t - Bt = B (Xt). XJ , J 2 F (T ),
J (y() z()) = max (y(t) z(t)) t2J t . BJ .
= B(XJ ), . . $
(2.1)
- -
2 < , J
. / ! yn() 2 XJ , n ! 1 yn(t) (Xt t ) t 2 J . 2 , (XJ J ) |
. $* % J t : XJ ! Xt, J t y = y(t), (XJ J ) (Xt t ) t 2 J . *! J;t1Gt 2 B(XJ )
Gt 2 B(Xt). *
1.2 J;t1Bt 2 B(XJ ) Bt 2 B(Xt). ) (1.2) , BJ B(XJ ). *, , Xt, t 2 T . * G |
(XJ J ). (XJ J )
G 6 -
(
. N?, . 104]). ' B" (y) = fz 2 XJ : J (z y) < "g $ % XJ . / ,
G 6 BJ . *! B(XJ ) BJ . 2 ? 2.2. 3 , % + $ $ $ % - % $ - % , & $$ ( $, $) < .
- . 2.3 ( $ $ ). (X ) | Q | B (X ). " " > 0 B 2 B (X ) ( " F" G" , F" B G" Q(G" n F") < ". 2 C F , F" = F . < G = fz 2 X :0 (z F ) < g, (z F ) = inf f(z y): y 2 F g. 3
G , G G < 0 22
T G = F . / , Q(G ) # Q(F ) ! 0
>0
. *! = (") , G" = G(") Q(G" n F" < ". B , Y
,
(
2.3, - . ? , B(X ) Y ,
Y , B(X ) - , (
. * Bn 2 Y , n 2 N. B " > 0 Fn " Gn " , Fn " Bn Gn " Q(Gn " n Fn ") < "2;n;1 . S F , n , Q S1 F n F < "=2 (F F" = n" 0 n" " " n6n0 n=1 6
). S1 S1 B G Q(G nF ) < ". < , Y X
G" = Gn " . ? F" n " " " n=1 n=1 . 2 2.4 (D ). (X ) | Q |
B(X ). B 2 B(X ) " " > 0 ( K" B , Q(B n K" ) < ".
S1
2 B n 2 N X = B1=n(yn m ), B"(y) = fx 2 X : m=1 (x y) < "g, yn m , m = 1 2 : : : , 1=n- ( . . y 2 X yn m, (y yn m) < 1=n). B Sjn
" > 0 jn , Q B1=n(yn m) > 1 ; "2;n;1 . * m=1
T1 Sjn B (y ). ? NR ]
R R" = " " 1=n n m n=1 m=1 X 1 jn 1 Q(X n NR"]) 6 Q(X n R") = Q
n=1 m=1
B1=n(yn m)
6
n=1
"2;n;1 = "=2:
3
NR"] | , . .
( . . " > 0 (
"- ), , 2=n- yn m, m = 1 : : : jn . * 2.3 F" B , Q(B n F") < "=2. ? K" = F" \ NR"]
Q(B n K" ) < ". 2 & , X = Rq NR" ]
2.4 , . ? & $ 1.7 ( !
). 2 C Q QJ (
, C = T;J1 BJ 2 CT , BJ 2 BJ (J 2 F (T )), (. (1.5)) Q(C ) = Q(T;J1 BJ ) = QT;1J (BJ ) = QJ (BJ ): *! $ $ , $+ Q % CT , -
&
Q(C ) := QJ (B ) = T;1J B J 2 F (T ) B 2 BJ : (2.2) 1 $. B , C = T;J1i Bi, Ji 2 F (T ), Bi 2 BJi , i = 1 2. *, QJ1 (B1) = QJ2 (B2). J = J1 J2 (2 F (T )). * 23
C = T;J1i Bi = T;1J (J;J1i Bi), i = 1 2, . . h
N
M M1 M2 M h;1 (M1) = h;1(M2) () M1 = M2,
J;J11 B1 = J;J12 B2. ,
F (T ) ( (1.9) V U T , U 2 F (T )), QJ1 (B1) = QJ J;J11 (B1) = QJ (J;J11 B1) = QJ (J;J12 B2) = QJ J;J12 (B2) = QJ2 (B2):
< , Q | CT , Q(XT ) = QJ (XJ ) = 1 J 2 F (T ). ; $$+ Q CT . Ci 2 CT , i = 1 2, C1 \ C2 = ?. L , , Ci = T;1J Bi , J 2 F (T ), Bi 2 BJ , i = 1 2, B1 \ B2 = ?. ?
Q(C1 C2) = Q(T;1J (B1 B2)) = QJ (B1 B2) = QJ (B1) + QJ (B2) = Q(C1) + Q(C2): C Q CT , , Q fCT g, . . BT , 1.5. * ! (2.2) QT;J1 = QJ BJ J 2 F (T ). F % Q
. ) , & $+ QT CT , . . Q(Cn) ! 0, Cn # ? n ! 1 (Cn 2 CT , Cn+1 Cn, n 2 N, Cn = ?). n * Cn = T;J1n Bn , Jn 2 F (T ), Bn 2 BJn , n 2 N. 0 ( (
Q CT ) %
$ $+ Jn Jn+1, n 2 N. B , Q(Cn) > "0 > 0 n ( n 2 N, Cn+1 Cn, n 2 N). 8 , ! Cn # ?. * , & $+ <+ &$ Cn = = T;1Jn Bn, Bn | $ XJn , Jn 2 F (T ) (- Jn Jn+1 , n 2 N). , Bn 2 BJn 2.4 XJn Kn Bn ,
QJn (Bn n Kn ) < "02;n;1 . < Hn = T;1Jn Kn 2 CT . ? n Q(Cn n Hn ) = Q(T;1Jn (Bn n Kn )) = QJn (Bn n Kn ) < "02;n;1 :
Tn
Tn
Tn
J;n1Ji Ki . < , Ln # ?, Ln = Hi = T;J1n J;n1Ji Ki = T;1Jn i=1 i=1 i=1 Ln+1 Ln Ln Hn Cn . < Ln ,
Dn = Tn = J;n1Ji Ki, XJn
i=1 Kn J;n1Jn Kn (
). B, , Cn Ci, i = 1 : : : n,
Q(Cn n Ln ) = Q Cn \
n i=1
Hi
n
=Q
(Cn n Hi ) 6
n
i=1
6Q
i=1
X n
(Ci n Hi) 6
i=1
Q(Ci n Hi) < "0=2:
/ , "0 6 Q(Cn) = Q(Ln) + Q(Cn n Ln) 6 Q(Ln) + "0=2, . . Q(Ln) > "0=2 n 2 N. 2$ , ; Cn = T;J1n Bn, Bn | $ XJn , Jn Jn+1, Jn 2 F (T ), n 2 N. 24
XT yn 2 Cn (Cn 6= ?, Bn 6= ? n, . . Q(Cn) = = QJn (Bn) > 0, (?) = 0 - ). ? T Jn yn 2 Bn, n 2 N. B n > m T;1Jn Bn = Cn Cm = T;1Jm Bm = T;1Jn (J;n1Jm Bm ) , , Bn J;n1Jm Bm, Jn Jm Bn Bm. *!
ynjJm = T Jm yn = Jn Jm T Jn yn 2 Bm n > m: 8 , B1 | (XJ1 J1 ), fn(1) j g N x1 2 B1, yn(1) j ! x1 (XJ1 J1 ) j ! 1, . . (2.1) yn(1)j (t) ! x1(t) j J1
(Xt t ) j ! 1 t 2 J1. ) fn(1) j g (2) fnj g, yn(2)j jJ2 ! x2 2 B2 (XJ2 J2 ) j ! 1. B
, m > 2 fn(jm)g fn(jm;1)g xm 2 Bm , yn(jm) jJm ! xm 2 Bm
(XJm Jm ) j ! 1, . .
yn(jm) (t) ! xm(t) (Xt t ) t 2 Jm m 2 N (j ! 1):
(2.3)
* , ( , ,
xm(t) = x(t) m 2 N t 2 U =
1 m=1
Jm :
fnj g, . . nj = n(jj). ?
ynj (t) ! x(t) (Xt t ) j ! 1 t 2 U
U Jm x = xm = (x(t) t 2 Jm) 2 Bm m 2 N: - y 2 T;1U x, x = (x(t) t 2 U ) 2 XU (T U XT XU ). ? m 2 N T Jm y = U Jm T U y = U Jm x 2 Bm
T1
. . y 2 T;1Jm Bm = Cm m. / , Cm 6= ?. * m=1 . 2 B ,
. * , , $ $$ $ $ ( ). * , , (Xt Bt) t 2 T {
. < X (t1 : : : tn) {
, ( (x(t1) : : : x(tn)), t1 : : : tn 2 T , x(tk ) 2 Xtk , k = 1 : : : n, n 2 N. !
(2.1)
t1 ::: tn (x y) = 1max (x(tk ) y(tk)): 6k6n tk ,
2.1 , - B(t1 : : : tn) := B(X (t1 : : : tn)) Bt1 ::: tn , " " Bt1 Btn , Btk 2 Btk k = 1 : : : n. 25
*! , X = fXt t 2 T g { . ., .. Xt : D ! Xt FjBt- t 2 T ,
1.2 t1 : : : tn 2 T n 2 N, (Xt1 : : : Xtn ) 2 FjB(t1 : : : tn). 3 B(t1 : : : tn), Pt1 ::: tn (C ) = P (! : (Xt1 : : : Xtn ) 2 C ) (2.4) (-..) . . X. ) (2.4) , n > 2, t1 : : : tn 2 T C = Bt1 : : : Btn , Btk 2 Btk , k = 1 : : : n, (i1 : : : in) (1 : : : n) 1: Pt1 ::: tn (Bt1 : : : Btn ) = Pti1 ::: tin (Bti1 : : : Btin ), 2: Pt1 ::: tn (Bt1 : : : Btn;1 Xtn ) = Pt1 ::: tn;1 (Bt1 : : : Btn;1 ). & 1 2 + 1 3, 3 , Btm = Xtm m = 1 : : : n Pt1 ::: tn (Bt1 : : : Btn ) ! tm Btm , .. 3: Pt1 ::: tm ::: tn (Bt1 : : : Xtm : : : Btn;1 ) = = Pt1 ::: tm;1 tm+1 ::: tn (Bt1 : : : Btm;1 Btm+1 : : : Btn ):
? 2.5. 2$ $ $+ ; & % $ t1 : : : tn, .. (
$ % , ( (, Pt t(B 0B 00) = P (Xt 2 B 0 Xt 2 B 00) = = P (Xt 2 B 0 \ B 00) = Pt(B 0 \ B 00)). $ 2.6. (X (t1 : : : tn) B(t1 : : : tn)), (Xtk Btk ), tk 2 T , k = 1 : : : n (n 2 N) Qt ::: tn , "( 1 2 ( P Q). ( (D F P ) .#. X , T D, Qt ::: tn 1
" .-..
1
2 B J T $ % BJ = fy 2 XJ : y(t) 2 Bt t 2 J g 2 BJ QJ (BJ ) := Qt1 ::: tn (Bt1 : : : Btn ) (2.5) t1 : : : tn | -
J . < (2.5) . B
, 1 QJ (BJ )
J ( , t 2 J
Bt). C B = BJ1 B = BJ2 , , I = J1 \ J2, BJi = fy 2 XJi : y(t) 2 Bt t 2 I y(t) 2 Xt t 2 Ji n I g, i = 1 2. * ( , ) 3 1 , , QJ1 (BJ1 ) = = QJ2 (BJ2 ) = QI (BI ), BI = fy 2 XI : y(t) 2 Bt t 2 I g. * Qt1 ::: tn | (X (t1 : : : tn) B(t1 : : : tn)), (2.5) , QJ
$ % XJ , , BJ . * ! 1 3 (1.10). *!
1.8 (
. . X , QJ = PX T;1J . ) (2.5) (1.11) , ! .-.. 2 8 $ Qt1 ::: tn . B n > 2 (i1 : : : in) n : T n ! T n Tn : X n ! X n, n(t1 : : : tn) = (ti1 : : : tin ) Tn(x1 : : : xn) = (xi1 : : : xin ): (2.6)
n : T n ! T n;1 Un : X n ! X n;1 : n(t1 : : : tn) = (t1 : : : tn;1) Un (x1 : : : xn) = (x1 : : : xn;1): (2.7) 26
0
. ) n (2.6) (2.7) ( . 2.7. 3 n > 2 = (t1 : : : tn) T , t1 : : : tn , 1 2 ( 1 3 ) Q = Qt1 ::: tn "( : A) Q = Q T;17 B) Q = Q U;1.
2 / Bt1 : : :Btn
X (t1 : : : tn), t1 : : : tn 2 T (n 2 N), 1.6 . 2 ? 2.8. B T R Q , = (t1 : : : tn) 2 T n, n 2 N, ( t1 < : : : < tn. B
s1 : : : sn B1 : : : Bn Qs1 ::: sn (Bs1 : : : Bsn ) := Qsi1 ::: sin (Bsi1 : : : Bsin ) si1 < : : : < sin . ? 1 Qs1 ::: sn 3 Qt1 ::: tn ( t1 : : : tn.
$ , ; $$ + , % + $+ $$$ & $ $ Rn ( % - % B(Rn)) $$ . ', # $ (.#.) Q (Rn B (Rn)) 'Q () :=
Pn
Z
Rn
expfi( x)gQ(dx) 2 Rn
(2.8)
( x) = k xk i2 = ;1. k=1 3 Q # $ F (x) := := Q((;1 x]), (;1 x] = (;1 x1] : : : (;1 xn]. x, F , . . ( :
F (x) = (2);n
lim
Z
!0+ (;1 x]
Z
dy d expf;i( y) ; 2jj2=2g'Q () Rn
(2.9)
jj2 = ( ), d | 2 . & F , F , , Q B(Rn). & , 'Q 2 2 L1(Rn B(Rn) d), (2.9) = 0 . ' ( $ % . * g : X ! Y B jA- , h : Y ! Rn h 2 AjB(Rn). ?
Z
X
h(g(x))Q(dx) =
Z Y
h(y)Qg;1(dy)
(2.10)
(2.10) (
(
(
, . 27
#" ! #" ! #" ! g
x
(X B Q)
z y = gx
h
(Y A Qg;1)
zhy
(Rn B(Rn))
. 3.1 8.#.
Y : D ! Rn (F j B(Rn)- ) -
'Y () = E expfi( Y )g 2 Rn: ) (2.8) (2.10) ,
Z
Z
Rn
'Y () = expfi( Y (!))gP (d!) =
@ 2.9. 3 (Rn B (Rn))n>1 ,
2 T n n > 2
expfi( z)gPY ;1(dz) = 'PY ():
(2.11) (2.12)
Q , = (t1 : : : tn ) 2 T n,
, " 2 Rn,
(a) ' (T) = ' (), (b) ' (U) = ' (U 0), T U (2.6) (2.7) ( X n = Rn), ' = 'Q , 2 T n, (U 0) = (1 : : : n;1 0) = (1 : : : n ) 2 Rn. ) , .. ' () & $+ $ $ $ , $ & .., 6 7 $ , $ .. ' $ $ 1 : : : n $ 6 7 $ $. 2
(Rn B(Rn)) (A) (B) 2.7 , n > 2
' () = 'Q ;1 () 2 Rn ' ( ) = 'Q ;1 ( ) 2 Rn;1:
(2.13) (2.14)
* 1.6 Y Rn, Q = PY . ? Q T;1 TY , PY T;1(B ) = P (Y 2 2 T;1(B )) = P (TY 2 B ). B,
'Y () = E expfi(TY )g = E expfi(Y T)g = 'Y (T;1)
(2.15)
T , ( ! ( T = T;1, T { T). / (2.15) (2.13) ' () = ' (T;1), 2 Rn, ! (a). J ,
'Y ( ) = E expfi(UY )g = E expfi(Y ( 0))g = 'Y (( 0)) 2 Rn;1: (2.16) ? , (2.14) ! (b). 2 28
? 2.10. * X (t) = (X1(t) : : : Xk (t)), t 2 T { Rk, .. X 2 FjB(Rk) t 2 T . B n 2 N t1 : : : tn 2 T t1 ::: tn = (X1 (t1) : : : Xk (t1) : : : X1(tn) : : : Xk (tn)) ( B(Rkn) Qt1 ::: tn . . t1 ::: tn (), = (1 : : : n ), (k) k j = ((1) j : : : j ) 2 R , j = 1 : : : n. / , 2.9 ( Q ,
(Rkn B(Rkn))n>1. ? (a) t1 : : : tn 1 : : : n ,
(b) n . < t1 ::: tn = (X1(t1) : : : X1(tn) : : : Xk (t1) : : : Xk (tn))
, (a) (b). 3 %< . 9 , $$ + (..) X = fXt t 2 N0 1)g $ , ;% n 2 N 0 = t0 < t1 < : : : < tn Xt0 Xt1 ; Xt0 : : : Xtn ; Xtn;1 $. @ 2.11. f'(s tP )g | # $ , "( Q(s t] 0 6 s < t < 1, B (R). 3 ( (D F P ) $ X = fXt t 2 N0 1)g ( , .#. Xt ; Xs '(s tP ) " 0 6 s < t < 1,
,
'(s tP ) = '(s uP )'(u tP ) 0 6 s < u < t < 1 2 R. X0
.
(2.17)
2 ' (2.17) , . .
. . . * (2.17). B , Xt, PX0 = Q. ? 0 = t0 < t1 < : : : < tn . . = (Xt0 Xt1 ; Xt0 : : : Xtn ; Xtn ;1)
' (0 1 : : : n ) = 'Q (0)'(t0 t1P 1) : : : '(tn;1 tnP n ): ? , 0X 1 0 1 0 Xt0 1 t0 1 0 0 : : : 0 B B B Xt1 C Xt1 ; Xt0 C 1 1 0 : : : 0C B C B C C B B C B C C B X X ; X t t t 1 1 1 : : : 0 2C = 2 1 C: B B C B B @ ... C A @: : : : : : : : : : : : : : : A B @ ... C A 1 1 1 : : : 1 Xtn Xtn ; Xtn;1
(2.18)
B 2 Rq, A = (ak m)qk m=1, ak m 2 R (k m = 1 : : : q), 2 Rq
'A () = E expfi( A )g = E expfi(A )g = ' (A):
(2.19) 29
*! (
, .-.. n > 1 0 = t0 < t1 < : : : < tn ( . .
't0 t1 ::: tn (0 1 : : : n) = ' (A) = 'Q ( 0)'(0 t1P 1) : : : '(tn;1 tnP n) = A, A | , ( (2.18), . . 0 = = 0 + : : : + n , 1 = 1 + : : : + n , : : : , n = n . , ,
't0 (0) = 'Q (0) X X n n 't1 ::: tn (1 : : : n) = 'Q j ' 0 t1P j : : :'(tn;1 tnP n ): j =1
(2.20)
j =1
< 2.9. 8 (a) 2.8, (b), , (b0), ( 0 ' ()
m , , .. (2.17) 1 6 m 6 n
'(tm;1 tmP 0 + m+1 + : : : + n )'(tm tm+1P m+1 + : : : + n ) = = '(tm;1 tm+1P m+1 + : : : + n ) m = 0 (2.20)
't1 ::: tn (1 : : : n ) = 't0 t1 ::: tn (0 1 : : : n ): 2
? 2.12. ? 2.11
Rm (m > 1), .. Q(s t] B(Rm). B ! , (
(2.18), Im { m- , k 2 Rm, k = 0 1 : : : m. ? 2.13. * ( N0 1). C T = N Xt (, Xt = 1 + : : : + t, t 2 N, f j g1j=1 | . *!
$ $$ %% $ . $ 2.14. '( $ fNt t > 0g, ,
1) N (0) = 0 . .7 2) $ ( 7 3) Nt ; Ns , 0 6 s < t < 1, m((s t]), m() | - B (N0 1)). 9 , m((s t]) = (t ; s), 0 6 s < t < 1, > 0, $ " .
2 / 3) '(s tP ) = 'Nt;Ns ( ) = em((s t])(ei ;1) 2 R: 8 (2.17) , m((s t]) = m((s u]) + m((u t]),
Q , 0. 2 30
$ 2.15. '( $ fW (t) t > 0g,
( ),
-
1) W (0) = 0 . .7 2) W () | $ ( 7 3) W (t) ; W (s) N (0 t ; s) t > s > 0. < ( .. . B , !
1), 2), 3). 2 /(
,2 ( (u;s) 2 (t;u) 2 (t;s) ; ; ;
1), 2), 3), 2.11, e 2 = e 2 e 2 , 0 6 s < u < t, 2 R, . . (2.17). 2 B W (t) = W (t) ; W (0) N (0 t), ! EW (t) = 0 t > 0. * 0 6 s 6 t ( , ) cov(W (s) W (t)) = cov(W (s) ; W (0) W (t) ; W (s) + W (s) ; W (0)) = = D(W (s) ; W (0)) = s: ? , EW (t) = 0 cov(W (s) W (t)) = minfs tg s t 2 N0 1): (2.21)
. *, % $ , 1, $ +; % +$ $ + $+ , $ (. N?]) $ +$ $ / . 1
.
2 H 1. A $ $; -
$ , $ + $ $ . * (Xt t)t2T {
. 0 , t N0 1) t 2 T ( et(x y) = t(x y)=(1+ t(x y)) x y 2 Xt, ! tP !
(Xt t) (Xt et) (Xt et) {
). Xt
Mt = (x(1t) x(2t) : : : ) ht : Xt ! N0 1]1, ht(x) = (t(x x(1t)) t(x x(2t)) : : : ):
N0 1]1 d(y z) =
1 X k=1
2;k jyk ; zk j
(2.22)
y = (y1 y2 : : : ), z = (z1 z2 : : : ). D. 2.1. * , ht Xt ht(Xt) N0 1]1, t 2 T. D. 2.2. * , Zt = Nht(Xt)] (N0 1]1 d), N]
. 31
* . 2.1, 2.2 1.30, QJ (XJ BJ ) PJ (ZJ AJ ) J 2 F (T ), AU { Q - ZU = t2U Zt U T . H 2. B P (ZT AT ) PJ J 2 F (T ), $ $ $$+ PU AU U 2 N (T ), $.. , $ &$ U T ,
PU Q;U 1V = PV
(2.23)
V U ( V 2 F (T ) ), QU V { ! " " ZU ZV . B
, 1.5 B 2 AT U 2 N (T ) BU 2 AU , B = Q;T 1U BU . *!
P (B ) = PU (BU ):
(2.24)
D. 2.3. * , ! , .. B = Q;T 1V BV BV 2 AV V 2 N (T )
PV (BV ) = PU (BU ) (2.23). D. 2.4. B , (2.24)
AT , ( PJ J 2 F (T ). H 3. E $$+ PU U 2 N (T ), $; ; $ (2.23). * U = ft1 t2 : : : g, ..
U 2 N (T ) . ZU
dU (y z) =
1 X k=1
2;k d(ytk ztk )
(2.25)
y = (yt t 2 U ), z = (zt t 2 U ), (2.25) , .. d( ) 6 1. D. 2.5. * , (ZU dU ) {
U 2 N (T ). D. 2.6. B ,
ZU (U 2 N (T ))
U (2.25). D. 2.7. B ,
(ZU dU ) U 2 N (T ), - B(ZU ) AU . L C (ZU P R)
, (ZU dU ). * HU {
C (ZU P R), ( H , (
J = J (H ) 2 F (T )
hJ (ZJ dJ ) ( J dJ = PJ (2.1), t = d t 2 J ) ,
H (z()) = hJ (zJ ) zJ = zjJ = (z(t) t 2 J ):
(2.26)
/ HU C (ZU R), .. , HG 2 HU , H G 2 HU . & NHU ] HU
C (ZU P R), sup-, ZU , .. z y 2 ZU H 2 NHU ] , H (z) 6= H (y) (
), ! ' { 9- (., ., N?, . 119]) NHU ] = C (ZU P R). 32
3 HU FU (H ) =
Z ZJ
hJ (zJ )dPJ
(2.27)
H 2 HU (2.26). ) (2.27) , hJ 2 B(ZJ )jB(R) B(ZJ ) = AJ . 2.7, , $. D. 2.8. B , (2.27) , .. RH (z()) = hI (zI ) I 2 F (T ) hI 2 C (ZI P R), ZI hJ (zI )dPI (2.27). < FU , (2.27), HU . *! FU & $ C (ZU P R). * !
FU (f ) > 0 f > 0 f 2 C (ZU R) FU (1 ZU ) = 1 ( f 2 HU !
). / , * (., ., N?, . 124])
FU (f ) =
Z
ZU
fdPU f 2 C (ZU R)
{ : -
(2.28)
PU { , %, -
ZU . * ! - (., ., N?, . 122]) , $ $ $ $ %, % &$ ;$ (. N?, . 123]). ', , , . H 4. &, $ $ PU U 2 N (T ), . * V U 2 N (T ). ? HV HU . C f 2 HV , (
J 2 F (T ) fU 2 C (ZJ P R) ,
f (zU ) = f (zV ) = fJ (zJ ) zU 2 ZU zV 2 ZV zJ = zU jJ = zV jJ : (2.27) (2.28)
FU (f ) = FV (f ) =
Z ZU
Z ZV
f (zU )dPU = f (zV )dPV =
Z ZJ
Z ZJ
fJ (zJ )dPJ
(2.29)
fJ (zJ )dPJ :
(2.30)
* 2 (. (2.10)), , (2.29) (2.30),
Z
ZV
f (zV )dPV =
Z
ZV
f (zV )dPU Q;U 1V :
(2.31)
/ / { , , (2.31) f 2 C (ZV P R). < 33
D. 2.9. * Q1 Q2 { B(X ), X { -
. B ,
Z
X
f (x)dQ1 =
Z
X
f (x)dQ2
(2.32)
f : X ! R, Q1 = Q2 B(X ). D. 2.10. B
Q BT , ( QJ J 2 F (T ). ? , . 2 L $ $ $ Xt, t 2 T , $ N0 1]
$ $ $$+ $ +$ $ 1.7 & . 22, " Xt : : : " J , Cn 6= Bn 6= ,
.. Q(Cn) = QJn (Bn) > 0 n 2 NP , () = 0 - ;1 0 . * U = 1 n=1 Jn (
)
Cn = U Jn Bn , n 2 N. ? Cn0 Cm0 , , ,
Cn = T;1U Cn0 Cm = T;1U Cm0 (n > m):
= \1n=1Cn = \1n=1 T;1U Cn0 = T;1U \1n=1 Cn0
0 ! \1 n=1 Cn = . ? , U Jn { U
N0 1] ( (2.22))
N0 1]Jn ( Jn ). / , fCn0 gn2N { N0 1]U . ' 0
\1 n=1 Cn 6= . * . ? 82.11. )
(X B) , (
X , ( ( X
, B = B(X ) .. B ( ). , E {
X , E {
E , (E E ) {
( ( ,
( E ! , X ). < ,
$ / $ $ $ % $ $ Xt t 2 T . * (X B) (Y A) {
. & (X B) (Y A) ,
D 2 A , (X B) (D AjD ) ( " " 1.29, AjD = A \ D = fAD : A 2 A). ,
K = f0 1gN - B. D. 2.12. * ,
(K B), d (. (2.22)) B = B(K ). * ! (K d) { (, , (K d) {
). , , (K S B(K )S ) (K B(K ))
S = 6 . ' $ % $ $. @ 82.13 (., ., N?, . 98]). (X B) (Y B(Y )), (Y B(Y )) { .
8 (K B(K )) 8 , < < (X B) : (N A(N)) X , : ((1 : : : jXj) A(1 : : : jXj)) , 34
A(M ) { - M , j j { . 3 $$, (X B ) { $ $ %
$ $ X $ $ $ $ , $ X { $, B = B(X ).
)
( N?]. ? $ $ $ . ,
BT (. 1.5), B $ , , , ,
. * X = fXt t 2 T g, (D F P ) t 2 T (
) (
(X ), - " T0 T , T0 { T
(
N 2 F , P (N ) = 0 t 2 T ! 2 D n N
X (t !) 2
\
G3t G2J
fX (s !) s 2 G \ T0g
(2.33)
NB ] B , J {
(X ). @ 82.14 (. N?, . 128]). (X ) { , (T ) { . $ X = fXt t 2 T g, (D F P ) t 2 T "( (X ) " # $ ". ! X { , # $ X .
(2.33) , N?, . 111], T = N0 1] X = R. < . N?].
( . * (T A) {
. / X = fXt t 2 T g, (D F P ) ( t 2 T
(X B), , (t !) 2 T D 7! Xt(!) 2 X (2.34) A FjB . , , X {
, B = B(X ). I T { $ &$ ( ) A { -
T , $ X, , . @ 82.15 (N?]). B2.14 $ X T , .. t 2 T " " > 0 lim P ((Xs Xt ) > ") = 0: (2.35) s !t
X ( # $ . D. 2.16. B , X {
,
(2.35) , ..
F .
35
D. 2.17. * , .. , -
( . D. 2.18. B , , AjB- .
? D. 2.19. * X = fXt t 2 T g { : D ! T , 2 FjA. B , Y (!) = X (!)(!) FjB- . D. 2.20. * , ,
( . D. 2.21. * B2.14 (T ). , ? ? % $ , ; $ $ -
$ . F +$ $ / . @ 82.22 (. N?, . 124]). X = fX (t) t 2 Na b]g {
$ , " > 0 C = C ( ") > 0 EjX (t) ; X (s)j 6 C jt ; sj1+" s t 2 Na b]: $
X ( . .
(2.36)
# $ .
D. 2.23. * , (2.36) " = 0,
B2.22, ( , ( ). ? , (. L . ',
U T (U T ), (T ) {
, "- "
S" T , t 2 U s 2 S" , (s t) 6 ". B , U B Ns "] = ft 2 T : (s t) 6 "g s 2 S": (2.37) C U "- , "- " "- S"min(U ), jS"min(U )j , jV j !
V . * N (" U ) = jS"min(U )j, . S (2.38) H (" U ) = log N (" U ) "-
U ( (2.38) 2). / N?],
(T ). * (
m 2 N a > 0 , U T , ( DU := supf(s t) : s t 2 U g < 1, " > 0 N (" U ) 6 maxfa(DU =")m 1g: (2.39) * (
b > 0, "- S"min(T ) (2.40) jS"min(T ) \ B Nt 5"]j 6 b t 2 S"min(T ) B N cdot] (2.37). 36
@ 82.24 (. N?, . 134] ). X = fX (t) t 2 T g { .#. (X d) t 2 T , (D F P ) (T ), "( (2.39) (2.40). , 2 (0 1) > m= > 0 E(d(X (t) X (s)) 6 ((t s)) s t 2 T: (2.41) ( # $ Y = fY (t) t 2 T g, "( , , " 2 (0 ; m=) t0 2 T
sup d(Y (t) Y (s))=(t s) = 0 . ., lim #0 (s t)6
n
E sup d(Y (t) Y (t0
a
t2T #
))
o
6 a(2DT ) =(2;m= ; 1)
(2.42) (2.43)
(2.39). ? 82.25. C D { Rm Rm+1 ( Rq ), (2.39) (2.40) . *!
B2.24 B2.22,
. B , $% $ % % $ .. X, $ &$ + ;, $ $ ; ;, ; ; , ; R+ = N0 1) $ ;, $ (0) 6 1. @ 82.26 (. N?]). X = fX (t) t 2 T g { .#. (X d) t 2 T , (D F P ) (T ), , M > 0 ! d ( X ( t ) X ( s )) 6 M s t 2 T (s t) 6= 0 (2.44) E (s t)
d(X (t) X (s)) = 0 . ., (s t) = 0:
Z
(2.45)
1
X(N (T "))d" < (2.46) +0 X { # $ , , (2.46) 0. .#. ( . . # $ .
X
D. 2.27. B2.26, X = fX (t), t 2 N0 1]mg t 2 T
(X )
( f R+ p > 1 E(d(X (t) X (s))p 6 f p(ks ; tk) (2.47)
Z +1
f (x;p=m)dx < +1 (2.48)
.. +1, (
.. . 37
? 82.28 (. N?]). 2$ + (2.48) $ $ +, .. ,
( ,
(2.48), , .. . B
. . X = fX (t) t 2 T g , n 2 N t1 : : : tn 2 T (X (t1) : : : X (tn)) . 1 3. ) B2.26 ( . @ 82.29 (. N?]). X = fX (t) t 2 T g { $ T , (.(1.28)).
Z p
1
log N (T ")d" < + +0 ( # $ , "( . .
(2.49) .
*
! + $ , $ "( . 1
T . B , ! " " "" . , -
(T ), "(, sup
Z1
t2T 0
j log (B Nt "])j1=2d" < 1
(2.50)
B N ] (2.37). * (B Nt "]) = 1 " > DT , (2.50)
0 DT , (2.50) ! 0. D. 2.30. (. N?, . 182]) B , ( (. /(
q > 1 C = fCk k 2 Ng ( T
- , sup DC 6 2q;k (2.51) C 2Ck
sup
1 X
t2T k=1
q;k j log (Ck (t))j1=2d" < 1
(2.52)
Ck (t) { (
)
Ck , t. @ 82.31 (. N?, . 193]). X = fX (t) t 2 T g { .#. (T ). -
( "( ,
lim sup #0
Z
j log (B Nt "])j1=2d" = 0:
t2T 0 * , .
38
# $
(2.53)
"( (. .)
B I, { ? . & , ( , ( Z ! 1 X (B Nt "]) d" +0
X , ., ., N?]. /
N?], . ' , , ( .
39
3. ! .
3 " . " . $ . 4 "- . 2 ( ). % " . # 0 5 . # " . 60 1], 60 1). 3 .
3 , 2, $ $ & %< . ', Y
Rn ( Y N (a C )), 2 Rn
'Y () = exp i(a ) ; 21 (C )
X n
n X 1 = exp i ak k ; ckm k m 2 k=1 k m=1
(3.1)
a 2 Rn, C = (ckm )nk m=1 |
! . < $ C ( C > 0) , (C ) > 0 2 Rn. 2 , : C > 0 C = C ( ) : n X
k l=1
ckl zk zYl > 0
(3.2)
z1 : : : zn ( z ). B (., ., N?, . 175]), C
a 2 Rn, , ( (3.1), . . Y . C C > 0, . . (C ) > 0 6= 0 2 Rn, Y ( 2 )
PY (x) = (2); n2 jC j; 21 expf;(C ;1(x ; a) x ; a)g jC j | C . B Y N (a C ) $ a C :
ak = EYk ckm = cov(Yk Ym ) k m = 1 : : : n (n > 1):
(3.3)
B
. . X (t), T (D F P ), , .-.. . B , (X (t1) : : : X (tn)) n 2 N t1 : : : tn 2 T ( ! (t1 : : : tn), ( , , , ( ! , ). 3.1. 9 Y =n(Y1 : : : Yn) Rn P k Yk , ( Y ) = k=1
= ( 1 : : : n) 2 Rn. 40
2 * Y N (a C ). ? 2 R (3.1) Eei( Y )
= Eei( Y )
= exp i(a ) ; 21 (C ) = exp i(a ) ; 12 (C ) 2
. . ( Y ) N ((a ) (C )) ( , = ( 1 : : : n )). < . * ( Y ) N (a 2). ? (. (3.3) n = 1)
a = E( Y ) = X n 2 = D( Y ) = D
k Yk = k=1
n X k m=1
n X k=1
k EYk = ( EY )
k m cov(Yk Ym ) =
(3.4)
n X k m=1
k m ckm = (C )
( j = 1 k = 0 k 6= j , , Yj { , ! EYj2 < 1, j = 1 : : : n). / , 2R 1 i (
Y ) 2 2 Ee = exp ia ; : (3.5) 2 * (3.5) = 1 a 2 (3.4), (3.1). 2 B
r(s t), T T , $ , (r(tk tm))nk m=1 > 0 n 2 N
t1 : : : tn 2 T .
@ 3.2.
# $ a(t), t 2 T , $ # $ r(s t), s t 2 T . T (D F P ) ( .#. X (t !) 2 R, a(t) = EX (t) r(s t) = cov(X (s) X (t)) s t 2 T .
2 B n 2 N = (t1 : : : tn) 2 T n (Rn B(Rn)) Q ,
( . . (3.1), a =(a(t1) : : : a(tn)) ckm = r(tk tm)
( ). ? (a) (b) 2.9. 2 ? ,
.., $ , (RT BT ), $ 6 : . < ,
. . X (t), ( t 2 T , n 2 N, t1 : : : tn 2 T 1 : : : n 2 R
n X
k m=1
cov(X (tk ) X (tm))k m = cov
X n k=1
k X (tk )
n X
m=1
m X (tm) > 0:
< , cov(X (s) X (t)) = cov(X (t) X (s)). *! , $ 3.2 ; r, ;$ $ + $ $ , %-
$ $$ + .. r. ? , $ $ + $$ + $ ( $$ +) . 41
, R(s t), s t 2 T , T |
, $ , n 2 N, t1 : : : tn 2 T z1 : : : zn 2 C n X
k l=1
zk zYlR(tk tl) > 0
(3.6)
(z = u ; iv z = u + iv, u v 2 R). * fX (t) t 2 T g | . ., ( t, EjX (t)j2 < 1 t 2 T . < . . L2-$ . = $ # $ !
r(s t) = E(X (s) ; EX (s))(X (t) ; EX (t)):
(3.7)
/ X = fX (t) = (t) + (t) t 2 T g C ( (t) (t) {
), n 2 N
t1 : : : tn 2 T ( (t1) (t1) : : : (tn) (tn)) ( ( (t1) : : : (tn ) (t1) : : : (tn))). @ 3.3. = $ # $ R(s t), s t 2 T , $ # $ L2 -$ fX (t) t 2 T g, $ # $ $ fX (t) t 2 T g.
2 C X (t), t 2 T , | L2- , n 2 N, t1 : : : tn 2 T
z1 : : : zn 2 C
X 2 n zk zYlr(tk tl) = E zk X (tk ) > 0: k l=1 k=1 n X
< . * R(s t), s t 2 T , | . * R1(s t) = Re R(s t) R2(s t) = Im R(s t), s t 2 T . B zk = uk +ivk, k = 1 : : : n,
(3.6)
n X
k m=1
R1(tk tm)(uk um + vk vm) +
X n
+i
k m=1
n X
k m=1
R2(tk tm)(uk vm ; vkum) +
R1(tk tm)(vk um ; uk vm) +
n X k m=1
R2(tk tm)(uk um ; vk vm) > 0:
(3.8)
) (3.6) n = 1 , R(t t) > 0 t 2 T . * n = 2, t1 t2 2 T z1 z2 2 C , (3.6)
jz1j2R(t1 t2) + z1z2R(t1 t2) + z1z2R(t2 t1) + jz2j2R(t2 t2) > 0: / , z1z2R(t1 t2) + z1z2R(t2 t1) {
. , z1 = z2 = 1 , R(t1 t2) + R(t2 t1) 2 R t1 t2 2 T . z1 = 1, z2 = i, , R(t1 t2) ; R(t2 t1) . ? , R(s t) = R(t s) s t 2 T . 1
R(s t) R1 (s t) = R1 (t s) R2(s t) = ;R2 (t s) s t 2 T . *!
(3.8) (C ) > 0, 42
= (u1 : : : un v1 : : : vn), ( ) | R2n, C
! : R (t t ) R (t t ) (3.9) C = ;1R k(t mt ) R2(tk tm) 2 k m 1 k m k m=1 ::: n: 2.10 (
( (t) (t)), t 2 T , R2 t, n 2 N t1 : : : tn 2 T ( (t1) : : : (tn) (t1) : : : (tn)) N (0 C ), C (3.9). X (t) = p1 ( (t) ; i (t)) t 2 T: (3.10) 2 ? s t 2 T ( , ;R2(t s) = R2(s t), s t 2 T ) EX (s)X (t) = 21 (R1(s t) ; iR2(t s) + iR2(s t) + R1(s t)) = = R1(s t) + iR2(s t) = R(s t): 2
@ 3.4. )
$ ,
"(: 1. (W (t) t > 0) { $7 2. EW (t) = 0 t 2 N0 1)7
2.15,
3. cov(W (t) W (s)) = minft sg t s 2 N0 1). 23 , (,
2.15,
(2.21), .. 2 3. B, !
. (W (t1) : : : W (tn)), 0 6 t1 < : : : < tn , , . . (. (2.18)) (W (t1) ; W (0) W (t2) ; W (t1) : : : W (tn) ; W (tn;1)) , ..
2) 3). B
, Rn Y N (a C ), , (3.1) (2.19), , AY N (Aa ACA), A =(ak m)nk m=1 ak m 2 R, k m = 1 : : : n. A% $, 1 2 3. *,
1), 2), 3),
2.15.
2.15 2.21 , r(s t) = minfs tg, s t 2 N0 1), . / , 3.2 (
(W (t) t > 0) . ?, r(s t) ,
:
r(s t) =
Z1 0
1 0 minfs tg](z) dz =
Z1 0
n 2 N 1 : : : n 2 R n X
k m=1
r(tk tm)k m =
Z1X n 0
k=1
1 0 s](z)1 0 t](z) dz
2
k 1 0 tk ](z) dz > 0: 43
-
* EW (0) = 0 DW (0) = cov(W (0) W (0)) = minf0 0g = 0, W (0) = 0 .. B, 0 6 t1 < : : : < tn 0 W 1 0 1 0 0 : : : : : : : : 01 0 W 1 0 0 B C ; 1 1 0 : : : : : : : : 0 B C B B C W ; W W t1 0 C B t1 C B C B C 0 ; 1 1 : : : : : : : : 0 = B B C . . B C . . @ . A @: : : : : : : : : : : : : : : : : : : : : : :A @ . C A: Wtn ; Wtn;1 0 0 0 : : : ;1 1 Wtn *! (W (0) W (t1) ; W (0) : : : W (tn) ; W (tn;1)) | . /-
$ $ $ $ + $ , $ + ( -
, . . . . P (3.1)). ? ,
cov(W (tk+1) ; W (tk ) W (tm+1) ; W (tm)) = = minftk+1 tm+1g ; minftk+1 tmg ; minftk tm+1g + minftk tmg = 0 k 6= m: (3.11) ) ( W (t) ; W (s) t s > 0 ($ $ $ , . . . . . . j ). , , E(W (t) ; ; W (s)) = EW (s) ; EW (t) = 0 D(W (t) ; W (s)) = t ; s t > s ( (3.11)). 2 *, ; $ ; $ +$ ! , 1 $ $. ? 3.5.
2.14,2.15 2.11 , $$ + Xt t > 0 $ 0 $
, $ $ (..) $ $ $ $+ ,$ . 0 ( * 2.
$ W (t) $ N0 1]. # $ Hk (t), t 2 N0 1], k = 0 1 : : : , H0(t) 1 H1(t) = 1 0 1=2](t) ; 1 (1=2 1](t) 2n 6 k < 2n+1 (n 2 N), an k = 2;n (k ; 2n ), Hk (t) = 2n=2(1 In k (t) ; 1 Jn k (t))
8
In k = Nan k an k + 2;n;1 ], Jn k = (an k + 2;n;1 an k + 2;n ] (. 3.2). / fHk g $ L2N0 1] % . ', L2N0 1]
Z1
hf gi = f (t)g(t) dt f g 2 L2N0 1]: 0
< fHk g , , ( Hk . (?, 1 0 1=2] = (H0 + H1)=2, 1 (1=2 1] = (H0 ; H1)=2, 44
qq
6
2n=2
Hk (t)
0
an k + 2;n 1
an k
-
t
;2n=2
. 3.2
1 0 1=4] = (1 0 1=2] + (1=2)H2 )=2, 1 (1=4 1=2] = (1 (0 1=2] ; 1=2H2 )=2, : : : , 1 an k an k +2;n;1 ] = = (1 an k an k +2;n] + 2;n=2 Hk )=2 2n 6 k < 2n+1 .) / , f 2 L2N0 1]
f=
1 X k=0
hf Hk iHk
(3.12)
(3.12) L2N0 1]. < ,
*
hf gi = < # $
Sk (t) :=
1 X k=0
>
Zt 0
hf Hk ihg Hk i:
(3.13)
(. . 3.3):
Hk (y) dy h1 0 t] Hk i t 2 N0 1]:
6 2;( n2 );1
Sk (t) (k > 1)
0
an k
an k + 2;n
1
-t
. 3.3
@ 3.6. f k k > 0g | N (0 1) (D F P ) ( " 1.9). t 2 N0 1], ! 2 D W (t !) =
1 X k=0
k (!)Sk (t):
(3.14)
W $ N0 1] ( . . , , N0 1]), "( . . .
' . 45
3.7. fa1k g1k=0 , ak = P = O(k" ) k ! 1 " < 1=2. ak Sk (t) -
k=0 N0 1] , , " N0 1] # $ " ( . . Sk (t) ).
2 B , X Rm := sup jakjSk (t) ! 0 m ! 1 t2 0 1] k>2m
(Sk () > 0 k). ) jak j 6 ck" k > 1 c > 0, ! t 2 N0 1] n > 1
X
2n 6k<2n+1
X
jakjSk (t) 6 c2(n+1)"
2n 6k<2n+1
Sk (t) 6 c2(n+1)"2;n=2;1 6 c2";n(1=2;") :
3 , t ( Sk , 2n 6 k < 2n+1 , , 0 6 Sk (t) 6 2;(n=2);1 k. * " < 1=2, ,
Rm 6 c2"
X
n>m
2;n(1=2;") ! 0 m ! 1: 2
3.8. (D F P ) ( ) k N (0 1), k = 0 1 : : : . " c > 21=2 . . ! 2 D N0 (! c), j k (!)j < c(log k)1=2
2 B N (0 1) x > 0 P (
> x) = (2);1=2
Z1 x
expf;
= (2);1=2
y2=2
gdy = (2);1=2
x;1e;x2 =2 ;
Z1 x
k > N0(! c):
Z1 x
(;1=y)d(e;y2 =2) =
y;2e;y2 =2dy
6 x;1(2);1=2e;x =2: (3.15)
0 , , P ( > x) x;1(2);1=2e;x2=2 x ! 1: / , x > 0 P (j j > x) 6 x;1(2=)1=2e;x2=2: *! c > 21=2 X X P (j k j > c(log k)1=2) 6 c;1 (2=)1=2 k;c2 =2(log k);1=2 < 1: k>2
k>2
2
(3.16) (3.17)
< P E {/ $ (. N?, . ??]),
( , P (Ak ) < 1, 0 k
Ak , . . P 46
T S n k >n
Ak = 0. 2
3 $; . 8 & $ 3.6. 2 F$+ .. $ $ W ( !) $ 3.7 3.8. & , (3.14) .. N0 1], t 2 N0 1] L2(D) = L2(D F P ). B
,
nX 2 nX +m +m nX +m nX +m E k (!)Sk (t) = Sk (t)Sl(t)E k l = Sk2(t) n m 2 N: k=n k=n l=n k=n P S 2(t) 6 (2;(n=2);1)2 < L2(D) , k 2n 6k<2n Pn t 2 N0 1]. < Z (t) = S (t). ) , t 2 N0 1] +1
n
k=0
k k
( ) Zn (t) L;! Z (t) n ! 1 Zn (t) ! W (t) .. n ! 1. < , Z (t) = W (t) .. ( .., , ! ). 2 ( ) ? , Zn (t) L;! W (t), n ! 1 (t 2 N0 1]). F ; ; W (t) t 2 N0 1]. * EZn (t) = 0 n 2 N t 2 N0 1], 2
jEW (t)j = jEW (t) ; EZn (t)j 6 (EjW (t) ; Zn (t)j2)1=2 ! 0 n ! 1: / , EW (t) = 0 t 2 N0 1]. B s t 2 N0 1], ,
(Zn (s) Zn (t))L2( ) ! (W (s) W (t))L2( ) = EW (s)W (t) = cov(W (s) W (t)) f k g E 2 = 1, k 2 N, (Zn (s) Zn(t))L2( ) = EZn (s)Zn(t) = (3.13) 1 X k=0
Sk (s)Sk (t) =
1 X k=0
n X k=0
Sk (s)Sk (t)E k2
!
1 X k=0
Sk (s)Sk (t) n ! 1
hHk 1 0 s)ihHk 1 0 t]i = h1 0 s] 1 0 t]i = minfs tg:
&, $ W | . 3.1. L Y = ( ), = (W (t1) : : : W (tn)), 2 Rn, t1 : : : tn 2 N0 1]. ? Y=
n 1 X X
m=1
m
k=0
k Sk (tm) =
Pn
1 X k=0
bk k
(3.18)
bk = bk ( 1 : : : n P t1 : : : tn) =
m Sk (tm), (3.18) .., m=1
L2(D) , ( !
. & , PN PN YN = bk k N (0 N2 ), N2 = b2k , EYN2 = N2 ! EY 2 = 2 k=0
k=1
47
D ( ! L2(D) ). 8 , YN ;! Y N ! 1 ( .., L2(D)), ,
'YN () = e
. . Y N (0 2). 2
;N2 2 2
! e ; = 'Y () 2 R 2 2 2
(
f(t !) = W (t !) ! 2 D0 W (3.19) 0 ! 2 D n D0 D0 D !, W ( !) . ? f | ! 2 D. * ! W f(t !)g D n D0. 3; % $ $+, t 2 N0 1] f! : W (t !) 6= W $ (D F P ) ( F
f W ( . . P (Wf(t) = W (t)) = 1 ). ? W t 2 N0 1]), N0 1]. B W N0 1) (
. 3.9. X = fX (t !) t 2 T g { $,
- T (D F P ). t 2 T X (t !) " X . ! X T , X .. C (T X ) ( T # $ X t 2 T )
, ..
C (x() y()) =
1 X n=1
2;n sup 1 +(x((xt)(ty)(yt))(t))
(3.20)
t2Kn
T = 1n=1Kn , Kn { , Kn Kn+1 n 2 N. 2*
(C (T X ) C ) (
). *! 2.2
1.2 , y 2 C (T X ) r > 0 NBr(y)] = fx 2 C (T X ) : C (x y) 6 rg A = f! : X ( !) 2 NBr (y)]g 2 F . T
M . ?
(
NBr (y)] = x 2 C (T X ) : < ,
1 X
)
2;n sup 1 +(x((xt)(ty)(yt))(t)) 6 r : t2Kn\M n=1
(t !) y(t)) 2 FjB(R)
n (!) = sup 1 +(X(X (t !) y(t)) t2Kn\M P ;n
f! : 1 n=1 2 n (! ) 6 rg 2 F ( , 6
). 2 48
f .!. C N0 1], B(C N0 1]) * 3.9 W 9 W . *
1.9, $ () (D F P ) fn , n 2 N, $ $ W fn( !) 2 C N0 1] $ +$+ .,. W f N0 1) +; ! 2 D PWfn = W B(C N0 1]). A W fn, .. W 8 f1(t !) > W < k f(t !) = X W fj (1 !) + W fk+1 (t ; k !) > : j=1 W 6
t 2 N0 1) t 2 Nk k + 1) k 2 N:
r r
(3.21)
W1(1) + W2(t ; 1) W1(1) 0
t 2
1
-
. 4.1
f, # (3.21), @ 3.10. $ W $ N0 1), .
f . < , W f(0) = 0. 2 ' W f | (, W f(t) ; *, W f(s) N (0 t ; s) 0 6 s < t < 1. C s t 2 Nk k + 1) ;W f(t) ; W f (s) = W fk+1 (t ; k) ; Wfk+1 (s ;Sk) N (0 t ; s). CS s 2 Nk k + 1) k > 0, W t 2 Nm m + 1), k < m, Ns t) = Ns k + 1) Nl l + 1) Nm t) ( Nl l + 1) = ?). 0
*!
1
s
r
k
k k +1
f(t) ; W f(s) = Wf(k + 1) ; Wf(s) + W
l l +1 . 4.2
r
l2?
t m m +1
X f f(l)) + Wf(t) ; W f(m): (W (l + 1) ; W
k
(3.22)
Pq Pq
? , 1 + : : : + q N ai i2 , 1 : : : q i=1 i=1 fk+1 : : : W fm+1 . C i N (ai i2), i = 1 : : : q. /.!. W 49
Il N0 1], k 6 l 6 m, ! jIlj fl : RjIlj ! R ( . . B(RjIlj) jB(R)- ), fk+1 (tj ) tj 2 Ik+1), : : : , fm+1(W fm+1(tj ) tj 2 Im+1) | . fk+1(W , (, ( (3.22). f J , W ( ( , ( ). 2
8 $ ; < W . ( % & fW (t) = (W1(t) : : : Wm(t)) t > 0g Rm, m fWk (t) t > 0g, k = 1 : : : m. 3 W1(t) : : : Wm (t) .!.
C N0 1) (3.20), Kn = N0 n], n 2 N.
.
3 , .. N0 1] , N0 1). *! ( $ ; % & N0 1], & ; . . * (D F P ) fX0 1 Y0 1P Yn k 1 6 k 6 2n n > 1g , X0 1 Y0 1 N (0 1), Yn k N (0 1), 1 6 k 6 2n n > 1 (!
1.9). < , n, Xn k , 1 6 k 6 2n n > 1, k, .. k = 2m ; 1 k = 2m
Xn+1 2m;1 = (Xn m + Yn m )=2 Xn+1 2m = (Xn m ; Yn m )=2:
P
< Sn 0 = 0, Sn k = kj=1 Xn j , 1 6 k 6 2n n > 1,
Bn(t) 0 6 t 6 1, , Bn (k2;n !) = Sn k (!) 1 6 k 6 2n t 2 Nk2;n (k + 1)2;n ] Bn(t) Bn (k2;n ) Bn ((k + 1)2;n ), k = 0 : : : 2n ; 1.
Bn (t) 0 6 t 6 1, n > 1, N0 1]. 2 , B1(t) = tX0 1 sup jB1(t)j = jX0 1j n > 1
t2 0 1]
jY j: sup jBn+1(t) ; Bn(t)j = 12 16max k62n n k
t2 0 1]
* (3.17), ,
2 2;n=2 ) 6 2n P ( > n2 ) 6 2n (2 );1=2e;n2 =2 P (16max j Y j > n n k n k 62
N (0 1). / , 0 { , (. . ??) . . ! (
N = N (!) , n > N sup jBn+1(t) ; Bn(t)j 6 n22;n=2;1 : t2 0 1]
50
) ,
B1(t !) +
1 X n=2
(Bn+1 (t !) ; Bn (t !))
. . ! , .. N0 1] B (t). / ( 3.1 ( (3.13) , n > 0 fBn(t) 0 6 t 6 1g . D. 3.1. ' EBn(t) cov(Bn(s) Bn(t)) s t 2 N0 1] n > 0. D. 3.2. B , B (t) { . D. 3.3. B , EB (t) = 0 cov(B (s) B (t)) = min(s t) s t 2 N0 1]. 8 3.2 3.3 , B (t) { .. N0 1]. < , % & &$ %$+ -
$ +$ $ $ $ , $ $ $ H . ?, *! (1934) ,
n ;1 1 2X X p t+ 2 sin(kt)
0
n=1 k=2n;1
k
k
(3.23)
0 1 : : : { ... N (0 1) , .. N0 1] ,
, ( N0 1]. ) ! \, ( L), 1 2t) X f (t) = sin(nn t 2 R 2 n=1 , . N?, . 10] ( 0 , r X n t ( ! ) 2 0 ( n ) W (t !) = p + k k(!) sin kt k=1 fnmg .. N0 ]. L (10.31) (10.48). 3 &$+ $ N0 1] % &-
$ $ $ 2.11 $ / (. B2.22). B
, E(Wt ; Ws )4 = 3(t ; s)2 s t > 0: D. 3.4. < , (
2.15, , A pA (x) = j det Aj;1p (A;1x) (3.24) Rn, ( p (x), A { n-
! , .-.. . 51
3 % $ % " $+" % & + $ x 2 R, 0. < ,
Wx(t !) = x + W (t !):
(3.25)
*, ! ( x 2 R),
. J , 0 = t0 < t1 < : : : < tn 6 1 n 2 N Qxt0 t1 ::: tn , " "
Qxt0 t1 ::: tn (B0 B1 : : : Bn) = Qx0 (B0)
Z
B1
dx1 : : :
Z
Bn
dxn
Yn k=1
ptk ;tk;1 (xk ; xk;1) (3.26)
x0 = x t0 = 0 Bk 2 B(R), k = 0 : : : n
pt(x) = (2t);1=2 expf;x2=(2t)g x 2 R t > 0 Qx0 (B ) = 1 B (x) B 2 B(R): (3.27)
D. 3.5. , Qxt ::: tn 0 < t1 < : : : < tn 6 1, Qx , = (s1 : : : sk ) 0 6 s1 < : : : < sk 6 1, k 2 N .-.. Xt t 2 N0 1]? 1
< R0
N0 1]. * , (
Xt t 2 R0, ( .-.. Qxt0 t1 ::: tn (tk 2 R0 k = 0 : : : n n 2 N), ! , , X (t !) = !(t) D, (
, R0. < Qx (RR0 BR0 ) . D. 3.6. B , Qx(C0) = 1, C0 { R0
. ? , , R0 ( 6 , ! ), & $ , N0 1]. *!
X (t) 0 6 t 6 1, ( .. N0 1]. < 6 , X .-.. ( R0), (3.26)? D. 3.7. B , X (t) (3.25) N0 1]. * ! PX B(C N0 1]) , Qxg, g { ! C N0 1] R0. * ! . D. 3.8. * (
), , + < 1=2, ..
jX (t) ; X (s)j 6 C jt ; sj s t 2 N0 1] C = const > 0:
(3.28)
D. 3.9. B , 1 lim sup W (t)=t1=2 = 1: t#0
B , 0 9 1=2. 52
(3.29)
( . @ 83.10 (E $). * Qn n 2 N, "( N0 t] t > 0 t(mn) = tm2;n m = 0 : : : 2n .
X
2n ;1
m=0
) (W (t(mn+1 ) ; W (t(mn)))2 ! t . .
n ! 1:
(3.30)
D. 3.11. B , (
(3.30) ( Qn n 2 N. < %% % &. 0 ( ) 2 (0 2] { ! W ()(t) t > 0 cov(W ()(s) W ()(t)) = 1 (s + t ; jt ; sj) s t > 0: (3.31) 2
* = 1 . D. 3.12. B , , ( (3.31), . (1 $ ( +%$ , . ??.) * X (t) t > 0 $ $ ( , n > 1, 0 6 t1 < : : : < tn h > 0 (X (t2) ; X (t1) : : : X (tn ) ; X (tn;1)) =D = (X (t2 + h) ; X (t1 + h) : : : X (tn + h) ; X (tn;1 + h))
=D
. ) , { ! (. D. 3.13. * X = fXt t > 0g { , 06s6t<1 E(Xt ; Xs) = (t ; s)c D(Xt ; Xs ) = f (t ; s) (3.32) c 2 R, f : R+ ! R+ (R+ = N0 1)). ? X { (. D. 3.14. * , 2 (0 2] E(Ws() ; Wt())2 = js ; tj s t > 0: (3.33) * (T ) {
. E (
) T
B L = fB L(t) t 2 T g (3.34) R(s t) = 21 ((s ) + (t ) ; (s t)): / , , , (3.34) . 53
D. 3.15. 9 . . X = fXt t 2 T g 2 T , X = 0 .. E(Xs ; Xt)2 = (s t) ( !
(3.34) ). C (T k k) {
( , Rd j j), 2 . . (3.35) R(s t) = 21 (ksk + ktk ; ks ; tk): (3.31) + { > . 1 . . V () = fV ()(t) t 2 Rd+ = N0 1)d g, cov(V ()(s) V ()(t)) = 12 (jsj + jtj ; jt ; sj) s t 2 Rd+: (3.36) ' + { ? $
X = fXt t 2 Rd+g , EXt = 0 cov(Xs Xt) =
Yd k=1
minfsk tk g
(3.37)
t = (t1 : : : td) s = (s1 : : : sd) 2 Rd+.
8 &, $ $ $ $. * Wk (t) t > 0 k = 1 : : : d { ( Wk (Dk Fk Pk ), (D F P ) = (D1 F1 P1): : :(Dd Fd Pd ). * Y (t) = W1(t1) Wd (td) t 2 Rd+ = N0 1)dg. ? EY (t) = 0 cov(Y (s) Y (t)) =
Q / , d
Yd k=1
minfsk tk g:
(3.38)
k=1 minfsk tk g . ? , 3.6. D. 3.16. S 2 { S Rd? ' ( fXt t 2 Rdg ( ) B = (a b] = (a1 b1] : : : (ad bd] Rd . .
Y (B ) =
X
(;1)k"kX ("1a1 + (1 ; "1)b1 : : : "dad + (1 ; "d)bd)
(3.39)
P " = ("1 : : : "d), ( 0 1, k"k = dk=1 "k . D. 3.17. B , fY (B ) B 2 Qg, Q { B = (a b], . ' . D. 3.18. B , n > 2 ( B1 : : : Bn 2 Q Y (B1) : : : Y (Bn) { . B 2 { S Y (B ) B 2 Q, , (
2.15 . 54
< A<$ { D % > 0
Vt = e;tW (e2t) t 2 R
(3.40)
W { . D. 3.19. B , fVt t 2 Rg { . D. 3.20. ' . ? ; $ $ , ; . ' . D. 3.21. * X = fXt t 2 Na b]g {
, t 2 Na b] (. (2.35)). ? X Na b], .. " > 0 ] = ](" )
,
P (jXt ; Xs j > ") 6 js ; tj 6 ] s t 2 Na b]:
(3.41)
D. 3.22. * . 3.21. ? X
, ..
lim sup P (jXtj > c) = 0:
c!1 t2 a b]
(3.42)
D. 3.23. * . 3.21. ? P ( sup jXtj < 1) = 1 t2M \ a b]
(3.43)
M {
- . 8: $ A$$ : Yk k = 1 : : : n {
2 N0 1), r > 0 Sk =
P (jSn ; Sk j > r) 6 k = 1 : : : n
(3.44)
1 P (jS j > c): P (1max j S j > r + c ) 6 k n 6k6n 1;
(3.45)
Pk Y , c > 0 j =1 j
< ' DN0 1)
, N0 1), ( t 2 (0 1). D. 3.24. B , $ $ N0 1)
$ .. ; $ $ $ $ DN0 1). D. 3.25. B , N0 1). 55
@ 83.26 ( J , ., ., N?]). fXt t 2 N0 1)g {
$ ( . " t 2 N0 1) 2 R
E expfiXt g = expft(ia +
a 2 R, {
Z1
;1
g( x) (dx))g
$
(3.46)
B(R),
(expfixg ; 1 ; i sin x)(1 + x2)=x2 x 6= 0 g( x) = 2 ; =2
(3.47)
x = 0:
(3.46) , .. # .
a
@ 83.27 ( , ., ., N?]). ( . " t 2 N0 1) 2 R E expfiXtg = expft(ia ;
22 =2 +
a 2 R, > 0, L { B(R), ,
L(f0g) = 0
Z1
;1
(eix ; 1 ; i sin x)L(dx))g
-
(3.48)
$ ( ( , )
Z1 ;1
x2=(1 + x2)L(dx) < 1:
(3.49)
a,
L, # "( (3.49), . D. 3.28. * , \ 2 . , ! a a 2 L? D. 3.29. , 2 Wx(t) = x + W (t), t > 0, W { , x 2 R? ) ( ( N?]. * ( ( . @ 83.30 ( 1K< { L+, ., N?]). 3 c > 0 -
lim T !1
sup
06t6T ;c log T
p jW (t + c log t) ; W (t)j= log T = 2c . .
L ( ! N?].
56
(3.50)
4. # # # $ %
4 " .. 60 1). 3 . 3 , - F . . . < . , , 60 t]. < . ' " . 2 .
/ ( , . @ 4.1. ' " $ $ W ## $ N0 1). 2 L Nk k + 1), k 2 f0 1 : : : g. B ! 2 D W ( !) s 2 Nk k + 1) s, ! , (
q l 2 N (q = q(! s), l = l(! s q(s !))), jW (t !) ; W (s !)j < l(t ; s) t 2 Ns s + q;1) Nk k + 1): (4.1) B l n i 2 N ( k)
j j ; 1 7l Al n i = ! : W k + n ! ; W k + n ! < n j = i + 1 i + 2 i + 3 : (4.2) L l q, ( (4.1). C 4=n < 1=q i = i(s n) (i 2 f1 : : : ng)
, k + (i ; 1)=n 6 s < k + i=n, j = i + 1 i + 2 i + 3 !,
r
k
r
s + q1
s
k + i;n1
i+2 i+3 k + ni k + i+1 n k+ n k+ n
-
k+1
. 4.3
( (4.1),
j j ; 1 j W k + ! ; W k + ! 6 W k + ! ; W (s !) + n n n 4 3 7l j ; 1 + W (s !) ; W k + ! 6 l + l = : T Sn
n
n
n
n
*! ! (4.1), ! 2 Al n i. * Dk = f! : W ( !) n>4q i=1 s 2 Nk k + 1)g ( s = k, ). ) ,
Dk
1 1
\
n
q=1 l=1 n>4q i=1
Al n i:
(4.3) 57
? , Bn 2 F , n 2 N,
P
\ 1 n=1
Bn 6 lim inf P (Bn ): n!1
*! q l 2 N, ( W ,
P
\
n
n>4q i=1
Al n i 6 lim inf P n!1
n i=1
Al n i 6
1 7l 3 6 lim inf P (Al n i) 6 lim inf n P W n < n = n!1 n!1 i=1 1 14 l 3 7l 3 n X
= lim inf n P jW (1)j < pn n!1
6 lim inf n p pn = 0: n!1 2
(4.4)
3 , W (t) N (0 t), t > 0, z > 0
Z x2 1 P (jW (1)j < z) = p e; 2 dx 6 p1 2z: 2 ;z 2 z
< (4.4) 6 , (4.2) ( W
N(j ; 1)=n j=n), j = i + 1 i + 2 i + 3. 8 , 6
, (D F P ) , (4.3), (4.4) , P (Dk ) = 0 k = 0 1 2 : : : . C D |
!, W ( !) S1 s 2 N0 1), D = Dk . / , P (D) = 0. 2 k=0
8 + < $ $ % & %$ $ $ $ + $&. 4.2. A1 A2 | F , . . P (A1 A2) ; P (A1)P (A2) = 0 " A1 2 A1 A2 2 A2: (4.5) fA1g fA2g, . . (4.5) A1 2 fA1g, A2 2 fA2g. 2 B A 2 fAg " > 0, A | F , (
B" 2 A, P (A 4 B") < ". ,
, ( , - . B
, D 4 D 1= ?, A 4 1B " = A 4 B"" S S ; n ; 1 P (An 4 Bn " ) < "2 n 2 N, P An 4 Bn " < 2 , n =1 1 1 1 S S S (A 4 B ). < n (n"=1) ,
. . An 4 Bn " n n" 0 P
S1n=1
n=1
n=1
< "=2. ? A 2 fA1g, B 2 fA2g " > 0 A" 2 A1 B" 2 A2,
P (A 4 A") < ", P (B 4 B") < ". 8 , AB 4 A"B" (A 4 A") (B 4 B" ), , jP (AB ) ; P (A)P (B )j < 4". 2 58
n=n0 Bn "
* . . X = fXt t 2 T g, . .
F jBt- .!. Xt : D ! Xt, t 2 T . B V T X (V ) = fXt t 2 V g := fXt;1(Bt) t 2 V g. / 1.1 1.4 X (V ) = X;1(BV ) (Xt = T t X, ! fXt;1(Bt) t 2 V g = fX;1(T;1t Bt) t 2 V g = fX;1(fT;1t Bt t 2 V g)g = X;1 (fT;1t Bt t 2 V g) = X;1 (BV )). '.#. fX (t) t 2 V g ( - E , X (V ) E . C . . X T R, F6t = = X ((;1 t] \ T ). @ 4.3 ( $). 3 " # a > 0 $ X (t) = W (t + a) ; W (a), t > 0, $, ( - F6a = fW (s): 0 6 s 6 ag. 2 < , X (0) = 0, X , ( X (t) ; X (s) = W (t + a) ; W (s + a) N (0 t ; s), 0 6 s 6 t. XT (T R) fx : (x(s1) : : : x(sn)) 2 B g si 2 T , i = 1 : : : n (si 6= sj i 6= j ), B 2 B(Rn) fx : (x(t1) : : : x(tn)) 2 B^ g, t1 < t2 < : : : < tn | ! ( s1 : : : sn, . . (s1 : : : sn) = (t1 : : : tn), B^ = T;1B , . (2.6). * , C = f! : (X (t1) : : : X (tn)) 2 B g, B 2 B(Rn), n 2 N, 0 = t0 6 t1 < : : : < tn. * j = W (tj + a) ; W (tj;1 + a), j = 1 : : : n. ? C = f! : ( 1 1 + 2 : : : 1 + : : : + n ) 2 B g = (( 1 : : : n ) 2 B~ ), B~ = H ;1B , n - H , ! 1 ( , , H ,
). J , m 2 N, 0 = u0 6 u1 < : : : < um 6 a, G 2 B(Rm) f(W (u1) : : : W (um)) 2 Gg = f( 1 : : : m ) 2 G~ g, G~ 2 B(Rm),
i = W (si) ; W (si;1), i = 1 : : : m. < 1.5 4.2. 2 C
: & %% $+ $ 4.3, $ $ $ a ; ? < , ! . * fFt t 2 T g, T R, | - F ($# $ %),
. . Fs Ft s 6 t (s t 2 T ) Ft F t 2 T . < : D ! T f1g fFt t 2 T g, f! : (!) 6 tg 2 Ft t 2 T . C (!) < 1 ..,
. 8 F F6t (t 2 T ) .
3 & $ $ @ 4.4. X = fXt t > 0g | .#. (X ) t > 0. X . . ! . " F X
F (!) := inf ft > 0: Xt (!) 2 F g (4.6) F6t = fXs : 0 6 s 6 tg,
( F (! ) = 1, Xt (! ) 2 = F t > 0). 2 0 ! 2 D~ , X (P (D~ ) = 1). 1. * G |
X G (4.6) F G. ? 0 6 t < 1 f! : G(!) < tg 2 F6t : (4.7) 59
& , f G < tg =
S fX 2 Gg, Q |
r + r
r2Q+
. B
, r < t, r 2 Q+ , Xr (!) 2 G, G (!) 6 r < t. < , G (!) < t. ? s = s(!) < t,
Xs(!) 2 G. * G |
, Xt(!) , Xz (!) 2 G z, s . / z r < t. B
(4.7) , fXr 2 Gg 2 F6r F6t r < t. 2. L
Gn = fx 2 X : (x F ) < 1=ng, n 2 N, (x F ) = inf f(x y): y 2 F g. * n = Gn , n (!) ! F (!) n ! 1 ! ( D~ ). < , n (!) 6 F (!) n 2 N, ! 2 D~ , . . F Gn . J , n (!) 6 n+1 (!), . . Gn+1 Gn , n 2 N. / , (
1(!) = nlim
(!) 2 N0 1] 1 (!) 6 F (!): (4.8) !1 n C 1 (!) = 1, F (!) = 1. L D0 = f! 2 D~ : 1 (!) < 1g. Xt(!) X n (!)(!) 2 NGn ] ( Gn ) n 2 N. *! X k (!)(!) 2 NGn ] k > n (Gk Gn , ! NGk ] NGn]), , Xt
X 1(!)(!) 2
1 \
NGn ] = F:
n=1
? , F (!) 6 1(!). 8 (4.8), ,
F (!) = 1 (!) = nlim
(!) ! 2 D~ : !1 n
(4.9)
3. ? , f F = 0g = fX0 2 F g (
F ). ) , f F 6 0g 2 fX0g = F60. *, 0 < t < 1
f! : F (!) 6 tg =
1 \
f! : n (!) < tg:
n=1
(4.10)
C n (!) < t, n 2 N, (4.9) F (!) 6 t. C F (!) 6 t, n (!) 6 F (!), n 2 N. C F (!) = 0, n (!) = 0, ! ! f n (!) < tg. *
! 2 D0 = f0 < F (!) 6 tg. B, n(!) < F (!) ! n 2 N. B ! 2 D0 n (!) ! F (!) > 0. *! n X n(!)(!) 2 @Gn ( Gn ), . . !
, ( Gn , ( Gn . B
, Gn |
, , , Xt(!) 2= Gn t < n (!). 8 , @Gn fx : (s F ) = 1=ng, , n(!) < n+1 (!) ! 2 D0 n ((X n(!)(!) F ) = 1=n, (X n+1 (!)(!) F ) = 1=(n + 1), ! n (!) 6= n+1(!)), , n (!) < F (!) n > 1 ! 2 D0. T1 < , f n (!) < tg 2 F6t (4.10). 2 n=1 * | - Ft, t 2 T . A F $+ %$ A, $ $ A \ f 6 tg 2 Ft ;% t 2 T . 2 , F - . ' , , $ %. 60
@ 4.5 ($ $). | F6t = fW (s): 0 6 s 6 tg, t > 0. Y (t) = W (t + ) ; ; W ( ), t > 0, $, - F
( F6t, t > 0). , W (t + (!) !) ; W ( (!) !) ! 2 D = f < 1g Y (!) = (4.11) 0 ! 2= D : 1 P
2 n (!) = k2;n 1 Ak n , A1 n = f 6 2;n g, k=1 Ak n = f(k ; 1)2;n < (!) 6 k2;n g k > 2. < , n (!) # (!) n ! 1
! 2 D . , , n 2 N n
F6t, . . t > 0
f n 6 tg = f 6 k2;n g 2 Fk2;n Ft k = maxfl : l2;n 6 tg: ( !) W (t + n (!) !) ! W (t + (!) !) n ! 1 t > 0 ! 2 D . *
n 2 N t > 0, z 2 R, ! 2 D 1
f! : W (t + n (!) !) 6 zg = f! : W (t + n (!) !) 6 z n (!) = k2;n g = k=1
=
1
k=1
f! : W (t + k2;n !) 6 z n(!) = k2;n g 2 F : (4.12)
3 , | ,
f = yg = f 6 yg n f < yg 2 F6y 1 S y 2 R, f < yg =
6 y ; k1 2 F6y (F6y; k1 F6y k=1 1 1 1 y > 0 k, y ; k > 0P y ; k < 0, 6 y ; k = ?). ' ( . 4.6. (M A) | , (N ) | - B (N ). Fn : M ! N " A j B(N )- n = 1 2 : : : . Fn(x) ;! F (x) 2 N x 2 M (n ! 1). F A j B (N )- . 2 2 ,
B 2 B(N )
fx : F (x) 2 B g =
1 \
1
\
k=1 m=1 n>m
fx : Fn(x) 2 B (1=k)g
B (") = fx 2 N : (x B ) < "g, (x B ) = inf f(x y): y 2 B g. <
1.2. 2 ? 4.7. * 4.6 (M A) P Fn(x) ;! F (x) n ! 1 P -. . x 2 M. ? F A jB(N )- , A | A P . 2 * Fn(x) ! F (x), n ! 1, x 2 M0, P (M0) = 1. z0 2 N Fen(x) = Fn(x) x 2 M0 Fen(x) = z0 x 2 M n M0, n 2 N. 61
? Fen(x) ! Fe(x) x 2 M, Fe(x) = F (x) x 2 M0 Fe(x) = z0 x 2 M n M0, Fe A j B(N )- . ? , B 2 B(N )
F ;1(B ) = fM0 \ Fe;1(B )g f(M n M0) \ F ;1(B )g 2 A
M0 2 A, Fe;1(B ) 2 A ( 4.6) (M n M0) \ F ;1(B ) M n M0, P (M n M0) = 0. 2 * 4.7 (4.12) Y (t) t. C F jB(R) - , ~ = .., . . ~ F j B(R) - (F ). < , Y () . 8 &, $ Y $ $ F $ %, $ Y | . ,
4.3, , A 2 F , n 2 N, 0 6 t1 < : : : < tm, B 2 B(Rm)
P (A \ f 2 B g) = P (A)P ( 2 B )
(4.13)
= (Y (t1) : : : Y (tm)). (4.13) B ( 2.3 B " > 0 F" B , P (B n F") < "). * (4.13) E1 A 1 f2Bg = E1 A E1 f2Bg:
(4.14)
B , f : Rm ! R
E1 A f ( ) = E1 A Ef ( ):
(4.15)
) (4.15) (4.14). fk (x) = '(k(x B )), '(t) = 1 t 6 0, '(t) = 1 ; t t 2 N0 1] '(t) = 0 t > 1, (x B ) = inf f(x y): y 2 B g, | . / (( B ) B ), ! 2 , , fk (x) # 1 B (x), k ! 1. / 2 A E1 A f ( n ) E1 A f ( ) = lim n n = (W (t1 + n) ; W ( n ) : : : W (tm + n ) ; W ( n)) ! n ! 1 ! 2 D . ? , 2 E1 A f ( n ) =
1 X k=1
E1 A f ( n )1 f n=k2;n g
=
1 X k=1
E1 A\f n=k2;ngf ( n k )
(4.16)
n k = (W (t1 + k2;n ) ; W (k2;n ) : : : W (tm + k2;n ) ; W (k2;n )). C |
fFt t 2 T Rg A 2 F , A \ f < tg = 1 S 1 = A \ 6 t ; q 2 F6t t 2 T A \f = tg = (A \f 6 tg) n (A \ q=1 \ f < tg) 2 F6t. *! A \ f n = k2;n g 2 F6k2;n A 2 F . * 4.3 - F6k2;n n k n k , 62
(W (t1) : : : W (tm)). ? , (4.16) ( : Ef (W (t1) : : : W (tm))
1 X k=1
E1 A\f n=k2;ng = Ef (W (t1) : : : W (tm))E1 A :
A = D, f , m 2 N 0 6 t1 < : : : < tm, Ef (Y (t1) : : : Y (tm)) = Ef (W (t1) : : : W (tm)):
(4.17)
? (4.15) . ) (4.17), , , Y W . / , Y | . 2
$ $ + & $ $ % &. * !
(
14). * | - F6t = fW (s): 0 6 s 6 tg. B ! 2 D = f < 1g (P (D ) = 1) $ %
Z (t !) =
(
W (t !) 0 6 t 6 (!) 2W ( (!) !) ; W (t !) t > (!)
r
6
(4.18)
W (t !)
(!)
X (t !) Z (t ! )
-t
. 5.1
Z (t !) = W (t !) ! 2 D n D . ', $ ;$ $ - % F .
@ 4.8 ( $ &). $ fZ (t) t > 0g
.
2 B t > 0 Z (t !) = W (t !)1 f >tg + +(2W ( (!) !) ; W (t !))1 f
C0N0 1) = ff 2 C N0 1): f (0) = 0g, (3.20), Kn = N0 n], n 2 N. 63
* Y (t !) | , (4.11). ( $ % X (t !) = W (t ^ (!) !), t > 0, t ^ s = minft sg (X (t !) = = W (t !) ! 2 D n D ). J , X ( !) | .!. C0N0 1). < ( b f g) h : Y ! C0N0 1), Y = N0 1) C0N0 1) C0N0 1),
h(b f () g())(t) = f (t)1 0 b](t) + (f (b) + g(t ; b))1 (b 1)(t): & , ! 2 D
h( (!) X ( !) Y ( !)) = W ( !) h( (!) X ( !) ;Y ( !)) = Z ( !):
? 4.9. C {.!.
(D F P )
(X B) = .., P = P B. *! % % $ $ D = D. ? , &: ) h 2 B(Y ) jB(C0N0 1)) (
, C0N0 1) (3.20)P . 2.1)P ) .!. ( X Y ) ( X ;Y )
(Y B(Y )). B
, PW = P( X Y )h;1 PZ = P( X ;Y )h;1 . ). H 2 C0N0 1) r > 0. 2.2 , A = f(b f g) 2 Y : (h(b f g) H ) 6 6 rg 2 B(Y ). B , t 2 M , M |
N0 1), At = f(b f g) 2 Y : 6 h(b f g)(t) 6 g 2 B(Y ) ;1 < < < 1. B , t th : Y ! R , t = 0 1) t, . . 4. 8 & $& %). ,
4.10. X (t ) F jB(R)- 2
..
t > 0.
1 X k;1
2n 1 N k2;n1 2kn ) ( (!)): ? n(!) " (!) ! 2 D (n ! 1), , W (t ^ n(!) !) ! W (t ^ (!) !) t > 0 ! 2 D (W , X , ! 2 D). 4.7 F j B(R)- W (t ^ n() ) (- % F $ $ ). B B 2 B(R), C = f! : W (t ^ n() ) 2 B g s 2 N(m ; 1)=2n m=2n ), n m > 1,
n (!) =
1
k=1
C \ f 6 sg = f 6 s n = k 2;n 1 W t ^ k 2;n 1 2 B g = k=1 k ; 1 k ; 1 k = n 6 < 2n W n ^ t 2 B 2 2 16mk6;m;11 m ; 1 2n 6 6 s W 2n ^ t 2 B 64
(4.19)
k ; 1 ( m = = ?). < , W 2n ^ t 2 B 2 ? k k ; 1 k k ; 1 2 F6 k;n F6s k 6 m, 2n 6 < 2n = < 2n n < 2n 2 F6 kn F6s m ; 1 S1 1 S 1
2
1
2
k 6 m ; 1 2n 6 6 s 2 F6s (f < ug =
6 u ; q 2 F6u u > 0P q=1 f < 0g = ?). 2 *
4.6, F A, , X ( !) F jB(C0N0 1))- .!. & , . . 2 F jB(R), s t > 0 f 6 sg\f 6 tg = = f 6 s ^ tg 2 F6s^t F6t. * 2.1, , ( X ( !)) F j B(R) B(C0N0 1))- .!. * 4.5 .!. ( X ) Y . / , P( X Y ) = P( X ) PY = P( X ) W. < , ;Y F , . . ;Y , P( X ;Y ) = P( X ) P;Y = P( X ) W = P( X Y ), . 2
A%
M (t !) = sup W (s !) t > 0: s2 0 t]
2 , M (t ) t > 0
. ., ..
Gtf = sup f (s) f 2 C0N0 1) s2 0 t]
(4.20)
C0N0 1) R. B ( .
@ 4.11 ( / ). (D F P ) .. fXt t 2 T Rg , .. Xt : D ! Xt , Xt 2 FjBt, t 2 T . ) F>t = fXs s 2 T \ Nt 1)g: - F 1 := \t2T F>t , .. P (A) " $ " A 2 F 1 ( T \ Nt 1) = , F>t := ). 2 3 + + A 2 F 1 &, $ A $ $ A. ) A 2 F>t t 2 T . B " > 0 4.2 (
A" %, ( F>t, .. A" fXt : : : Xtn 2 B g, B 2 B(t1 : : : tn), t 6 t1 < : : : < tn ( T ), n 2 N, , P (A4A") < " ( 1
4.1 2.5). / ,
jC (A A) ; C (A A")j 6 2P (A4A") < 2" C (A D) = P (AD) ; P (A)P (D) A D. / 4.2 , - B(t1 : : : tn) F>t t > tn. *! A A", .. C (A A") = 0. ? , C (A A) = P (A) ; P (A)2 = 0 ( "). 2
4.12. a > 0. a(!) = inf ft ( # $
> 0 : W (t !) = ag { Ft = fW (s) 0 6 s 6 tg). 65
2 ) 4.2 , a { . 8 &, $ a(!) < 1 .. B a > 0 p P ( a < 1) > P ( sup W (t) > a) > P (W (n) > a n ..) > t2 0 1)
> P (lim sup n;1=2W (n) > a) > lim sup P (n;1=2W (n) > a) = P ( > a) > 0 n!1
n!1
(4.21)
N (0 1) " .." ( n). 3
,
. . Yn n 2 N c 2 R
P (lim sup Yn > c) = P (\1n=1 m>n fYm > cg) = nlim P (m>n fYm > cg) !1 n!1
P (m>n fYm > cg) > P (Yn > c) n 2 N,
P (lim sup Yn > c) > lim sup P (Yn > c): n!1
n!1
? , c 2 R
n X p fW (n)= n > c ..g f X =pn > c ..g 2 F 1 k=1
k
- F 1
Xk = W (k) ; W (k ; 1) k 2 N: B
,
flim sup n!1
n X k=1
n X p p X = n > cg flim sup X = n > cg 2 F k
n!1 k=m
k
>m
m 2 N. *
p
P flim sup W (n)= n > cg 2 f0 1g: n!1
) (4.21) , ! ( P ( a < 1)) 1. 2 @ 4.13. t x y > 0
P (W (t) < y ; x M (t) > y) = P (W (t) > y + x): (4.22) 2 C y = 0, (4.22) (
P (W (t) < ;x) = = P (W (t) > x). * y > 0. * 4.12 y = inf fs > 0: W (s !) = yg F6t = fW (s): 0 6 s 6 tg. * Z (t !) (4.18) = y . < , y (!) = = inf fs > 0: Z (s !) = yg F6(Zt ) = fZ (s): 0 6 s 6 tg, ! y (!) y (!) y > 0. & , f y 6 tg = fM (t) > yg t y > 0. *! B 2 B(C N0 1)), t > 0 P ( y 6 t W () 2 B ) = P ( sup W (s) > y W () 2 B ) = P (W () 2 B~ \ B ) s2 0 t]
66
B~ = G;t 1 (Ny 1)) 2 B(C N0 1)), . (4.20). * 4.8, , P (y 6 t Z () 2 B ) = P (Z () 2 B~ \ B ) = P (W () 2 B~ \ B ): ) , .!. ( y W ) (y Z ) . / , x 2 R, t y > 0 P (y 6 t Z (t) < y ; x) = P ( y 6 t W (t) < y ; x): (4.23) W W ( y (!) !) = y y > 0, ! 2 D . *! t > y (!) Z (t !) = 2W ( y (!) !) ; W (t !) = 2y ; W (t !). ? , y > 0 x 2 R (4.23) P (M (t) > y W (t) < y ; x) = P (y 6 t Z (t) < y ; x) = = P (y 6 t W (t) > y + x) = P ( y 6 t W (t) > y + x) = = P (M (t) > y W (t) > y + x): (4.24) C x > 0, P (M (t) > y W (t) > y + x) = P (W (t) > y + x) (4.24) (4.22). 2 $ 4.14. t y > 0 P (M (t) > y) = 2P (W (t) > y): (4.25) 2 x = 0 (4.22). ? P (W (t) < y M (t) > y) = P (W (t) > y) P (M (t) > y) = P (M (t) > y W (t) < y) + P (M (t) > y W (t) > y) = = P (W (t) > y) + P (W (t) > y) = 2P (W (t) > y) (, P (W (t) = y) = 0 y 2 R t > 0). 2 & , M (t) ( 6. $ 4.15. 3 y > 0 0 6 a < b < 1 P ( sup jW (t) ; W (a)j > y) 6 4P (W (b ; a) > y) 2P (jW (b ; a)j > y): (4.26) a6t6b
2 4.3 P ( sup jW (t) ; W (a)j > y) = P ( sup jW (s)j > y) 6 06s6b;a a6t6b 6 P ( sup W (s) > y) + P (06inf W (s) 6 ;y): s6b;a 06s6b;a
* , sup (;W (s)) = ; inf W (s) t > 0 ;W s2 0 t] s2 0 t] ,
(4.25). 2 B t > 0 Log t = ln(t _ e). < (4.26)
(
.
@ 4.16 ( $ ). ' "
$
(t) lim sup (2t LogWLog t)1=2 = 1
(4.27)
W (t) lim inf = ;1: t!1 (2t Log Log t)1=2
(4.28)
t!1
67
6 (1;")p2t Log Log t
(1+")p2t Log Log t
~ 0
W (t)
t
t0(" !)
. 5.2
? 4.16 , p ( $ % (1 + ") 2t Log Log t ( " > 0, t0(" !)). 1 p $ % $ % (1 ; ") 2t Log Log t (. . 5.2). B
! ( ) 3, , ! ( h . ' ( , . @ W0(t) = W (t) ; tW (1) t 2 N0 1]: (4.29) '
, W0(0) = W0(1) = 0, (4.29).
.
D. 4.1. * ,
4.1, , fW (t) 0 6 t 6 1g .. ,
( 9 > 1=2 (. . 3.8 ??). D. 4.2. * 3.8 B2.24 , ( ) q 2 N, 0 6 s < t < 1 E(W (t) ; W (s))2q = (2q ; 1)!!(t ; s)q : (4.30) 68
< H (!) ! 2 D ( , D = C N0 1))
t 2 N0 1),
9 . 8 3.8, 4.2 , P (H = N0 1)) = 1 < 1=2. 8 4.1 , P (H = ) = 1 > 1=2. ) ?? , P (t 2 H1=2) = 0 t > 0. < B! N?] 1983 . , P (H1=2 6= ) = 1. ? , $$ +-
( ) $ ; % & $ . 1 J. . / (. N?]). @ 84.3. X { . ., "( EjX j < 1.
( " ( ) W (t) t > 0, . . ,
X =D EX + W ( ): (4.31) ! EX 2 < 1, ( E E = DX: (4.32) 2 * (D F P ) X fW (t) t > 0g (
). B , EX = 0 X { ( 0). * X a b (a < 0 < b, EX = 0). C P (X = a) = p P (X = b) = 1 ; p (4.33)
EX = 0
(4.34) p = b ;b a 1 ; p = b;;aa : 4.4
a b = inf ft > 0 : W (t) 2 fa bgg
. B, a b , .. a b < 1 .. (! 4.12). 0 , , E ak b < 1 k 2 N. B m 2 N f a b > mg fjW (n) ; W (n ; 1)j 6 b ; a n = 1 : : : mg: P (jW (n) ; W (n ; 1)j 6 b ; a n = 1 : : : m) = P (j j 6 b ; a)m N (0 1). *! ( ) (4.35) P ( a b = 1) = 0 E ak b < 1 (k 2 N): 8 , : W ( a b) = a W ( a b) = b ( 1). ) (4.35), P (W ( a b) = a) = pa b P (W ( a b) = b) = 1 ; pa b: (4.36) 69
B, EW ( a b) = 0
(4.37)
(4.36) , pa b = b=(b ; a) , (4.33), (4.34),
X (4.31) . $&$ 3 + : 1 2 : : : ...
{ ( N) - Fn = f 1 : : : n g, n 2 N, Ej 1j < 1 E < 1. ? E( 1 + : : : + ) = E E 1: B
, Ej 1 + : : : + j = =
1 X n=1
1 X n=1
E(j 1 + : : : + j1( = n)) =
E(E(j 1 + : : : + n j1( = n)jFn )) = =
1 X n=1
P ( = n)Ej 1 + : : : + n j 6
1 X n=1
1 X n=1
(4.38) E(j 1 + : : : + n j1( = n)) =
E(1( = n)E(j 1 + : : : + njjFn )) =
1 X n=1
nP ( = n)Ej 1j = E Ej 1j:
(4.39)
3
, ( (., ., N?]). ( . J (4.39) ( ) (4.38). D. 4.4. B $ $&$ 3 + . J , , ( (4.38), D 1 < 1, E( 1 + : : : + ; E 1)2 = E D 1:
(4.40)
? n 2 N n m = W (m=n) ; W ((m ; 1)=n), m 2 N
a(nb) = inf fm : n 1 + : : : + n m 2= (a b)g:
(4.41)
D. 4.5. B , n 2 N a(nb) { - Fk(n) = f n 1 : : : n k g, k 2 N E a(nb) < 1.
) ! , (4.35) (4.38), , n 1 + : : : + n a(nb) = W ( a(nb)=n) E n 1 = 0, EW ( a(nb)=n) = 0:
(4.42)
D. 4.6. 8 , , (4.43)
a(nb)=n ! a b .. n ! 1: 70
) 4.6 , EW ( a(nb)=n) ! E(W a b) n ! 1: (4.44) B ! ,
. . f 2 Rg ,
Z
lim sup j jdP c!1 2 fj j>cg
= 0:
(4.45)
D ) (., ., N?, . ??]), n ;! ( ) f n g , E n ! E n ! 1 (4.46) ( , n > 0, (4.45) (4.46)). < .. . D. 4.7. B , sup Ej j < 1 > 1 2
f 2 Rg. D. 4.8. / ( (4.35), (
, jW ( a(nb)=n)j 6 b + a + j n a(nb) j
, fW ( a(nb)=n) n 2 Ng . 8 (4.43), 4.8, (4.44). (4.42) (4.37). ? , (4.33) ,
X =D W ( a b): (4.47) L % . < F (x) = P (X 6 x). 8 , EX = 0 , . . X , c=
Z
(;1 0]
(;y)dF (y) =
Z
(0 1)
zdF (z) 6= 0:
(4.48)
* f : R ! R { . ?
cEf (X ) = c +
Z (;1 0]
f (y)dF (y)
< Ef (X ) = c;1
Z
Z1
Z
;1
f (x)dF (x) =
(0 1)
zdF (z) =
Z
Z
Z
(0 1)
(0 1)
f (z)dF (z)
dF (z)
Z (;1 0]
Z
(;1 0]
(;y)dF (y)+
dF (y)(zf (y) ; yf (z)): (4.49)
dF (z) dF (y)(z ; y) f (y) z ;z y + f (z) z;;yy : (0 1) (;1 0]
(4.50) 71
' (D0 F 0 P 0) (Y Z ) R2 ,
P 0((Y Z ) 2 B ) = c;1
ZZ
(z ; y)dF (z)dF (y):
B \f(;1 0] (0 1)g
(4.51)
2 , (4.51) . ?, ! , (4.50),
f 1. B y < 0 < z, (4.36), py z = z=(z ; y) (4.47)
f (y) z ;z y + f (z) z(;;yy) = Ef (W ( y z )):
(4.52)
1 y 6 0 < z, 0 b = 0 b > 0. L (D^ F P ) = (D F P ) (D0 F 0 P 0): ? , (4.51), (4.52), (4.37) (2.10),
(4.50) Ef (X ) = Eef (X ) = E0Ef (X ) = E0Ef (W ( Y Z )) = eEf (W ( Y Z ))
(4.53)
E0 P 0, eE { Pe = P P 0. *! 2.9
PeX = PeW ( Y Z ):
(4.54)
f(t !e) := W (t !), B
(4.31) ( ), W 0 t > 0 !e = (! ! ) 2 De , ,
(D^ F P ). B , (D F P )
E P . D. 4.9. )
( 4.4), , E a b = ;ab:
(4.55)
$+ $+ EX 2 < 1.
(4.55) I , Ee Y Z = E0E( Y Z ) = E0(;Y Z ):
(4.56)
* (4.56), (4.51) (4.48), Ee Y Z = E(;Y Z ) = = = 72
Z
Z (;1 0] (;1 0]
Z
(;1 0]
dF (y)(;y)
dF (y)(;y) ;y +
y2dF (y) +
Z
(0 1)
Z
Z
(0 1)
(0 1)
dF (z)z(z ; y)c;1 =
dF (z)c;1z2
=
z2dF (z) = EX 2 = EeX 2 : 2
@ 84.10 (, M?]). X1 X2 : : : { $ -
, . - ,
. . Tk , k 2 N fW (t) t > 0g ,
fXk k 2 Ng =D fW (Tk ) ; W (Tk;1) k 2 Ng $ Tk ; Tk;1 , k 2 N (T0 0) EXk2 < 1, E(Tk ; Tk;1) = EXk2.
(4.57) ,
2 B
B4.3. * Fk .. Xk (k 2 N). * (D F P ) (Yk Zk ), k 2 N, , (Yk Zk ) (4.51), F Fk . *
f(Yk Zk ) k 2 Ng fW (t) t > 0g. * Tk = inf ft > Tk;1 : W (t + Tk;1) ; W (Tk;1) 2= (Yk Zk )g k 2 N:
(4.58)
(4.58) (Yk Zk ) , Yk Zk . ? fTk k 2 Ng { . 2 * W (t) = (W (1)(t) : : : W (q)(t)) q- % & ( .. W (j)(t) t > 0, j = 1 : : : q, , , q .!. C N0 1)).
(C N0 1])q ,
kx()k = sup jx(t)j t2 0 1]
j j { Rq . <
gn(t) = p W (nt) t 2 N0 1] n 2 N 2nLog Logn
(4.59)
Logz = (log z) _ 1 z > 0 Log Logz = Log(Logz) ( , (4.59) , n > 3).
@ 84.11 ( + $ , H$ -). fgn g (C N0 1])q -
: , . .
K = fx : x(t) =
R
R
Zt 0
h(s)ds s 2 N0 1]
R
Z1 0
jh(s)j2ds 6 1g
t t t 0 h(s)ds = ( 0 h(1)(s)ds : : : 0 h(q) (s)ds). 3 , K " # $ x , x(j )(0) = 0 j = 1 : : : q
Z1 0
jdx=dtj2ds 6 1
dx=dt = (dx(1)=dt : : : dx(q)=dt). 73
? 84.12. ? B4.11 ( , -
)
gT (t) = p W (Tt) T > 0 t 2 N0 1] 2T Log LogT T ,
gTn (), fTng { , Tn > 0 Tn ! 1 n ! 1. D. 4.13. B ,
K (" h ") { (C N0 1])q. D. 4.14. * C (fxng) {
fxng
(X ). * h { X
(Y ). B , C (fhxng) = h(C (fxng)).
h(x()) = x(1)(1) x 2 (C N0 1])q: ? B4.11 4.14 ( 6 ) ( ( 4.16. $ 84.15. W (t) { . " ftn g, p tn ! 1 n ! 1, fW (tn )= 2tn Log Log tn n 2 Ng " 1 N;1 1], .. "- " . 2 B 4.11. 2 , $& "C (fgn g) = K .." + ; ; $< : 1) " > 0
P (gn 2= K " ..) = 0
(4.60)
B " { ! "-
B , " .." (fAn ..g = \n j>n Aj ), 2) x 2 K " > 0 1
gnk ( !) 2 fxg" k > N
(4.61)
fnk g N ", x ! !. ) , + " > 0 % $+ P (gn 2= K ") % +< n. L N0 1] i=m i = 0 1 : : : m, m . I g 2 (C N0 1])q ( ) gb (C N0 1])q, ( (i=m g(i=m)) i = 0 1 : : : m. B r > 1
P (gn 2= K ") 6 P (r;1bgn 2= K ) + P (r;1 bgn 2 K kgn ; r;1bgn k > ") =: p1 + p2: B,
74
Z 1 1 dbg 2 p1 = P (r;1gbn 2= K ) = P ( r dt dt > 1) = P (2md > 2r2Log Logn) 0
(4.62)
Z 1 1 dbg 2 q X m 2 X ;1 2 ( j ) ( j ) dt = q = (2Log Log n ) g ( i=m ) ; g (( i ; 1) =m ) md n n 0 r dt j =1 i=1
gn(t) = (gn(1)(t) : : : gn(q)(t)), 2d { , ( - d (d 2 N) . ', 2d 8 zd=2;1e;z=2 < p2d (z) = : 2d=2 ;(d=2) z > 0 0 z < 0: R ;() = 01 x;1e;xdx > 0. *! ( ) d=2;1 ;x=2 P (2d > x) 2xd=2;1;(e d=2) x ! 1: / , c1 ( c , ( n) n
p1 6 c1 expf;rLog Logng:
(4.63)
B,
p2 6 P (r;1bgn 2 K (1 ; r;1)kr;1 bgn k > "=2) + P (kgn ; gbn k > "=2): (4.64) r = r(") 1, , n P (r;1 bgn 2 K (1 ; r;1)kr;1gbn k > "=2) = 0 (4.65) r;1bgn 2 K kbgn k 6 r. B
, x() 2 K , 06s6t61
Z t Z t !1=2 dx du 6 (t ; s)1=2 jx(t) ; x(s)j = du du 6 (t ; s)1=2 dx s s du R R R t dx du = ( t dx du : : : t dx q du). &
(1)
s du
( )
s du
s du
4.15,
P (kgn ; bgn k > "=2) 6
6q
m X i=1
P(
sup
t2 (i;1)=m i=m]
m X i=1
P(
sup
t2 (i;1)=m i=m]
jgn (t) ; gn ((i ; 1)=m)j > "=4) 6
p
p
jw(t) ; w((i ; 1)=m)j > ("=4) m=q 2Log Logng) 6
6 qmc2 expf;Log Logng"2m=(16q)g 6 c3 expf;rLog Logng
(4.66)
w { m > 16qr";2, ,
pqg: fy 2 Rq : jyj > "g fy 2 Rq : 1max j y j > "= i 6i6q
(4.67) 75
? , c > 1, % nk = Nck ], N] { , (4.62) { (4.66)
X k
P (gnk 2= K ") 6 c4
X k
expf;rLog Lognk g < 1:
$ +, E { / $ gnk 2 K " .. ! k > N (" c !). L $ $+ gn n 2 Nnk nk+1]. 8 (4.67), P (n 6max kgn ; gnk k > ") 6 k n6 nk +1
! w ( n ) p w ( n ) 6 qP nk 6max ; p2n Logk Logn > "= q 6 p n6nk 2nLog Logn k k !
+1
6 qP
kwp(n) ; w(nk )k > "=(2pq) + max nk 6n6nk+1 2nLog Logn
p ; 1 = 2 ; 1 = 2 qP n 6max (2nLog Logn) ; (2nk Log Lognk ) kw(nk )k > "=(2 q) 6 k n6nk " p +1
6 qP
sup
s t2 0 nk+1 ] js;tj6nk+1 ;nk
jw(s) ; w(t)j > 2pq 2nk Log Lognk +
+qP sup jw(t)j > 2p" q ((2nk Log Lognk );1=2 ; (2nk+1 Log Lognk+1 );1=2) =: q(p3 + p4 ): t2 0 nk ] (4.68) ? ,
4.15. D. 4.16. B y > 0, 0 6 a < b < 1 6 b ; a
P
sup
s t2 a b] js;tj6
p c0(b ; a) ;y =16 jw(s) ; w(t)j > y 6 y e : 2
) (4.68) 4.16 , c 1, = (c) > 1 k 2 N
pi 6 c5 expf; Log Lognk g i = 3 4: / 0 { , , , gn 2 K 2" . . ! n > N (" !). < , " > 0 . 2$ , (4.60)
$ . 8 & (4.61). D. 4.17. < 6 ,
(4.61) , x 2 K ,
Z 1 dx 2 dt = a2 < 1 0 dt
P (gn 2 fxg" ..) = 1. 76
(4.69)
& , fgn 2 fxg"g = fkgn ; xk < "g f1max jg (i=m) ; x(i=m)j+ 6i6m n +
jx(t) ; x(s)j + 1max 6i6m
sup
s t2 0 1] js;tj61=m
sup
s2 (i;1)=m i=m]
jgn (s) ; gn (i=m)j < "g:
* x (. (4.69)) sup jx(t) ; x(s)j 6 a=pm s t2 0 1] js;tj61=m
, 1 Bn = f1max jg (i=m) ; gn ((i ; 1)=m) ; (x(i=m) ; x((i ; 1)=m))j < "=(8m)g: 6i6m n
3 , gn 2 K "0 "0 > 0 . . ! ( ! ) n > N ("0 !), ! p jgn (t) ; gn (s)j 6 jt ; sj + 2"0 s t 2 N0 1]: (4.70) 0 , , .. Bn n ml (l 2 N). F $+ $, $ Bn i > 2, % ;$ $+ %$ Bml , l 2 N. 8 (4.70), = "=(8mpq), ,
P (Bn ) > P
j
max max g(j)(i=m) 26i6m 16j 6q n m q
>
;
gn(j)((i
; 1)=m) ;
(x(j) (i=m)
;
x(j)((i
; 1)=m))j < >
YY p i=1 j =1
P ( 2mLog Lognjx(j)(i=m) ; x(j)((i ; 1)=m)j <
p
< w(1) < 2mLog Logn(jx(j)(i=m) ; x(j)((i ; 1)=m)j + "= ): ( . D. 4.18. B 0 6 u < v < 1
Zv 2 1 p e;s =2ds > p1 e;u2 =2(1 ; e;(u2;v2 )=2): 2 u v 2 * , Z 1 dx 2 q X m X ( j ) ( j ) 2 lim m (x (i=m) ; x ((i ; 1)=m)) = dt dt m!1 0 j =1 i=1 ( ,
x K ), , P (Bn ) > expf; Log Logng (4.71) 2 (a 1), = (x() m)P l. < , l P (Bml ) = 1 (4.71) $ 1 X E { / $ : A1 A2 : : : P (An) = 1, n=1
P (An ..) = 1. 2 C( ( $ + $ / { $ (. N?]), , { * { 1 { I. 77
@ 84.19. (t) t > 0 { "( # $ , (t) ! 1 t ! 1.
0 I () < 1 p P (w(t) > t(t) ..) = 1 I () = 1
Z 1 (t)
I () = expf;(t)2=2gdt: t 1 & , ! , 4.16, p I () , (t) = (1 + ") 2tLog Logt, " > 0 " < 0,
( { # $ , P (w(t) > (t) ..) = 0, { P (w(t) > (t) ..) = 1, tn !p1 w(tn)). ? B4.19 " = 0, .. (t) = 2tLog Logt, , (
) . B4.19
: &$ + $ % % , & (4.59), $ fn(t) = pWn(nt(n)) t 2 N0 1] n 2 N { , B4.19. @ 84.20 (N?]). C (ffn g) = KR . ., KR R = R(), " # $ x ,
Z1
jdx=dtj2ds 6 R2
0 ( R = 0, KR = 0 , .. -# $ R = , KR = (C N0 1])d). R2() = inf r > 0 : I ( r) <
1
fg
f
Z 1 (t)
, ",
1g
expf;r(t)2=2gdt: I ( r) = t 1 D. 4.21. N. N?]] B , (t) R();1 = lim inf t!1 2Log Log t ( 0;1 = 1 1;1 = 0). ) , (I&*2). ', N?], I&*2, . N?] . < I&*2 , , . N?]. D. 4.22. * a = inf ft > 0 : w(t) = ag, a 2 R. B , D
a = a2 1. D. 4.23. * U { ! w (0 t). B , rx 2 P (U 6 x) = arcsin t x > 0: 78
5. '#
3 , , . . , . . $ . . = . " . $ . $ . = . . 3 " " . . ' . = . >1 ? " { > . " L1() F P ). = '. .
. + { $+ + $ % + %< % $ $ , ; % &. *
(D F P ) fFt t 2 T Rg, .. (
- Ft F , t 2 T (Fs Ft s 6 t, s t 2 T ). 9 , (Xt Ft)t2T , Xt : D ! R, , s t 2 T , s 6 t ( 1) Xt 2 FtjB(R), , fXt t 2 T g fFt t 2 T g, 2) EjXtj < 1, 3) E(XtjFs) = Xs .. C 3) , E(XtjFs) > Xs .., (Xt Ft)t2T . & 3) E(Xt jFs ) 6 Xs .. . < , (Xt Ft)t2T $ $ $ $ + $ , (;Xt Ft)t2T { % $ (! ). 3 , , . & , 3) , s t 2 T , s 6 t A 2 Fs
Z
A
Xs dP =
Z
A
XtdP:
(5.1)
/
(5.1)
"6". C (Xt Ft)t2T { ( ) (Gt)t2T , Gt Ft, Xt 2 GtjB(R) t 2 T , (Xt Gt) { ( ). 1 " "
: - A1 A2 F E(E( jA1)jA2) = E(E( jA2)jA1) = E( jA1) .. (5.2) ,
(Gt)t2T " # $ " FtX = fXs s 6 t s 2 T g, t 2 T . "3 " , .. FtX Gt t 2 T
1). , , ,
, (Xt t 2 T ) { ( ). ? ,
. S
, .. (D F Ft P ) # . , , 79
P
P . ? # (D F Ft P ). C , , (Xt Ft P ) { , ,
3) P -.. < Rm, , 1),2),3) . B T = Z+ 3) ( ) t = s + 1. ) , ( ), E(]XnjFn;1 ) = 0 .., ]Xn = Xn ; Xn;1 n 2 N: (5.3) * ( n Fn)n2Z+, n { - Fn (n 2 Z+), - ", E( n jFn;1) = 0 .., n 2 N. ? , (5.3) , (Xn Fn)n2Z+ {
, (]Xn Fn)n2Z+ { - (]X0 := 0). . 1. * fXt t 2 T Rg { (, EXt = c t 2 T (c = const). ? (Xt)t2T { . B
, s 6 t, s t 2 T
(Ft)t2T E(XtjFs) = E(Xt ; Xs + Xs jFs) = E(Xt ; Xs) + Xs .. (5.4) 3 , Xt ; Xs { Fs (. 4.2) E( jA) = E , A E (
. / , $ $ . fNt ; ENt t 2 R+g, fNt t > 0g { , $ & $ $ . B
(5.4) ,
. , t Xt Ft Xt ; Xs Fs. , Sn = 1 + : : : + n, n 2 N, , 1 2 : : : { E n = 0, n 2 N. & , Fn = fS1 : : : Sn g = f 1 : : : n g, n 2 N. 2. Q* n , n 2 N { E n = 1 n. * Xn = nk=1 k , Fn = f 1 : : : ng, n 2 N. ? (Xn Fn)n2N { ( 6 ). loc 3. * (D F ) Q P , (Ft)t2T Q P , .. Qt := QjFt Pt := P jFt , t 2 T . B , Q P Ft t 2 T . *
(D A). ', (
" ), (A) = 0 (A) = 0. 1 , (
AjB(R)- g (g > 0 -..), R , d=d , , (B ) = B gd B 2 A. ) , gt = dQt=dPt . ? (gt Ft)t2T { . B s 6 t, s t 2 T , B 2 Fs ( B 2 Ft), ,
Z
B
gs dPs = Qs(B ) = Q(B ) = Qt(B ) =
Z
B
gtdPt: loc
< (5.1). & , Q P , Q P . 4. L ( + $ . * ("n)n2N { , ( 1 -1 ( (D F P )). / , "n = 1 n- 80
, "n = ;1 . * Vn { !
n- , n > 2
( ) "1 : : : "n;1 V1 : : : Vn;1 . ?
, V1 f"n gn2N. * F0 = fV1g, Fn = fV1 "1 : : : "ng, n 2 N. ) , V = (Vn )n2N , .. Vn 2 Fn;1jB(R), n 2 N ( (Vn Fn;1)n2N). , V0 const, F0 = f Dg Fn = f"1 : : : "ng, n 2 N. / ( , ) n
Xn = Xn;1 + Vn "n =
n X k=1
Vk ]Yk n 2 N (X0 = 0)
(5.5)
]Yk = Yk ; Yk;1 , Y0 = 0, Yk = "1 + : : : + "k , k 2 N. * (Xn )n2Z+ V Y
(V Y ). / , EVk < 1, k 2 N ( Vk > 0 ), (5.5) E(]Xn jFn;1) = Vn E"n .. n 2 N Vn { "n Fn;1 n 2 N. ? , (Xn Fn )n2N { , E"n = 0, .. P ("n = 1) = P ("n = ;1) = 1=2 n 2 N, (Xn Fn)n2N { , P ("n = 1) > 1=2 n 2 N (
). /
, ( ) . L , V1 = 1 Vn = 2n;1 1 f"1 = ;1 : : : "n;1 = ;1g n > 2: B , ( ) ( . ( , : D ! f0 1 : : : 1g. *, {
- (Fn)n2Z+. B
, f = 0g = 2 F0 f = ng = f"1 = ;1 : : : "n;1 = ;1 "n = 1g 2 Fn n 2 N: C , P ( = n) = 2;n , n 2 N, , , P ( < 1) = 1. * ! X = 1, ..
f = ng, n 2 N
X = Xn = ;
n;1 X k=1
2k;1 + 2n;1 = 1:
) , P (X = 1) = 1 EX = 1, , EXn = 0 n 2 N . L , ( $ , ) . ' , . < , (5.5) , (. 5. * (Ft)t2T { { . * Xt = E( jFt), t 2 T . ? (5.2) , (Xt Ft)t2T { . 1 + .
H ; &$+ $$+ % $ $ $ 81
5.1. (Xt Ft)t2T {
. h : R ! R { # $ , Yt = h(Xt ) t 2 T . (Yt Ft)t2T { . ! # $ h , , (Xt Ft)t2T { .
2 8
k (. N?, . 250]) , s 6 t (s t 2 T ) h(Xs ) = h(E(XtjFs)) 6 E(h(Xt )jFs): (5.6) C (Xt Ft)t2T { h ,
(5.6)
h(Xs ) 6 h(E(Xt jFs)) .. 2
/ ( , & 8 % . @ 5.2 (8 %). (D F P ) ( n) $ (Xn )n2Z+ , # $ (Fn )n2Z+ . ( (. .) X = M + A, .. Xn = Mn + An n 2 Z+, (Mn Fn )n2Z+ { , (An Fn;1 )n2Z+ { $ A0 0 F;1 = f Dg. 9 , (Xn Fn )n2Z+ , $ A , .. ]An > 0 . . n 2 N. 2 C X , ]An = E(]Xn jFn;1)
n 2 N (5.3), (A0 = 0)
An =
n X k=1
E(]Xk jFk;1) n 2 N
(5.7)
. B X ,
( A (5.7) A0 = 0. ? M = X ; A , 1),2), , E(]Mn jFn;1) = E(]Xn jFn;1) ; ]An = 0 .., n 2 N:
(5.8)
8
, ( , (5.7) (5.8). 2 * , ; $+ $ 5.2 . * S0 = 0, Sn = "1 + : : : + "n, "1 "2 : : : { , P ("n = 1) = P ("n = ;1) = 1=2, n 2 N. L B ( 5.1) Xn = jSn j, (F0 = f Dg, Fn = f"1 : : : "ng, n 2 N), n 2 Z+. ) ]Xn = jSnj ; jSn;1j, n 2 N. ? ]Mn = ]Xn ; ]An = ]Xn ; E(]XnjFn;1 ) = jSn j ; E(jSn jjFn;1) n 2 N:
(5.9)
? ,
jSnj = jSn;1 + "nj = (Sn;1 + "n)1 fSn;1 > 0g + 1 fSn;1 = 0g ; (Sn;1 + "n)1 fSn;1 < 0g:
(5.10)
*!
82
E(jSn;1 + "njjFn;1) = E((Sn;1 + "n )1 fSn;1 > 0gjFn;1 )+ +E(1 fSn;1 = 0gjFn;1 ) ; E((Sn;1 + "n )1 fSn;1 < 0gjFn;1 ) =
= Sn;1 1 fSn;1 > 0g + 1 fSn;1 = 0g ; Sn;11 fSn;1 < 0g n 2 N:
(5.11)
3 , E("njFn;1) = E"n = 0. ) (5.9) { (5.11) ,
Mn = / (5.7),
An =
n X k=1
n X k=1
(sgnSk;1 )]Sk
8 1 x > 0 < sgnx = : 0 x = 0 ;1 x < 0: E(]Xk jFk;1) =
n X k=1
(E(jSk;1 + "k jjFk;1) ; jSk;1j):
) (5.11) , E(]Xk jFk;1) = 1 fSk;1 =0g k 2 N, ! An = Ln (0),
Ln (0) = #fk 1 6 k 6 n : Sk;1 = 0g
.. Ln (0) { ! fSk g06k6n;1 . ? ,
jSnj =
n X k=1
(sgnSk;1)]Sk + Ln (0)
(5.12)
@ . ) (5.12) , ELn (0) = EjSnj:
(5.13)
* ,
r
ELn (0) 2 n n ! 1: (5.14) ? (
) ( ( , !
(5.14) 6.7. ? 5.3. 5.2 , A0 0 A0 = 0 .. (
). B , , A1 = E(]X1jF0) + A0 F0jB(R) { . S , ;$, $ $$ $ $ (D F P ) +$ (Ft)t2T R $ & , .. - Ft P -
F ( (D F )). < N = fA 2 F : P (A) = 0g. B - A A = fA Ng. * Ae = fA N A 2 A N 2 Ng. < , Ae - , Ae A. / , A N Ae, ! A Ae. / , A = Ae. * F t = fFt Ng, ( ) , ( (Ft F t t 2 T ). 83
5.4. (Xt Ft)t2T {
( ).
( ).
(Xt F t)t2T
{
2 < , (Xt F t)t2T
1) 2) ( , Ft F t, t 2 T ). s 6 t, s t 2 T , E(XtjF s ) = Xs .. B
, Xs 2 F sjB(R) A 2 F s B 2 Fs C 2 N , A = B C ( ! F BC = ). * (5.1),
Z
A
Xs dP =
Z
B
Xs dP =
Z
B
XtdP =
Z
A
XtdP:
(5.15)
3 , (
E P (C ) = 0, E 1 C = 0. B
(5.15)
"6". 2 C (Xn Fn)n2Z+ { , ( (An Fn;1)n2Z+, ( B (A0 = 0, F;1 = f Dg), (" " ( ). * M = (Mn Fn)n2Z+ { , .. EMn2 < 1, n 2 Z+. ? 5.1 , M 2 = (Mn2 Fn)n2Z+ { . * B Mn2 = mn + hM in , mn { , hM in { , M . ) (5.7) n 2 N
hM in =
n X k=1
E(]Mk2
jFk;1) =
n X k=1
E(Mk2
;
Mk2;1
jFk;1) =
n X k=1
E((]Mk )2jFk;1) (5.16)
E(Mk Mk;1 jFk;1) = Mk;1E(Mk jFk;1) = Mk2;1 , k 2 N. / $ $$ &$ $ $+ . , M0 = 0 Mn = 1 + : : :+ n , n 2 N, 1 2 : : : { P n 2 E k < 1 (k 2 N), (5.16) , hM in = k=1 D k = DMn , k 2 N. $ X = (Xn )n2Z+, NX ]n =
n X k=1
(]Xk )2 n 2 N (NX ]0 = 0):
(5.17)
? % $ $ $ $ +$ ( ) $ , + ; $. * (Xn Fn)n2Z+ { , .. (Xn )n2Z+ (Fn)n2Z+. 3 ! . 3 , 6 k .. k 2 N. * X (!)(!) = 0, (!) = 1 (P ( = 1) = 0 ).
@ 5.5 ( % %, 8 %). (Xn Fn)n2Z+ { , ( . "( :
84
1)
(Xn Fn)n2Z+ {
2)
EX = (>)EX " > . .7
( )7
,
3)
E(X jF ) = (>)X ^ , { , { - F = fA 2 F : A \ f 6 ng 2 Fn ng.
9 # ">" ( ) " 1).
2 1) ) 2). * 6 k .. 6 .., k 2 N. ? X ; X = = = =
k X m=1
k X m=1
k X
m=1
k m X X;1
m=1 j =0
1 f =mg1 f=jg (Xm ; Xj ) =
1 f =mg1 f<mgXm ;
1 f<mg 1 f =mgXm ;
1 f<mg1 f >mgXm ;
k X m=1
k;1 X
k X
j =0
m=1
1 f=jg1 f >jgXj =
1 f=m;1g 1 f >mgXm;1 =
1 f<mg 1 f >mgXm;1 =
k X m=1
* (5.18) k X
m=1
1 f<m;1g 1 f >mgXm;1
/ , E(X ; X ) =
k;1 X
m=0
n X m=1
1 f<mg1 f >m+1gXm
1 f<mg 1 f >mg]Xm: k X m=1
(5.18)
1 f<mg1 f >m+1gXm:
EN1 f<mg 1 f >mgE(]XmjFm;1)] = (>)0:
2) ) 3). * = ^ A 2 F. 2 , A = 1 A + 11 A A = 1 A + 11 A = A ^ k = A ^ k , 6 6 k ! 2 D. *! EX = (>)EX . / ,
? ,
EX 1 A + EXk 1 A = (>)EX1 A + EXk 1 A :
E(E(X jF)1 A) = EX 1 A = (>)EX1 A:
m=0 EjXm j < 1 6 k .. , E(X jF ) 2 F jB (R) (
). 3) ) 1). n m, 0 6 m 6 n (m n 2 Z+). < , F = Fm, ! 3) 1). 2 $ 5.6. (Xn Fn)n2Z+ { ( ) , " "( n , ( .. n 6 n+1 . . n 2 Z+ n 6 kn . . kn 2 N) (X n F n )n2Z+
P 3 , EjX j 6 k
( ).
85
$ 5.7 ( + + $ , 8 %). (Xn Fn)n2Z { . " N 2 N u > 0 uP (0max X > u) 6 EXN 1 f max Xn > ug 6 EXN+ (5.19) 6n6N n +
06n6N
uP (06min X 6 ;u) 6 ;EX0 + EXN 1 f min Xn 6 ;ug 6 ;EX0 + EXN+ : n6N n 06n6N
(5.20)
2 < = minfn : Xn > ug ^ N . ? EXN > EX = EX 1 A + EX 1 A > uP (A) + EXN 1 A 9 A = f0max X > ug. / , 6n6N n uP (A) 6 EXN ; EXN 1 A = EXN 1 A 6 EXN+ : B
(5.20) , = minfn : Xn 6 ;ug^N . 2 * (
, . $ 5.8. (Xn Fn )n2Z+ { EjXn jp < 1 p > 1 n 2 Z+. u > 0 N 2 Z+ ;p EjXN jp : P (0max j X j > u ) 6 u n 6n6N
2 B , (jXn jp Fn)n2Z+ { p > upg: 2 f0max j X j > u g = f max j X j n n 6n6N 06n6N
$ 5.9. (Xn Fn)n2Z { $ . EjXn jp < 1 n 2 f0 : : : N g, N 2 N p 2 (1 1). +
k 0max jX jk 6 (p=(p ; 1))kXN kp 6n6N n p , ,
k kp = (Ej jp)1=p.
(5.21)
2 0 ( ( 5.1) Xn > 0 .. n 2 Z+. * p (. N?, . 223]), EN(0max X 6n6N n
)p] = p
Z1
Z1 0
up;1P (0max X > u)du 6 6n6N n
up;2E(XN 1 f max Xn > ug)du = 0 06n6N p p;1 ) 6 p (EX p )1=pNE(max X )p ](p;1)=p: = p ; 1 E(XN (0max X ) n n 6n6N p;1 N &
(5.19) uP (0max X > u) 6 EXN 1 fmax06n6N Xn>ug 6n6N n
6p
86
, B = inf fn : Xn 2 B g
B 2 B(R). 3 , > 0 .., (
E , E =
Z1 0
E( 1 f>ug)du:
(5.22)
*
N?] . 223 1. < P m = j=1 cj 1 Aj , . 2 '
k max06n6N jXn jk1 .
8 $ % $ %$ N (a b) "
" % $ $ a b $ 0 N . . * (Xn Fn)n2N { -
(a b) { . < ( ! ) k , k 2 N, 0 = 0
2m;1 = minfn : n > 2m;2 Xn 6 ag 2m = minfn : n > 2m;1 Xn > bg m 2 N ( k j , j > k, ,
(
). (. . 5.1)
N (a b) =
0
2 > N maxfm : 2m 6 N g 2 6 N:
5.10 ( , 8 %). 3
- -
+ + EN (a b) 6 E(XN ; a) 6 EXN + jaj : (5.23) b;a b;a 2 * N (a b) (Xn Fn)n2N N (0 b ; a) ((XN ; a)+ Fn)n2N, , a = 0 Xn > 0, n 2 N. * X0 = 0, F0 = f Dg. * i 2 N
i = 1 f m < i 6 m+1 mg: ?
bN (0 b) 6
N X
(Xi ; Xi;1 )i:
i=1 S & , fi = 1g = m { ff m < ig n f m+1 < igg 2 Fi;1, i 2 N.
*!
bEN (0 b) 6
6
N X i=1
N X i=1
EN(Xi ; Xi;1 )i] =
E(E(Xi jFi;1) ; Xi;1 ) =
N X i=1
ENi(E(Xi jFi;1) ; Xi;1)] 6
N X i=1
(EXi ; EXi;1 ) = EXN : 2
@ 5.11. (Xn Fn)n2N {
, supn EjXn j < 1. " $ ( X1 = lim Xn , EjX1 j < 1. n!1
87
inf X , X = lim sup Xn . B P (X < X ) > 0. * 2 * X = lim n!1 n n!1
fX < X g = a b2Q a
, P fX < a < b < X g > 0 a < b. 5.10 N 2 N EN (a b) 6 (EXN+ + jaj)=(b ; a): < 1(a b) = Nlim (a b). ? !1 N
E1(a b) 6 (sup EXN+ + jaj)=(b ; a): N
? , (Xn Fn)n2N sup EXn+ < 1 , sup EjXnj < 1 n
n EXn+ 6 EjXn j = 2EXn+ ; EXn 6 2EXn+ ; EX1.
(5.24)
/ , E1(a b) < 1
.., P fX < a < b < X g > 0. ? , P fX < X g = 0. * I EjX1 j 6 supn EjXnj < 1. 2 ? 5.11 ( , (;Xn Fn)n2Z+ { , (Xn Fn)n2Z+ { ). * ! sup EjXn j < 1 n ( { ). $ 5.12. (Xn Fn)n2N { $ , supn EjXn jp < 1 p 2 (1 1). ( X1 = nlim X . ., Lp. !1 n
2 *
5.9
E(sup jXn jp) 6 (p=(p ; 1))p sup EjXn jp: n2N
n 2N
* 5.11 Xn ! X1 .. n ! 1. ? , jXn ; X1 jp 6 2p;1(jXn jp + jX1jp) 6 2p sup jXn jp: n2N
(5.25) (5.26)
8
(5.25), (5.26) . 2
3 $ $ $ $ $ ( . 5.13 ($ ! +$ { 3 $ ). * f k(n) P k n 2 Ng, ( -
, ( P P (1) (1) S n ; 1 (n) E 1 = > 0. * S0 = 1, S1 = 1 Sn = k=1 k n > 2 ( 0k=1 k(n) := 0, n 2 N). 3 , Sn n- ( n- ). J , & ;%
$ $ $ $(1) $$ 1 , $ n = 0 $ $ + . 8 , . 88
& , (S0 : : : Sn;1) n f k(n) k 2 Ng. 3
Xn = Sn = n n 2 N &, $ % ;$ $
Fn = fX0 : : : Xn g, n 2 Z+. ) n 2 N
X
Sn;1
E(Xn jFn;1) = E(
SX n;
= ;n E
1
k=1
1 X ; n = 1 fS j =1
k(n) S0 : : : n;1 =j g
? ,
j X k=1
k=1
k(n)jX0 : : : Xn;1 )= n =
j 1 X X ( n ) ; n Sn;1 = E 1 fSn;1=jg k S0 : : : Sn;1 =
j =0
j
E( k(n) S0 : : : Sn;1 ) = ;n+1
k=1
1 X j =1
j 1 fSn;1=jg = Sn;1 = n;1 :
sup EjXn j = sup EXn = 1:
n2Z+
n2Z+
/ ( 5.11), Xn ! X1 .. n ! 1 EX1 < 1. <
, n ! 1
Sn ! 0 .., < 1 Sn ! 1 .., > 1: = 1 ( (., ., N?, x36]), Sn ! 0 .. n ! 1. 2 A< $+ $ $ $ L1(D F P ). @ 5.14. (Xn Fn )n2Z+ { . "( -
:
Xn = E(X jFn), n 2 Z+, X { 7 2) fXn g 7 3) ( X1 , EjX1 ; Xn j ! 0 n ! 17 4) supn EjXn j < 1 X1 = lim Xn ( ( . . n!1 5.11) Xk = E(W X1 jFk ), k 2 Z+. , (Xk Fk )k2Z+ f1g, F1 = n2Z+ Fn, . 2 1) ) 2). B n 2 Z+ a > 0, b > 0 EjXnj1 fjXn j > ag 6 EjX j1 fjXnj > ag = 1)
*! ,
= EjX j1 fjXn j > a jX j 6 bg + EjX j1 fjXnj > a jX j > bg 6 6 bP (jXn j > a) + EjX j1 fjX j > bg 6 6 ab EjXn j + EjX j1 fjX j > bg 6 ab EjX j + EjX j1 fjX j > bg: lim sup EjXnj1 fjXn j > ag 6 EjX j1 fjX j > bg:
a!1 n
89
< , b > 0 . 2) ) 3). L fXn g , supn EjXn j < 1. / , .. (
X1 = nlim X . ' ( .. !1 n L1(D F P ). 3) ) 4). ) 3) , supn EjXn j < 1. *! .. (
Y = nlim X , EjY j < 1. ' Y = X1 .. B n k 2 Z+ !1 n EjE(Xn jFk ) ; E(X1 jFk )j 6 EjXn ; X1 j:
' E(Xn jFk ) = Xk .. n > k. / , Xk = E(X1 jFk ) .., k 2 Z+. B
4) , X1 2 F1jB(R) 4.7 ( Fk , k 2 N , ! - F1 ). 4) ) 1) { . 2 * $ $ Q . * 1 2 : : : { ... , P ( 1 = 0) = P ( 1 = 2) = 1=2. * Xn = nk=1 k . ? fXn = 0g fXn+1 = 0g P (Xn 6= 0) = 2;n n 2 N , , Xn ! 0 .. n ! 1. , , EjXn ; 0j = EXn = 1, n 2 N. < 5.14. @ 5.15 ( ). EjX j < 1 (Fn)n2N { "( - 1 (D F P ). F1 = _n=1 Fn . n ! 1
Xn := E(X jFn ) ! E(X jF1 ) . . L1(D F P ): (5.27) 2 (Xn Fn )n2N { $ $ 5.14. *! .. (5.27). / Xn ! X1 .. ( L1(D F P )) n ! 1. * Xn 2 FnjB(R), Xn 2 F1jB(R), n 2 N, , , 4.7, X1 2 F 1jB(R), F 1 { F1. B
, X1 = E(X jF1) .. , EX 1 A = EX1 1 A A 2 F1, E(X jF1) = E(X jF 1) .. 2 , A = 1 n=1 Fn F1 = (A). *! (.
4.2) " > 0 A 2 F1 A" 2 A
, P (A]A" < ". ? , A" 2 Fn n > N , N = N (" A). B, jEX 1 A ; EX 1 A" j 6 EjX j1 AA" jEX1 1 A ; EX11 A" j 6 EjX1 j1 AA" : * , Ej j1 B ! 0 , P (B ) ! 0,
,
( . * n > N
jEX 1 A" ; EX1 1 A" j = E(E(X jFn )1 A" ) ; EX1 1 A" j 6 6 EjE(X jFn) ; X1 j = EjXn ; X1 j ! 0 (n ! 1): 2 ) , , , . A +$ $ $ ( ) . ?, (
. $ 5.16. (Xs Fs)06s6t { , "( . . . " c > 0
P ( sup Xs > c) 6 EXt+ =c: 06s6t
90
(5.28)
2 B u > 0, Xs , 0 6 s 6 t, f sup Xs > ug = 1n=1 06k2;n6t fXk2;n > ug fXt > ug: 06s6t
L,
(5.19) ( X0 : : : XN Xs1 : : : Xsm , s1 < : : : < sm)
P (Nn=1 06k2;n 6t fXk2;n > ug fXt > ug) 6 EXt+ =u: < , P (Nn=1 Bn) ! P (1 n=1 Bn ), N ! 1. < u # c. 2 $ 5.17. (Xs Fs)06s6t { ( .. EXs2 < 1 s 2 N0 t]), "( . . . c > 0
P ( sup jXs j > c) 6 c;2EXt2: 06s6t
(5.29)
2 ,
5.8, , (Xs2 Fs)06s6t { . * ! (5.28). 2
. D. 5.1. * f k gk2N { P , Fn = f 1 : : : n g, Xn = nk=1 k , n 2 N. B , (Xn Fn)n2N { ,
E n+1 fn( 1 : : : n ) = 0
(5.30)
fn : Rn ! R n 2 N. ) (5.30) (
. B
k , k 2 N, ( ) ,
covg( n+1 )fn( 1 : : : n ) = 0 n g : R ! R fn : Rn ! R. D. 5.2. B , ( n Fn)n2N { - , (Sn m Fn )n>m { m 2 N,
Sn m =
X
16i1 <:::
i1 im n > m:
,
n Fn = f 1 : : : n g, n 2 N. D. 5.3. * a b . ) , ( c . < Xn (n 2 N) { ( n- ( c (X0 := a=(a + b)). B , (Xn )n2Z+ { . 91
D. 5.4. * (D F P ) = (N0 1] BN0 1] ), { 2 . * fTn n 2 Ng
{ ( N0 1], , Tn 0 = tn 0 < : : : < tn mn = 1 Tn Tn+1, n 2 N, mn > 2. - , ]n 1 = N0 tn 0], ]n k = (tn k;1 tn k ], 2 6 k 6 mn, n 2 N. B f : N0 1] ! R
Xn (!) =
mn X f (tn k ) ; f (tn k;1)
tn k ; tn k;1
k=1
1 n k (!) n 2 N:
B , (Xn Fn)n2N { . D. 5.5. * ( f
2 , .. jf (x) ; f (y)j 6 Ljx ; yj x y 2 N0 1] L > 0. * Tn , 06max (t ; tn k;1 ) ! 0, n ! 1. B , k6mn n k _1n=1Fn = BN0 1] fXn gn2N . ?
5.16 , Xn ! X1 .. L1 n ! 1, X1 2 BN0 1]jB(R). / , Na b] N0 1]
Zb a
Xn d = EXn 1 a b] ! EX1 1 a b] =
Zb
Zb a
X1 d n ! 1:
(5.31)
< 6 , Xn d ! f (b) ; f (a) n ! 1. < (5.31) , a f { , ( X1 . D. 5.6. * fXn n 2 Ng {
, (D F P ), fn0(x1 : : : xn) fn1(x1 : : : xn) {
2 (Rn B(Rn)), n 2 N. L 0 (X (! ) : : : X (! )) f n Ln (!) = f 1(X1 (!) : : : Xn (!)) n 2 N n
1
n
, fn1 { (X1 : : : Xn ), n 2 N , , fn0(z) = 0, fn1(z) = 0 ( Ln(z) := 0). B , (Ln Fn)n2N { , Fn = fX1 : : : Xn g, n 2 N. D. 5.7. * (Xt Ft)t2T { , .. ((ReXt ImXt), Ft)t2T { R2. ? E(XtjFs) = Xs .. s 6 t, s t 2 T (E(Y + iZ jA) := E(Y jA) + iE(Z jA) Y Z - A F ). * EjXtj2 < 1 t 2 T . B , fXt t 2 T g { (, .. E(Xt ; Xs)(Xu ; Xv ) = 0 v < u 6 s < t, v u s t 2 T . D. 5.8. * f n gn2N {
P n itS . * Sn = k=1 k , Xn = e n =EeitSn , n 2 N, t 2 R. B , (Xn Fn)n2N { () , Fn = f 1 : : : n g,Pn 2 N. < (. N?, . 547]), (
1k=1 k ! : 1) .., 2) , 3) (
( ). & , 0 1 , .. 0. 92
D. 5.9. * fYt t 2 T Rg { () ( , EjYtj2 < 1, t 2 T . B , (
h : T ! R , Xt = jYtj2 ; h(t) { Ft = fYs s 6 t s 2 T g, t 2 T , , EjYt ; Ys j2 = h(t) ; h(s) s 6 t s t 2 T: (5.32) 85.10. fYt t 2 T Rg { $ -
( .
EevYt < 1 Eev(Yt;Ys) < 1
v2R
s 6 t (s t 2 T ):
(5.33)
Zt = fv (t)evYt , t 2 T , fv : T ! R. $ fZt t 2 T g fv (t) 0, t 2 T , , s 6 t (s t 2 T )
E(evYt =evYs ) = EevYt =EevYs
(5.34)
fv (t) = c=EevYt t 2 T
(5.35)
c { . 2 C Zt t 2 T { , EZt = EZt0 t t0 2 T . *! fv (t)EevYt = fv (t0)EevYt0 t 2 T:
C fv (t0) = 0, fv (t) 0, t 2 T . * fv (t0) 6= 0, fv (t) 6= 0 t 2 T , , , (5.35). & ,
Ft = fZs s 6 t s 2 T g fYs s 6 t s 2 T g t 2 T: 8 ( fYt t 2 T g (5.33), s 6 t (s t 2 T )
E(Zt jFs) = ffv((st)) E(ev(Yt;Ys)evYs fv (s)jFs) = Zs ffv((st)) Eev(Yt;Ys ): v
v
? , (Zt Ft)t2T { , Eev(Yt ;Ys) = ffv ((st)) s 6 t (s t 2 T ): (5.36) v / (5.35) (5.34). A% $. 8 (5.34) (5.35), , (5.36). 2 D. 5.11. (5.34), ) fYt t 2 R+g { , ) fYt t 2 R+g { ? D. 5.12. N. N?, . 166]] * fWt t > 0g { m- f (x) { Rm, .. f (x) x 2 Rm ! x. B , f (Wt) {
. 93
3 $ 5.11 $ $ $ D. 5.13. N , ] B (Xn Fn)n2Z supn EjXn j < 1 , (
Yn Zn ( (Fn)n2Z ) , Xn = Yn ; Zn, n 2 Z+. D. 5.14. N L] * (Xn Fn)n2Z { . ? (
Xn = Mn +Rn, (Mn Fn)n2Z { (Rn Fn)n2Z { $ , .. , Rn ! 0 .. n ! 1. A%< $ $ $ , % $ $ ( % $ ). D. 5.15. * (Xn Fn)16n6N { h : R ! R+ { (
. ? u 2 R t > 0 P (1max X > u) 6 Eh(tXN )=h(tu) (5.37) 6n6N n +
+
+
+
+
(
, Eh(tXN ) < 1). D. 5.16. * (Xn Fn)16n6N { . ? e (1 + EjX j log+ jX j) E 1max j X j 6 (5.38) n N N 6n6N e;1 log+ x = log x x > 1 log+ x = 0 x < 1. B (5.38) , ( . D. 5.17. NB ] * 1 : : : N {
P n E k = 0, k = 1 : : : N . * Xn = k=1 k , k = 1 : : : N . ? E 1max jX j 6 8EjXN j 6n6N n
(5.39)
* 0 , B! 9, ( (
\ 3 { & , . 1 ( N?]. @ 85.18 (E +). X = (Xn Fn)n2Z+ { X0 = 0. p > 1 Ap Bp , ( X , ,
p
p
Apk NX ]nkp 6 kXnkp 6 Bpk NX ]nkp n 2 Z+ (5.40) P NX ]n = nk=1 (]Xk )2 { $ X (NX ]0 = 0). 9 (5.40) Ap = N18p3=2 =(p ; 1)];1 , Bp = 18p3=2 (p ; 1)1=2. = , 5.9 , Xn = max jXk j p > 1 06k6n
p
p
Apk NX ]nkp 6 kXnkp 6 Bpk NX ]nkp Ap = Ap, Bp = (p=(p ; 1))Bp . 94
(5.41)
D. 5.19. * (. N?, . 532]), (, p = 1
(5.40) . / 8,,
(5.41) p = 1 A1, B1, ( X .
8 $ % % , $, @ 85.20 (E + { 8, { ! ). X : N0 1] ! N0 1] { "( # $ , N0 1), , X(0) = 0, X(;1) = X(1) X(2t) 6 cX(t) t > 0 c > 0. 0 < A < B < 1, ( c, , " X = (Xn Fn )n2Z AEX(S1 ) 6 EX(X ) 6 B EX(S1) P (]Xk )2)1=2, ]Xk = Xk ; Xk;1 , (X0 = 0), k 2 N. X = supn jXn j, S1 = ( 1 k=1 D. 5.21. * fWt t > 0g { m- . B , (kWtk Ft)t>0 { ( .. ), k k { Rm. ) 5.15, h(x) = etx ( p t > 0), , s > 0 x > ms +
sd ;m=2
P ( sup kWtk > x) 6 ex2 e;x2=(2s): (5.42) t2 0 s] D. 5.22. N. N?, . ]] * fWt t > 0g { m- , m > 3. B , jWtj ! 1 .. t ! 1. B , 1 Wt 0 . < 6 , m = 1
. ' $ $ + +$ $, $ $ . * (D F P ) {
N { , ( . B - A F A = fA Ng. * (Ft)t2R+ { (
- (D F P ). - Ft+ = \s>t Fs, t 2 R+. 3 5.5, $ % $ , % $ $+ - % Ft ( , N F0). ? , , N Ft+ t 2 R+. B ( .. f < tg 2 Ft t 2 R+) - (5.43) F + = fA 2 F : A \ f 6 tg 2 Ft+ t 2 R+g: & , F + . D. 5.23. * n, n 2 N { = inf n n . B , { , F + = \n F n+ . C n { , < n
f < 1g, F + = \n F n . 85.24. (Xt Ft)t2R+ { , "( . . . { $ ( # $ (Ft)t2R+ ) , 6 6 c . ., c { .
E(X jF+ ) = X . ., (5.44) - F + (5.43) ( f = 1g f = 1g, "( " , X = 0 X = 0). 95
2
Tn = 2;n Z+ = fk2;n k 2 Z+g, n 2 N. < , (Xu Fu)u2Tn { . < (n) = 2;n N2n +1], (n) = 2;n N2n +1], n 2 N, N] { . < , (n) (n) 2 Tn (n) # , (n) # n ! 1 ! 2 D. , , (n) 6 (n) .., n 2 N. & , (n) (n) { (Fu)u2Tn , k 2 Z+ f (n) 6 k2;n g = f < k2;n g 2 Fk2;n ( (n), n 2 N). *! , 5.5 ((
, !
Z+, Tn) E(X (n)jF(n)) = X(n) .., n 2 N: (5.45) m 2 Z+ E(Xc(m)jF(n)) = X(n) .. n > m c(m) = 2;m N2mc + 1]. L
5.14, , E(X jA)
- A F , 2 R (X {
). 8 Xt t 2 R+ , X(n) ! X .. L1(D F P ) n ! 1. / , X { . * E(Xc(m) jF (n)) = X (n) .. n > m, , X (n) ! X .. L1(D F P ) n ! 1. 5.23 F+ = \n F(n), ! X 2 F+ jB(R). / ,
(5.44) , EX 1 A = EX 1 A A 2 F+ . * F+ F(n) n 2 N, (5.45) , EX (n)1 A = EX(n) 1 A
n 2 N. < n !
L1 X (n) X X(n) X . 2
? 85.25. B ( B5.24) ( (5.44) EX = EX . * ! ,
B5.24, 5.23, , (5.44) F+ = \n F(n). A% $ $ / { % (1.18). ) , Yt = y0 + ct ; St t > 0 St =
Xt X j =1
j
P
(5.46)
fXt t > 0g (1.17) ( 0j=1 j = 0). $+ Xt = Nt, t 2 R+, fNt t > 0g { $$ . ? 8.5 ( )
, fNt t > 0g (1.17), f j gj2N { , . ? Nt t > 0 .. , , !
fYt t 2 R+g. ', ...
f j gj2N ( f j gj2N, f j gj2N fNt t > 0g ( - (D F P )). 85.26. 9 (5.46) $ fYt t 2 R+g ( .
96
2 & , fZt t 2 R+g {
( h(t), t 2 R+, {
, fZt+h(t) t 2 R+g { (. *! ( fSt t > 0g. 2.11 ! ( ), (2.17). * (v) = Eeiv1 , v 2 R. 8 ( f j gj2N, 0 6 s < t v 2 R Eeiv(St;Ss ) = =
1 X k m=0
E expfiv =
X
k+m j =k+1
1 X k=0
1 X
k m=0
Eeiv(St;Ss )1 fNs = kg1 fNt ; Ns = mg =
j gP (Ns = k)P (Nt ; Ns = m) =
P (Ns = k)
1 X
; s))m e;(t;s) = e(t;s)((v);1): 2 (5.47) ((v))m ((t m !
m=0
& ,
(v) = Eev1 < 1 v > 0: (5.48) 1 j , j ( 1 > 0 ..,
, (5.48) v 6 0). , (5.47), , 0 6 s < t v 2 R Ee;v(Yt;Ys ) = e(t;s)g(v) g(v) = ((v) ; 1) ; vc: (5.49) ) (5.49), , Y0 = y0, , Ee;vYt = etg(v);vy0 , t 2 R+. / , (5.33) (5.34) , 5.10 v 2 R Zt = e;vYt;tg(v) Ft = fZs 0 6 s 6 tg fYs 0 6 s 6 tg t 2 R+:
3 $
= inf ft > 0 : Yt < 0g inf ft > 0 : Yt 2 (;1 0)g: (5.50) * fYt t > 0g .. ,
(;1 0) { , 4.4 (
), , .. f < tg 2 Ft t 2 R+. B5.25 ( = 0 6 ^ t 6 t, , 0 t ^ { ) t v 2 R+ e;vy = EZ0 = EZt^ > E expf;vYt^ ; (t ^ )g(v)g1 f 6tg > 0
> E expf;vY ; g(v)g1 f 6tg > Ee; g(v)1 f 6tg > 06infs6t e;sg(v)P ( < t): * (5.51) , Y 6 0 .. ) , v > 0 t > 0 P ( 6 t) 6 e;vy0 sup esg(v): 06s6t
(5.51) (5.52) 97
$% $+, $% c > a a = E 1 > 0: (5.53) ? g0(v) = 0(v) ; c g0(0) = a ; c < 0. , , g00(v) = 00(v) > E 12 v > 0, 00(v) = E 12ev1 ( E 1 > 0 , E 12 > 0). / , (
v0 > 0, , g(v0) = 0. v = v0, (5.52) P ( 6 t) 6 e;v0 y0 t 2 R+. < , P ( < 1) 6 e;v0 y0 : (5.54)
2$ , @ 85.27. = { + (1.18)
$ (1.17) > 0, j , j 2 N, " " (5.48) (5.53). $ (" # (5.54), y0 { , v0 > 0 ( ) g (v ) = 0, v 2 R+ (# $ g (5.49)).
&, % $$ $ $+ & +$ $. * (Gt)t2T R { "( - (D F P ), .. Gt Gs F s < t, s t 2 T . * (Xt )t2T {
, (Gt)t2T . ? (Xt Gt)t2T
( , ), (Xt Gt)t2U U = ;T = f;t t 2 T g ( , ). , (Xn Gn)n2Z+ { , E(Xn jGn+1) = Xn+1 , n 2 Z+. &
"=" ">", ( "6" { ). * Rm (m > 1). D. 5.28. * 1 : : : N { ...
Rm (m > 1) Ek 1k < 1, k k P { . * Xn = (1=n) nk=1 k , Gn N = fXn : : : XN g. B , (Xn Gn N )16n6N { . D. 5.29. * f n n 2 Ng {
( .. N 2 N (i1 : : : iN )
f1 : : : N g ( i1 : : : iN ) =D ( 1 : : : N )). * m 2 N g : Rm ! R. L U -
Un m = (Cnm);1
X
16i1 <:::
g( i1 : : : im ) n > m:
* Gn m = fUk m k > ng, n > m. B , (Un m Gn m)n>m { . & Cnm = n!=(m!(n ; m)!). @ 85.30. (Xn Gn)n2Z+ { . . . ( X1 = limn!1 Xn 2 N;1 1). ! , , ( lim EXn = c > ;1 (5.55) n!1 X1 2 (;1 1) Xn Xn > 0 . . n 2 Z+ Lp(D F P ) n ! 1.
98
! Xp1
L1(D F P ) n ! 1. ! EX0 < 1 p 2 (1 1), Xn ! X1
2 B (a b) N 2 N N (a b)
(. . ??) (XN GN ) : : : , (X0 G0). ? EN (a b) 6 E(X0 ; a)+=(b ; a) 5.10. / , 1(a b) := Nlim (a b) < 1 .. !1 N ( (
). ? ,
5.11 , .. (
X1 = limn!1 Xn . ? , (Xn+ Gn)n2Z+ . *! , E(nlim X + ) 6 lim inf EXn+ 6 EX0+ < 1: !1 n n!1
< P (X1 = 1) = 0. *
. $+ (5.55). * EXn > EXn+1 , EXn+ > EXn++1 n 2 Z+, EjXn j = 2EXn+ ; EXn 6 2EXn+ ; c 6 2EX0+ ; c n 2 Z+: ) , supn EjXnj < 1. 8 & ; $ $+ fXn n 2 Z+g. B " > 0 m = m(") 2 N , EXn < c + " n > m. ? > 0 n > m ( (5.1) ), EjXn j1 fjXn j > g = EXn 1 fXn > g ; EXn 1 fXn 6 ;g = = EXn 1 fXn > g + EXn 1 fXn > ;g ; EXn 6 6 EXm 1 fXn > g + EXm 1 fXn > ;g ; c = = EjXm j1 fjXn j > g + EXm ; c 6 EjXmj1 fjXnj > g + " < 2" ( n > m), supn P (jXn j > ) 6 ;1EjXn j ! 0 ! 1. < , Xn ! X1 L1(D F P ) n ! 1. J X1 2 L1,
, , jX1 j < 1 .. C Xn > 0 .. n 2 Z+ EX0p < 1 p 2 (1 1),
5.9, XN : : : X0 E(0max X p ) 6 (p=(p ; 1))pE(X0p ): 6n6N n 8 N , E( sup Xnp) < 1. ,
n2Z+
5.12, Xn ! X1 Lp(D F P ) n ! 1. 2 ? 85.31. 8 (5.55) . $ 85.32. EjX j < 1 (Gn)n2N { "( - (D F P ). G1 = \1 n=1 Gn . n ! 1
Xn := E(X jGn) ! E(X jG1)
. .
L1(D F P ):
(5.56)
2 < , (Xn Gn)n2N { . B5.31 Xn ! X1 .. L1 n ! 1. 2 ( ), X1 2 G1jB(R). ? A 2 G1 EX 1 = lim E(E(X 1 AjGn)) = EX 1 A: 2 EX1 1 A = nlim !1 n A n!1
*, $ ( 5.17) $ $ 0 1 / ( 4.11). * f k gk2N { 99
Rm (m > 1), Fn = f k k 6 ng, Gn = f k k > ng, n 2 N. B A 2 G1 = \1 n=1 Gn P (A) = E1 A = E(1 A jFn) ! E(1 AjF1) = 1 A .. n ! 1 (5.57) F1 = _1 n=1 Fn = f k k 2 Ng. / , P (A) 0 1. 3 , A 2 F1, G1 F1 , A Fn n 2 N. : N ! N (( 2 Q(N)), (n) 6= n ( ( ) n 2 N. * f k gk2N {
(D F P ). ? = ( 1 2 : : : ) FjB(R1) { ! ( ). * = ( 1 2 : : : ) = (1 2 : : : )
N. < F - G = f ;1(B ) B 2 B(R1) : P ( ;1 (B )]( );1(B )) = 0 2 Q(N)g: @ 85.33 ( 0 1 J+;$$ { ,& ). f k gk2N { ... . - G , .. 0 1. D. 5.34. ( 5.29) * B5.30 , Un m ! U1 m .. n ! 1: / ( B5.33 , f k gk2N { ...
, U1 m = Eg( 1 : : : m). D. 5.35. * (Xn Gn)n2Z+ { , (5.55). B , X1 6 E(Xm jG1), m 2 Z+, G1 = \1n=1 Gn (X1 = limn!1 Xn (
.. L1(D F P ) B5.30). ) , (Xn Gn)n2Z+ f1g { . * , (Xn Gn)n2Z+ { ,
(Xn Gn)n2Z+ f1g { . D. 5.36. 3 B5.30 (5.55)?
$ 85.37 ( % +< , / ). { ... Rm (m > 1), E 1 = a 2 Rm.
1 2 : : :
n X 1 Xn := n k ! a . . L1(D F P ) n ! 1: k=1 0 ( a = 0 2 Rm. 5.28 2 (Xn Gn N )16n6N { ( ) , Gn N = f n : : : N g, n = 1 : : : N . , Xn = E(X1jGn N ) .. n = 1 : : : N . * 2 ( ), N , Xn = E(X1jGn) .., Gn = f k k > ng.
B5.32, E(X1jGn) ! E(X1 jG1) .. L1(D F P ) n ! 1, - G1 = \1n=1 Gn 0 1 , (5.57). / , E(X1jG1) = 0 .. 2 @ 85.38 (E , , 8 %%). fXn n 2 Ng { (D F P ) , Xn ! X1 . . n ! 1 E(supn jXn j) < 1. (Fn )n2N { - F , " , " ( F1 = _1 n=1 Fn F1 = \1n=1 Fn).
lim E( X jF ) = E( X jF ) . . L1 (D F P ): (5.58) n k 1 1 n k!1
100
2 < U = mlim sup E(Xn jFk ) V = mlim inf E(Xn jFk ) !1 !1 k n>m k n> m
( U V (
.. f sup E(Xn jFk )g k n>m fk inf E( X jF ) g ). * Y = sup X , m 2 N . & , E j Y j 6 E(sup n k m n n n jXn j) < 1, n>m n>m m 2 N. / , m 2 N 5.16
B5.32 E(Ym jFk ) ! E(Ym jF1) .. L1(D F P ) k ! 1:
(5.59)
* Xn 6 Ym n > m, E(Xn jFk ) 6 E(Ym jFk ) .. n > m, k 2 N. ? (5.59)
U 6 mlim sup E(YmjFk ) 6 lim sup E(YmjF1) .. !1 m!1
k >m
) , Ym # X1 , E(YmjF1) # E(X1 jF1) .. m ! 1 ( ). J , V > E(X1 jF1). *! U = V .. B L1- (5.58). 2 2 ; &$ $ % $ x5, . 7 N?]. * , .
@ 85.39 (N?, . ??]). (Xn Fn)n2Z+ { fAn g { . A1 = limn!1 An . fXn g . . fA1 < 1g Xn = o(f (An )) n ! 1 " # $ f : R+ ! R+, "( "
Z1 0
( , # $
(1 + f (t));2dt < 1
f (t) = t1=2(log+ t), > 1=2).
*, $ B5.38 &$ $+ , ! . D. 5.40. * (Ft)t2R+ { ( - ( )
(D F P ). ? F t+ = Ft+, .. -
+$ $ $ . 2 < , Ft F t, ! Ft+ F t+ Ft+ F t+ = F t+ ( .. F t N { , F t+ N ). A% $. Ft+ Ft+h h > 0. / , Ft+ F t+h, Ft+ F t+. 2 9 , ( ) (Xt Ft)t2R
, (Ft)t2R . D. 5.41. * (Xt Ft)t2R { ( ) ( .. . * t 2 R+ (
= (t) > 0
, E(sups2 t t+] jXs j) < 1 ( , sups2 t t+] jXs j
). ? (Xt F t+ ) { ( ). +
+
+
101
5.4 5.40 F t = Ft, t 2 R+. L 0 6 s < t sm # s m ! 1 tn # t n ! 1 , sm < t m 2 N tn < t + (t), n 2 N. ? , B5.38
2
E(Xt jFs+) = mlim E(Xtn jFsm ) > (=) mlim X = Xs .., n!1 !1 sm
(5.60)
(5.60) (=) . 2 D. 5.42. 8 , ( (, .. , .. ). ( $ + +$ $. @ 85.43 (. N?, . 16]). X = (Xt Ft)t2R+ { # $ (Ft)t2R+ . $ X -
# $ " , , ,
# $ t 7! EXt R+ R . ! # $ Yt , t 2 R+, ( , , t 2 (0 1) ( , $ cadlag RCLL) # $ (Ft)t2R+ . (Yt Ft)t2R+ { .
! , (
( , , , ., . N?], N?, ?], N?]). < . N?].
102
6. $ #) *# " . " .. @. $ . . " C (T X ). ! " " " . # . . { . 3 = ' ( ), . " $ . 2 .
! ( ) $ + % , . F $ % $ . * (X ) |
- B(X ) Qn Q | (X B(X )). 9 , Qn, n 2 N, Q ( Qn ) Q), f 2 Cb(X R), . . f : X ! R,
Z
X
f (x)Qn(dx) !
Z
X
f (x)Q(dx) n ! 1
(6.1)
( , f 2 Cb(X C ), . . f : X ! C P , ..
, , , Q", " > 0). 2$ f Q ( ) % % $+ $ & hf Qi. C (
,
,
6.1. hf Qi = hf Qei f 2 Cb(X R). Q = Qe.
2 * , (4.15), , Q(F ) = Qe(F ) F . 2.3 , Q = Q~ B(X ). 2 @ 6.2. Q Qn, n 2 N, | (X ). - Qn ) Q (n ! 1) "( : 1. lim sup Qn(F ) 6 Q(F ) " F 2 B(X )7 n!1 2. lim inf Q (G) > Q(G) " G 2 B(X )7 n!1 n 3. nlim Q (B ) = Q(B ) " B 2 B(X ) , Q(@B ) = 0, !1 n @B | $ B (@B B (X )). : B , # "( 3 , " Q- . 2 * , (6.1) + $ , $ ;% f 2 Cb(X R) lim suphf Qni 6 hf Qi: n!1
(6.2)
B
, (6.2) f ;f , lim infhf Qni > hf Qi. n!1 * Qn ) Q. 8 & $& 1. B F " > 0 f"F (x) = '((x F )=") 2 Cb(X R), '(t) = 1 t 6 0, '(t) = 1 ; t 103
0 < t < 1 '(t) = 0 t > 1. ? Qn(F ) = h1 F Qni 6 hf"F Qni, 1 F 6 f"F ( (1.15)). (6.2) lim sup Qn(F ) 6 lim suphf"F Qni 6 hf"F Qi: n!1
n!1
< , 2 hf"F Qi ! h1 F Qi = Q(F ) " ! 0. $+ 1. 8 &, $ Qn ) Q. '
(6.2) 0 < f (x) < 1, af (x)+ b, a > 0, b 2 R. k 2 N
Fi = fx : f (x) > i=kg, i = 0 : : : k. i ; 1 i < Ci = k 6 f (x) < k , . . Ci = Fi;1 n Fi, i = 1 : : : k. ? k X i;1 i=1
k Q(Ci) 6
Z
X
f (x)Q(dx) 6
k X i i=1
k Q(Ci):
(6.3)
Pk Pk Pk ? , i Q(Ci) = i (Q(Fi;1) ; Q(Fi)) = 1 + 1 Q(Fi). Jk k i=1 i=1 k i=1 k (6.3)
Z k k X 1X 1 1 k i=1 Q(Fi) 6 f (x)Q(dx) 6 k + k i=1 Q(Fi): X
(6.4)
&
(6.4) Qn, k k X X 1 1 1 1 lim suphf Qni 6 k + lim sup k Qn (Fi) 6 k + k Q(Fi) 6 k1 + hf Qi: n!1 n!1 i=1 i=1
8 k, (6.2). / , Qn ) Q. 1 $$+ 2 1 ( ). &, $ 1 $ 3 . < B
B , NB ] | . ? , 1 2,
Q(NB ]) > lim sup Qn(NB ]) > lim sup Qn(B ) > n!1 n!1 > lim inf Q (B ) > lim inf Q (B ) > Q(B ): (6.5) n!1 n n!1 n ' Q(NB ]) = Q(B ) = Q(B ), Q(@B ) = 0 (NB ] n B 2 @B B n B 2 @B ), ! (6.5) 3. $+ 3, &, $ 1.
F F " = fx 2 X : (x F ) < "g, " > 0. & , @F " fx : (x F ) = "g, ! @F " \ @F = ? " 6= . ) , Q(@F ") > 0
". *! "k # 0
, Q(@F "k ) = 0, k 2 N. ? lim sup Qn(F ) 6 limn!1 Qn (F "k ) = Q(F "k ) n!1 k. < , Q(F "k ) ! Q(F ) k ! 1. 2 *
(D F P ), (Dn Fn Pn), n 2 N, .!. X : D ! X Xn : Dn ! X , . .
F jB(X ) Fn jB(X )- . /.!. Xn ( " X (
104
D Xn ! X ), PXn ) PX n ! 1 (. (1.5)). * (2.10), , D Xn ! X , Enf (Xn ) ! Ef (X ) f 2 Cb(X R) ( f 2 Cb(X C )). & En Pn . @ 6.3. (X B(X )), (Y B(Y )) | h | D X Y . ! Xn ! X (Xn X " X ),
D # Y B(Y )) n ! 1.
D h(X ) n ! 1: h(Xn ) ! (6.6) : Qn ) Q (X B (X )), Qn h;1 ) Qh;1
2 g : Y ! R. ? gh 2 2 Cb(X R) . / , Eg(h(Xn )) ! Eg(h(X )), n ! 1, (6.6). * 1.6 .!. Xn , X , Qn = PXn , n 2 N, Q = PX .
? ,
Qnh;1 = Ph(Xn) Qh;1 = Ph(X ): 2 6.4. Qn ) Q n ! 1 , fnk g N fn0k g ", Qn0k ) Q k ! 1. 2 ' . B . B ,
, Qn 6 )Q n ! 1. ? (
f 2 Cb(X R) fmk g N , " > 0 jhf Qmk i ; hf Qij > " k 2 N: , fm0k g fmk g , Qm0k ) Q k ! 1. * . 2 /
fQ 2 Rg (X B(X )) , Qn (n 2 N) ( Q0n (( ! !
). < , ( . 2 ( , ( , ) , ( . * !
@ 6.5. 3 , fQngn2N (X B (X )) , , - # $ H Cb (X R), 1) ( limn!1 hh Qn i h 2 H, 2) " Q Q (X B (X )) -
-
e
Q = Qe B(X ).
hh Qi = hh Qei
h2H
(6.7) 105
2 ' ( H = Cb(X R) 6.1). 8$ $ $+. ) fnk g N fn0k g , Qn0k ) Q k ! 1. ? hf Qn0k i ! hf Qi f 2 Cb(X R), 1) , lim hh Qn i = hh Qi h 2 H: (6.8) n!1
B , Qn 6 )Q n ! 1. ? 6.4 fmk g N , Qmk ) Qe k ! 1 Qe 6= Q. ) 6.8 , hh Qi = hh Qei h 2 H. / 2) Q = Qe. * . 2
L $ $+ X = fXt t 2 T g X (n) = fXt(n) t 2 T g, n 2 N, (D F P ) (Dn Fn Pn) ( t 2 T
Xt. / 1.4 X 2 FjBT X (n) 2 FnjBT n 2 N. C
X (n) X BT , % $+ % - % $ (
) $ $ . 3 % + $& $+, $ XT $ $ ;, $ BT = B(XT ).
! , , ( 1)
XT . , T {
C (T X ) {
T
X ( t 2 T ), , .. C (3.20). < B(C (T X )) -
(C (T X ) C ). * BT (C (T X )) { - C (T X ), .. - , " " t;11::: tk (B ), B 2 B(X k), t1 : : : tk 2 T , k 2 N (
X k (2.1)), t1 ::: tk : C (T X ) ! X k t1 ::: tk x = (x(t1) : : : x(tk)) x 2 C (T X ): (6.9) @ 6.6. C (T X ) { - . "( : a) B (C (T X )) = BT (C (T X )), b) Q Q,
, ..
e
B(C (T X )) " -
e t;11::: tk Qt;11::: tk = Q
Q = Qe B(C (T X )),
t1 : : :
tk 2 T k 2 N
(6.10)
c) fQn gn2N B (C (T X )) n ! 1 , ( " , .. t1 : : : tk 2 T k 2 N Qnt;11::: tk gn2N .
2 a) B(C (T X )) BT (C (T X )) 2.2 3.9,
, BT (C (T X )). / , " " B(C (T X ))
1.2 t1 ::: tk t1 : : : tk 2 T , k 2 N. *! BT (C (T X )) B(C (T X )). 106
b) C Q Qe " " C (T X ), - BT (C (T X )), , a) BT (C (T X )). c) ' 6.3 (6.9). B
H = fh = gk t1 ::: tk gk 2 Cb(X k R) t1 : : : tk 2 T k 2 Ng: / (2.10) h = gk t1 ::: tk 2 H hh Qni = hgk Qnt;11::: tk i: (6.11) / fQnt1 ::: tk gn2N (6.11) 1)
6.5. 8 2) 6.5. ? (6.11) Q, Qe Qn , 6.1 , (6.10). & , b) Q = Qe BT (C (T X )). 2 ) , $+ ; , $ C (T X ) -
$ % ; $+ .-.. A % $ $& , $ ; . 6.7. * Qn xn () 2 C N0 1], . 6.1, . . Qn = xn , n 2 N ($ -% x(A) = 1 A (x)). * Xn (t !) = xn(t) ! 2 D = C N0 1], t 2 N0 1] (F = B(C N0 1]))P Qn = PXn . 6 1 x (t) n
0
1
1=2n 1=n
t
. 6.1
2 , B 2 B(Rk), k 2 N t1 : : : tk 2 N0 1] xn t;11::: tk (B ) = xn f! 2 C N0 1]: (!(t1) : : : !(tk )) 2 B g = 1 B (xn(t1) : : : xn(tk )): * n xn(ti) = 0, i = 1 : : : k. *! ! n 1 B (xn(t1) : : : xn(tk )) = 1 B (0 : : : 0) = x0 t;11::: tk (B ) x0(t) = 0, t 2 N0 1]. C (
Q, xn ) Q (
Q 6.1), k 2 N t1 : : : tk 2 T Qnt;11::: tk ) Qt;11::: tk n ! 1: ' b) 6.6 , Q = x0 . C N0 1]
F = fx() : supt2 0 1] x(t) > 1g. < , x0 (F ) = 0 xn (F ) = 1 n 2 N. / 6.2 ( 1) xn ) x0 n ! 1 . < , $$ + $$+; $ $ $$. /
fQ 2 Rg
(X ) , " > 0 K" , Q(K" ) > 1 ; " 2 R. * ( (
3). 107
@ 6.8 ( ). !
, . ! X , .
$+ T = N0 1]. *
C N0 1] , !
!
. B x : N0 1] ! R > 0 ](x ) = sup jx(s) ; x(t)j: s t2 0 1] js;tj<
) J { J, ( C N0 1] (., ., N?, . 302]),
@ 6.9. ' fQn n 2 Ng C N0 1] , 1. 3 " " > 0 (
M = M (") > 0 , Qn(x(): jx(0)j > M ) 6 " n 2 N: 2. 3 " " > 0 ( " = (" ) > 0 m0 = m0(" ) , Qn(x(): ](x ) > ") 6 n > m0: ? 6.10. 2 ( 20. B " > 0 (
N (" ) 2 N , i ; 1 ! N X Qn i;1sup i x(s) ; x N > " 6 n > n0(" N ): i=1 N 6s6 N ? , , $ -
, $ %$+ % . ' . * Xn i , i = 1 : : : mn, n 2 N, | -
n- , EXn i = 0, i = 1 : : : mn, mn P n2 i = 1, n2 i = EXn2 i . C Yn i (i = 1 : : : mn) | i=1 ! , , Xn i = (Yn i ; EYn i )=Bn , m Pn Bn2 = DYn i . i=1 Pj Pj < tn 0 = 0, tn j = n2 i ( tn kn = 1), Sn 0 = 0, Sn j = Xn i , i=1 i=1 j = 1 : : : mn. A & ! 2 D ; Sn (t !), t 2 N0 1], ; ; (tn j Sn j ), j = 0 : : : mn, . . Sn(t !) = Sn i;1 + t(t ;;tnt i;1) Xn i t 2 Ntn i;1 tn i] (6.12) ni n i;1 (. . 6.2). & , n i > 0, , Xn i = 0 .. I (6.12) , Sn (t !) t 2 N0 1] , Sn ( !) ! 2 D, 3.9 Sn() | .!.
C N0 1]. * Pn = PSn () B(C N0 1]). 108
6
0
qq qq
Sn i
Sn i;1
Sn kn
tn i;1 tn i . 6.2
-
1
t
@ 6.11 ( ).
i = 1 : : : mn, n 2 N, mn X i=1
EXn2 i 1 fjXn ij>"g ! 0
+ :
"
">0
-
n ! 1:
Xn i , (6.13)
D
Pn ) W (n ! 1), W | 9 . 3 , Sn () ! W n ! 1, W | $.
* ! , . B ... X1 X2 : : : p B D (i=n Si= n), i =0 : : : n. 8
Sn() ! W () $ 3 { . *, $ 6.11 ; $ % % ; $ + ; + ; $
( 2 ). B
, h(x()) = x(1) | ! C N0 1] R. *!
6.3 n ! 1 D h(W ()) . . S (1) = S D h(Sn ()) ! (6.14) n n mn ! W (1) N (0 1): 1 6 , 6.12 # $ $ . , , 6
( , . . ). 2 $$+ $$ $, $ + h(Sn()) % $ $ &, h(W ()), % -
, $; ; % , % , ,$ h &$ %$+ ;% $% &, C N0 1]. ? , -
! , ,
( 2 . ', Yn , ( ;1 +1 1=2 ( ), p : Xn i = Yi= n, i = 1 : : : n, n 2 N. 2 B
6.11 6.6. $+ .-.. Sn (). *, k 2 N 0 6 t1 < : : : < tk 6 1 ; 1
Pn t1 ::: tk ) Wt;11::: tk (n ! 1), , ,
D (W (t ) : : : W (t )): (Sn (t1) : : : Sn(tk )) ! 1 k 8 & $ tj , j = 1 : : : k, & n % & < ; ft 6 tj g. ) $ tn i, i = 1 : : : mn, . . t(jn) = i=1max ::: mn n i
109
2 ( ( 6 2), max 2 ! 0 n ! 1 (6.15) 16i6m n i n
(n) ! t n ! 1 ! 16max ( t ; t ) ! 0 n ! 1 . / , t n i n i ; 1 j j i6mn j = 1 : : : k. C t(jn) = tn l, l = l(j n), jSn(tj ) ; Sn(t(jn))j 6 P 6 jSn (tn l+1) ; Sn(tn l)j 6 jXn l j. (6.15) Sn (tj ) ; Sn(t(jn)) ! 0 n ! 1 j = 1 : : : k. ? , $ $ $ $+, $
D Zn := (Sn(t(1n)) : : : Sn (t(kn))) ! Z := (W (t1) : : : W (tk )) n ! 1: (6.16) B
, , Zn Yn Z | D P Rk, Zn ! Z , Yn ! 0 ( . . D
Yn ), Zn + Yn ! Z ( 1 6.2). B
(6.16) + $ + + $ (
2). L Rk n i , i = 1 : : : mn, n- (n 2 N), E n i = 0 Ek n i k2 < 1 n i, kk { Rk. < Bn2 i = D n i , . . m m Pn Pn n i , Sn = n i , Bn2 = DSn = Bn2 i.
@ 6.12 ( mn X i=1
i=1 i=1 %). n i Bn2 ! B 2 ( ) n ! 1
Ek n ik21 fkn ik>"g ! 0 " " > 0
(6.17)
n ! 1:
(6.18)
D N (0 B ) n ! 1: Sn ! (6.19) & , cov(Sn(t(in)) Sn(t(jn))) = minft(in) t(jn)g ! minfti tj g n ! 1 (i j = 1 : : : k). *! Zn Z , ( (6.16), , DZn ! DZ ! . B, t(1n) = tn l1 , : : : , t(kn) = tn lk , li = li(n) 2 2 f1 : : : mng, Zn = n 1 + : : : + n lk , n i , i = 1 : : : lk(n), | Rk: 0S (t(n))1 0 1 0 1 0 1 0 1 Xn 1 Xn l1 0 0 n 1 (n) C BX C B B C B C B X X 0 C Sn(t2 )C B n 1 C n l1 C B n l1 +1 C B B B C = + : : : + + + : : : + B C B C B C B C B . . . . . . . . . . @ . A @ . A @ . A @ . A @ . C A: Xn 1 Xn l1 Xn l1 +1 Xn lk Sn(t(kn))
k
k
k
k
k
n l1 n l1+1 n lk Zn n 1 * , n i ( ,
) Xn i, (6.13)
X
lk (n) i=1
110
Ek n i k fkn i k>"g 6 k 21
mn X i=1
EjXn ij21 fjXn i j>"=pkg ! 0:
) , .-.. . $$+ fPn g. B ! 6.13. 1 : : : m | . . E i = 0, Pi i2 = D i < 1, i = 1 : : : m. Si = j (i = 1 : : : m), d2m = DSm.
j =1 p " > 2 p 2)dm ): (6.20) P (1max j S j > d ) 6 2 P ( j S j > ( ; i m m 6i6m
jS j > dm g = 2 * Aj = fmax jSij <dm jSj j > dmg. ? A = f1max 6i6m i
i<j m S = Aj , Ai \ Aj = ? i 6= j . j =1
p
p
P (A) = P (A \ fjSmj > ( ; 2)dm g) + P (A \ fjSmj < ( ; 2)dmg) 6
p
6 P (jSmj > ( ; 2)dm ) +
m X j =1
p
P (Aj \ fjSmj < ( ; 2)dm g):
p & , pAm \ fjSmj < ( ; 2)dmpg = ?. B 1 6 j < m Aj \ \fjSmj < ( ;p 2)dm g Aj \fjSm ; Sj j > 2dmg, p , jSj j > dm , jSmj < ( ; p2)dm, , jSm ; Sj j > jSj j ; jSmj > 2dm . / Aj fjSm ; Sj j > 2dm g . . ! m . )
S , ( , P (A) =
p
P P (A )):
j =1
j
X
m;1
j2+1 + : : : + m2 P (A) 6 P (jSmj > ( ; 2)dm ) + P (Aj ) 6 2d2m j =1 m X;1 p p 6 P (jSmj > ( ; 2)dm ) + 12 P (Aj ) 6 P (jSmj > ( ; 2)dm ) + 21 P (A): 2 j =1
? $, $ Sn(0) = 0, n 2 N. ,$ 1 $ 6.9 (2) . 20 6.10. & N 2 N. * t(1) n i tn i |
tn j 6 (i ; 1)=N tn j > i=N , i = 1 : : : N . ) (6.15) , n (1) 1=N 6 t(2) n i ; tn i 6 2=N i = 1 : : : N:
B,
i ; 1 ! pn i := P i; sup i Sn (t) ; Sn N > " 6 N 6t6 N
(6.21)
1
6P
i ; 1 sup Sn (t) ; Sn N 6 sup i; 6t6 i t2 t t 1
N
N
max
(2) j : tn j 2 t(1) n i tn i ]
(1) (2)
ni ni
jSn(tn j ) ;
!
j > 2" (6.22)
Sn(t(1) n i)
i ; 1 (1) Sn(t) ; Sn(t(1) 6 n i ) + Sn (tn i ) ; Sn N ] 111
6 2 sup jSn(t) ; Sn(t(1) n i )j = 2 (2) t2 t(1) n i tn i ]
max(1)
j : tn j 2 tn i tn i ] (2)
jSn(tn j ) ; Sn (t(1) n i )j:
(1) (2) (1) * 6.13 , q D(Sn(t(2) n i ) ; Sn (tn i )) = tnqi ; tn i , p (2) (1) p (1) (2) (1) pn i 6 2P (jSn (t(2) n i ) ; Sn (tn i )j > ( ; 2) tn i ; tn i ), = "=(2 tn i ; tn i ) > 2
(6.21) i = 1 : : : N , N > 16";2 n . * (2) t(1) n i = tn ji , tn i = tn ri , ji = ji (n), ri = ri (n). ' i- n- l
Ji(n) = fl : ji < l 6 rig, i = 1 : : : N . < Sn(i) = P (1) = Xn l = Sn (t(2) n i ) ; Sn (tn i ). ? (6.21) (6.13)
l2Ji(n)
9 0 12 8 mn < = X @ Xn l A X j X j " n l 2 ! E q 1 :q > " 6 N EjXn ij 1 jXn i j > p N ( i ) ( i ) i=1 DSn DSn l2J n
0
( )
i
n ! 1. / , 2 p ( k = 1) i = 1 : : : N > 0 n > n0( N ) > 2
q p p ( i ) ( i ) P jSn j > ( ; 2) DSn ; P (j j > ; 2) < =(2N )
(6.23)
N (0 1) (, y*? p p p p ). / (6.21) i = 1 : : : N ; 2 > " N=(2 2) ; 2. *! , N = Np(" ) (3.17), P (j j > ; 2) < =2N . )
(6.22){(6.23) , PN pn i < . 2 i=1 ? , ( $ / . * 1 2 : : : | ...
, ( .. F (x). < (1.14) ! Pn , ! , B = (;1 x], ! Fn(x !), n 2 N, x 2 R.
@ 6.14 (/ ). ! - f ng #.. F , z > 0 n ! 1 1 X p P ( sup j n(F (x !) ; F (x))j 6 z) ! 1 ; 2 (;1)k+1 e;2k z = K (z): (6.24) ;1<x<1
2 2
n
k=1
(1 (D F P ) .) 2 i = F ;1( i), i 2 N, F ;1(t) = inf fs : F (s) > tg, t 2 R, & $ $+ 1 2 : : : N0 1], . ? , , z>0 P ( sup jYn(t)j 6 t) ! K (z) n ! 1 (6.25) 06t61
Yn (t) = pn(Fn(t);t). < h(x()) = sup jx(t)j. C
t2 0 1]
(6.25) , Yn () 112
6.3. '
, Yn .!. C N0 1] (
/ DN0 1], . N?, . 3]). < ! . * (1) : : : (n) , 1 : : : n, . . ! 1(!) : : : n(!) ( 1 1 : : : n ). * ! G(t !), t 2 N0 1], ( (i)(!) i=(n + 1)), i = 0 : : : n +1, (0)(!) = 0, (n+1) (!) = 1. ) , Gn ( !) | .!. C N0 1]. ? , sup jFn(t);Gn(t)j 6 n1 1, Fn(t !) = i=n 06t61 p t 2 N (i)(!) (i+1)(!)), i = 0 : : : n. < Zn (t) = n(Gn (t) ; t), t 2 N0 1]. ? sup jYn (t) ; Zn (t)j 6 p1n . *! h(Yn ())
06t61 h(Zn ()) (h(Yn ()) ).
6
rr r r rrr r r r r Gn(t)
2 n+1 1
n+1
(0) = 0
-
9> = >1=n
-
(1)
-
(i)
-
9> = > 1=n
-
9> =1=n >
-
Fn(t)
(i+1)
(n) (n+1) = 1
t
. 6.3
8 $ +$ $ / $ $ ;
$& : Zn !D W0 C N0 1] n ! 1, W0 |
(. (4.29)), P ( sup jW0(t)j 6 z) = K (z), z > 0. * !
06t61 , 6.11 ( .-.. ), !
, . N?, . 2, x13]. <
. <
Q"(C ) = P (W () 2 C j W (1) 2 N;" "]) " > 0
(6.26)
C 2 B(C N0 1]), W () | .
6.15. Q" ) W 0 " ! 0+, W 0 |
C N0 1].
-
6.2 , lim sup Q"(F ) 6 P (W 0 2 F ) "!0+ F 2 B(C N0 1]). B 0 6 t1 < : : : < tk 6 1, k 2 N, (W 0(t1) : : : W 0(tk )) W (1). B
, (W 0(t1) : : : W 0(tk) W (1)) . , , EW 0(t)W (1)=EW (t)W (1);tEW (1)2 =minft 1g;t =0,
2
113
t 2 N0 1], . ? , C 2 B(C N0 1]) B 2 B(R) P (W 0() 2 C W (1) 2 B ) = P (W 0 2 C )P (W (1) 2 B ): (6.27) * , ( B(C N0 1]) (. 6.6 )), , (6.27) B 2 B(R) C 2 B(C N0 1]) (.
4.2) , , P (W 0() 2 C j W (1) 2 N;" "]) = = P (W 0() 2 C ) " > 0. & , (W () W 0()) = sup jW (t) ; (W (t) ; tW (1))j = jW (1)j t2 0 1]
| C N0 1]. F 2 B(C N0 1]). ) , jW (1)j < W () 2 F , , W 0 2 F = fx 2 C N0 1]: (x F ) < g. B " < P (W 2 F j jW (1)j 6 ") 6 P (W 0 2 F j jW (1)j 6 ") = P (W 0 2 F ): ? , lim sup Q"(F ) 6 W 0 (F ): (6.28) "!0+
T
< (6.28) # 0 , F = F . 2 >0 2 6.15 ,
( W (1) = 0). ? , W 0 h;1(B ) = W 0 (h;1(B )) = P (W 0 2 h;1(B )) = P (h(W 0) 2 B ) Q"h;1 (B ) = P (W () 2 h;1(B ) j W (1) 2 N;" "]) = P (h(W ) 2 B j W (1) 2 N;" "]) B 2 B(R), " > 0, h(x()) = sup jx(t)j x() 2 C N0 1]. *! 6.15 t2 0 1]
6.3
P (supt2 0 1] jW (t)j 6 z jW (1)j 6 ") z > 0: (6.29) P ( sup jW 0(t)j 6 z) = "lim !0+ P (jW (1)j 6 ") 06t61 ?, (6.29) z > 0, , , ( (6.29), . . K (z), z. 3 , , (
(6.29) (., , N?, . 96]), " ! 0+,
6.14. /. B6.14 . 2
.
D. 6.1. B , Qn ) Q n ! 1 ( B(X ), (X ) {
) , (6.1)
- $ , .. f : X ! R, kf k1 := sup jf (x)j < 1 L(f ) := supfjf (x) ; f (y)j=(x y)g < 1: (6.30) x2R
114
x6=y
D. 6.2. * f : X ! Y , X , Y {
( f ). * ,
Df = fx : f
xg 2 B(X ). B , Qn ) Q , (6.1)
f 2 B(X )jB(R) , Q(Df ) = 0. D. 6.3. * X = Rm . 0 , Qn ) Q, (6.1) f 2 C01(Rm), .. ?
A % $ (6.1) $ $ ;% ( ) . ! D. 6.4. * Q Qn n 2 N { (X B(X )). B , 6.2 , Qn(X ) ! Q(X ) n ! 1.
(X A)
N A "( ,
N
A. * X {
. ,
M B(X ) "( ( ) , Q Q1 Q2 : : :
, Qn(B ) ! Q(B ) B 2 M Q(@B ) = 0 Qn ) Q. D. 6.5. * ,
, ( ,
, ( . < 6 ,
. D. 6.6. ( 6.7). B ,
1 R ( 1 X ((x1 x2 : : : ) (y1 y2 : : : )) = 2;k 1 +jxjkx; ;ykyj j k k k=1
- ) Qn ) Q n ! 1
, . ) , R1 $ , ; ; $+. D.p 6.7. (
(5.14)). * p D D j j Sn= n ;! N (0 1) n ! 1 , , j S j = n ;! n 6.3. , , fjSnj=pngn2N . *p p ! EjSn j= n ! Ej j = 2=, n ! 1. B Rm F '
(. . ??).
D @ 86.8 (., ., N?, . 344]). ' - n ;!
Rm " "( : 1. Fn (x) ! F (x) n ! 1 x 2 Rm,
# $ F .
2. 'n () ! ' ()
n ! 1 2 Rm. = , n Rm 'n () ! '() 2 Rm, # $ ' 0 2 Rm, '() = ' () D n ;! n ! 1. 115
D D. 6.9. N , { 8] B , n ;! (Rm B(Rm)) D , (a n) ;! (a ) n ! 1 a 2 Rm ( ) { Rm. 1 , & $+ $ $ Rm ; % $ .
9
(
. @ 86.10 (8 %). $ X = fXt t > 0g t Rm ( . ! $
(. .), $ Y = fYt = Xt ; X0 t > 0g . 0 6 s 6 t < 1 E(Xt ; Xs) = Mt ; Ms D(Xt ; Xs ) = Gt ; Gs (6.31) # $ M : R+ ! Rm, "( # $ G : R+ ! Rm2 (R+ = N0 1)). I ! X0 $ & , $ fXt t > 0g % $ ( 6 ). C X
B6.10 Na b],
fYt = Xt ; Xa t 2 Na b]g,
Xet = Xt+a t 2 N0 b ; a] Xet = Xb t > b. ?
B6.10
, ( , 2. *, $$ ( 6.11) $ sup p t2 0 1] W (t). * Sn () Xn i = Xi= n, i = 1 : : : n, X1 X2 : : : , P (X1 = ;1) = P (X1 = 1) = 1=2. D. 6.11. (
4.14 B , j P (0max S > j ) = 2P (Sn > j ) + P (Sn = j ) (6.32) 6k6n k
S0 = 0, Sk = X1 + : : : + Xk , k > 1 fXn g
. ? , z > 0 Sk > z) = P ( max S > j ) = 2P (S > j ) + P (S = j ) p P ( sup Sn(t) > z) = P (0max n n n n n 6k6n n 06k6n k t2 0 1] p p jn { , z n ( jn = ;N;z n], N] { ). * y*?, , z > 0 p p P (Sn > jn ) = P (Sn= n > jn = n) ! P ( > z) N (0 1). ? , . 1 ( ! . D D. 6.12. * n ;! , n , n 2 N
, F (x) R. ? sup jFn (x) ; F(x)j ! 0 n ! 1: x2R
116
B (6.32) (4.25) D. 6.13. B , 6.11
Xi , i 2 N, Sk = X1 + : : : + Xk , k > 1, max P (Sn = j ) ! 0 n ! 1: j
* , - X1 X2 : : : ..., P (X1 = ;1) = P (X1 = 1) = 1=2 S0 = 0, Sk = X1 + : : : + Xk , k > 1. < mn = 0min S Mn = 0max S 6k6n k 6k6n k W (t) t > 0, m = t2inf W (t), M = sup W (t). 0 1] t2 0 1] h : C N0 1] ! R3, h(x()) = (t2inf x(t) sup x(t) x(1)): 0 1] t2 0 1]
6.3 6.11,
p
p
p
D h(Sn ()) = (mn= n Mn = n Sn= n) ;! (m M W (1)) (6.33) p Sn (t) 0 6 t 6 1 { (k=n Sk = n), k = 0 : : : n. ' , M , (., ., N?, . 18-21]) ( (6.33) ( . @ 86.14 ( ). 3 a < 0 < b, a < r < s < b N (0 1)
P (a < m 6 M < b r < W (1) < s) =
;
1 X k=;1
1 X
k=;1
P (r + 2k(b ; a) < < s + 2k(b ; a)) ;
P (2b ; s + 2k(b ; a) < < 2b ; r + 2k(b ; a)):
) , & $$ + -
$ $$ $ $ $ $. ?,P y*? ... , Xk k 2 N, D X n ! 1, X N (0 1). 0 1, n;1=2 nk=1 Xk ;! P P
(
( 6 ) . . Y , n;1=2 nk=1 Xk ;! Y n ! 1 ( , , r).
A% $ $ $ $$ $ $ (X A). < , hf Qni hf Qi f . < B(X R) {
, ( AjB(R) { f : X ! R, k k1 , . (6.30). * P = P (X ) {
(X A). B P Q 2 P
P Q kP ; Qkvar = supfjhf P i ; hf Qij : f 2 B(X R) kf k1 6 1g: D. 6.15. * , (P Q) = kP ; Qkvar P , " P , U (Q ) = fP 2 P : (P Q) < g, Q 2 P , > 0 (
. N?, . 2, x5]). $
117
D. 6.16. B , (P Q) = 2 sup jP (A) ; Q(A)j: A2A
D. 6.17. * P Q 2 P P , Q 2 P ( .. P Q -
P (
, = (P + Q)=2). * g = dP=d, h = dQ=d. B , (P Q) = kg ; hkL1 (), L1() = L1(X A ). * , (P Q) 6 2,
, P ?Q ( , .. A 2 A , P (A) = 1 Q(A) = 0). D. 6.18. B , (P (X ) ) { $ $ $. B , !
% +, X { $. S (P (X ) ), X { ? ) $ $ % + $$. @ 86.19. Sn = Pnk=1 Xk , X1 : : : Xn { , P (Xk = 1) = pk P (Xk = 0) = 1 ; pk , k = 1 : : : n. Y { P . . = p1 + : : : + pn . jP (Sn 2 B ) ; P (Y 2 B )j 6 nk=1 p2k ;% B R. B
! N?, . ], . B , @ 86.20 (H). Qn n 2 N 1 { (X A), Qn - (X A). dQn =d ! dQ1 =d . . ( ). Qn Q1 $ . / = { 8
d(P Q). ) ,
. 6.17, p d2(P Q) = 12 h(pg ; h)2 i: D. 6.21. * , d { P (X ),
( . & , d 8 1/2. 1 2 (0 1) H (P P Q) = h(g) h1; Qi: ? d2(P Q) = 1 ; H (1=2P P Q). p D. 6.22. B , 2d2 (P Q) 6 kP ; Qkvar 6 8d(P Q) P Q 2 P (X ). B , P = P1 : : : Pn , Q = Q1 : : : Qn,
H (P P Q) =
Yn
k=1
H (P Pk Qk ):
* d H N?, . 3, x9,10]. ?
$ , $ $$ $ $. ! 118
D. 6.23. * Xn ! X1 .. n ! 1, Xn : D ! X , Xn 2 FjB(X ) n 2 N f1g. ? L(Xn ) ) L(X1 ), n ! 1. 2 2 Ef (Xn ) ! Ef (X1 ) f 2 Cb(X R). 2
8
, ( ! ,
. .,
( ). < & $ % $$+ +$ $,
@ 86.24 (). (X ) { , Qn ) Q1 n ! 1. ( (D F P ) Xn : D ! X , Xn 2 FjB(X ), L(Xn ) = Qn n = 1 2 : : : 1 Xn ! X1 . . n ! 1. 2 $ ; $ + ; % X . B k 2 N X Gk m , m 2 N 2;(k+1) , m Gk m = X . 3 2;(k+1) 2;k , X Bk m
, Qn(@Bk m) = 0 k m 2 N n 2 N f1g ( 6 ). * Dk 1 = Bk 1 m > 2 Dk m = Bk m n mr=1;1 Bk r . * k 2 N
Dk m m 2 N X , diamDk m 6 2;k , m 2 N (diamD = supf(x y) : x y 2 Dg, D 2 X ) Qn (@Dk m) = 0 n k m. 3 &$ Si1 ::: ik = \kj=1 Dj ij ( { N). <
( k X , 1) Si1 ::: ik \ Sj1 ::: jk = (i1 : : : ik ) 6= (j1 : : : jk )P 2) j Sj = X , j Si1 ::: ik j = Si1 ::: ik P 3) diamSi1 ::: ik 6 2;k k i1 : : : ik 2 NP 4) Qn(@Si1 ::: ik ) = 0 k i1 : : : ik 2 N, n 2 N f1g. ? % $$+ ; $ +$+ (D F P ), D = N0 1), F = B(N0 1)) P = j j, j j 2 ( , , $ - % F ). B n 2 N 1 k 2 N N0 1) ](i1n)::: ik = Na(i1n)::: ik b(i1n)::: ik )
(n) i1 : : : ik 2 N ]i ::: ik = Qn(Si 1
1
::: ik ),
,
](1n) = N0 Qn(S1)) ](2n) = NQn(S1) Qn(S1) + Qn(S2)) : : : J ](in) ](inj) ( , ](in)) .. ) ](i1n)::: ik . B k 2 N
Si1 ::: ik xi1 ::: ik n 2 N f1g D = N0 1)
Xnk (!) = xi1 ::: ik ! 2 ](i1n)::: ik 119
( Qn(Si1 ::: ik ) = 0, ](i1n)::: ik = { Na a), ). < , Xnk 2 FjB(X ) k n. & , (Xnk (!) Xnk+m (!)) 6 2;k ! 2 N0 1) k n m (6.34)
2) 3) . X (
Xn (!) = klim X k (!) ! 2 N0 1) n 2 N f1g (6.35) !1 n
4.6 Xn 2 FjB(X ) n 2 N f1g. *
4) 3 6.2, , k i1 : : : ik 2 N
(n) (1) Qn(Si ::: ik ) = ]i ::: ik ! ]i ::: ik = Q1(Si 1
1
1
1
::: ik )
n ! 1:
/ , Q1(Si1 ::: ik ) > 0 (, ](i11:::) ik 6= ), (1) ! 2 ]i1 ::: ik (B { !
B ), nk (!) , ! 2 ](i1n)::: ik n > nk (!) ( 6 ). ' ! Xnk (!) = X1k (!) , (6.35), (Xn (!) X1 (!)) 6 (Xn (!) Xnk (!)) + (Xnk (!) X1k (!)) + (X1k (!) X1 (!)) 6 2;k+1
(1) n > nk (!). * D0 = \1 k=1 i1 ::: ik ]i1 ::: ik . < , P (D0 ) = jD0 j = 1 Xn (!) ! X1 (!) ! 2 D0 n ! 1. ? X1 2 FjB(X ) 4.6. A$ + $+, $ L(Xn ) = Qn n 2 N f1g. *
n m i1 : : : im k > m
P (Xnk
(n) 2 Si ::: im ) = ]i ::: im = Qn (Si 1
1
1
::: im ):
(6.36)
;% $$ &$ G X & $ $+, %: $ &$ Si ::: im ( m 1
i1 : : : im). * !
( 3,4 N?, . 2, x5]). GN ! G, GN N
. / , P (Xnk 2 G) > P (Xnk 2 GN ) N , (6.37), lim inf P (Xnk 2 G) > lim inf P (Xnk 2 GN ) = Qn(GN ): k!1 k!1
? , lim infk!1 P (Xnk 2 G) > Qn(G). / 2 6.2 , L(Xnk ) ) Qn k ! 1 n 2 Nf1g. ' Xnk ! Xn .. k ! 1 n, ! . 6.22 , L(Xnk ) ) L(Xn ). <
. 2 ? B6.24
. 6.2. ) !
( ..), D X , X , EX ! EX . B X = R, Xn ;! n n 86.25. (X ) { , Qn ) Q P .
supfjhf Qni ; hf Qij : f 2 GC g ! 0 n ! 1 (6.37) GC { # $ f : X ! R, "( kf k1 6 C . 120
2 * (D F P ) ( , !) -
, ( B6.24. ? ( (2.10)), , supfjEf (Xn ) ; Ef (X1 )j : f 2 GC g ! 0 n ! 1: " > 0 = (") > 0,
, jf (x) ; f (y)j 6 " f 2 GC , (x y) 6 . ?
jEf (Xn ) ; Ef (X1 )j 6 Ejf (Xn ) ; f (X1 )j1 f(Xn X1) 6 g + + Ejf (Xn ) ; f (X1 )j1 f(Xn X1 ) 6 g 6 " + 2CP ((Xn X1 ) > ): P < , Xn ! X1 .., Xn ;! X1 , ..
P (! : (Xn (!) X1 (!)) > ) ! 0 > 0 n ! 1 (Xn X1 )
( X ), 0 .., . 2 ? $ % $. $+ ( ) (X ) { + $ $. B B X " > 0 B " = fx 2 X : (x B ) < "g, (x B ) = inf f(x y) : y 2 B g.
P (X ) + {
(P Q) = inf f" > 0 : P (B ) 6 Q(B ") + " Q(B ) 6 P (B ") + " B 2 B(X )g: (6.38) D. 6.26. B , ( ) P (X ). B ,
( ( ) (P Q) = inf f" > 0 : P (F ) 6 Q(F ") + " F Xg: (6.39) D. 6.27. B , (P (X ) ) {
. @ 86.28. Qn ) Q , (Qn Q) ! 0 (n ! 1). 2 * (Qn Q) ! 0. ? (6.41) " > 0 F X Qn(F ) 6 Q(F ") + " n > n("). / , lim supn!1 Qn(F ) 6 Q"(F ) + " " > 0. < "
1 6.2. A% $. * Qn ) Q. / f"F (),
6.2, " > 0 F {
X , , 0 6 f"F () 6 1 jf"F (x) ; f"F (y)j 6 ";1(x y) x y 2 X ( 6 ). / , " > 0 M" = ff"F ()
F Xg G1, GC
B6.25. ) (6.37) , " > 0 ]n(") = supfjhf Qni ; hf Qij : f 2 M"g ! 0 n ! 1: (6.40) B " > 0 F , , 1 F () 6 f"F () 6 1 F " (), ](n") Q(F ") > hf"F Qi > hf"F Qni ; ](n") > h1 F Qni ; ]n(") = Qn(F ) ; ](n"): (6.41) < , ](n") 6 " n > n0(") (6.39). 2 121
D. 6.29. L
BL
X , kf kBL = kf k1 + L(f ), . (6.30). B ,
kP ; QkBL = supfjhf P i ; hf Qij : f 2 BL kf kBL 6 1g $ P (X ), $+ $ $ & , $ % $. 0 , P Q 2 P (X )
kP ; QkBL 6 2(P Q) '((P Q)) 6 kP ; QkBL '(t) = 2t2=(t + 2), t > 0. @ 86.30 (H$ , M?]). : + { = 0 { ( "( ). E ,
(P Q) = inf f{(X Y )g
(6.42)
(X Y ) (
) , L(X ) = P , L(Y ) = Q,
{ (X Y ) = inf f" > 0 : P ((X Y ) > ") < ".
/ (6.42) ( - ( .. ), ( , ( ! (X Y ), ( .. L(X ) L(Y )) , ! X Y . ) ( N?], N?]. < , $ $ + $ . ', + <
& $+ $ + ; $ $ $$ 8 { . * Xn i i = 1 : : : mnP n 2 N { , , EXn i = 0, EjXPn ijs < 1 s > 2. 0 PXn i , mi=1n n2 i = 1, n2 i = DXn i . * Ln s := mi=1n EjXn i js. 1 { +
( s=2 P P m m n n s 2 , Ln s = i=1 EjXn i ; EXn i j = i=1 n i
). 2 ( ),
+
Ln s ! 0 n ! 1
2 (6.18). & , 2 , y*? (. 2),
" " ": " > 0 max P (jXn i j > ") ! 0 n ! 1:
16i6mn
(6.43)
' y*? (6.43) N?], N?]. 122
@ 86.31 (E, N?]). Pn { Sn(), # "( 6.11. - ( ) Xn i i = 1 : : : mn n 2 N, s 2 (2 3] "( $ :
(Pn W) 6 cL1n=s(s+1) (6.44) W { 9 C N0 1], c n. *
! N?], N?], (, (6.44) $ + $ $+; % $ &$ c. * ( + +$ $ % $ $$ $ $. <
C (T X ) ](f ) = supf(f (x) f (y)) : d(x y) 6 g f 2 C (T X ) > 0 d { T , {
X . ' , > 0 ](f ) C (T X ) R ! B(C (T X ))jB(R)-. @ 86.32 ( ). .. X (n) n 2 N C (T X ), T { , X { ,
,
lim lim sup E(](X (n) ) ^ 1) = 0: #0 n!1
L
C (T X ), X {
, T {
( ,
). / C (T X )
(. (3.20)). D. 6.33. B , X (n) !D X , .. L(X (n)) ) L(X )
D
C (T X ) , Xj(Kn) ! XjK C (K X ) K T , , YjK Y = fYt t 2 T g YjK = fYt t 2 K g. 1 , . 6.9 6.11
D. 6.34. * 1 2 : : : { ... Rm, E 1 = 0, Ek 1k2 < 1, k k { . B , X Xt(n) = n;1=2 k + (nt ; Nnt]) nt]+1 t > 0 n 2 N: k6nt
n ! 1 m { (N] ). D. 6.35. * X (n) n 2 N { Rd t 2 Rd
X . ? C (Rd X ), > 0 E((Xs(n) Xt(n))) 6 kt ; skd+ s t 2 Rd k k { Rd. 123
B6.8 , $ $$ $ $ $ + % &$ Rm. * X {
X {
. 8 # $ Q 2 P (X ) ! X : D ! X , X 2 FjB(X ), , L(X ) = Q, 'Q : X ! C ,
'Q(x) = E expfihX xig x 2 X hy xi
x ! y 2 X . * C {
X , ..
fx 2 X : (hx z1i : : : hx zn i) 2 B g B 2 B(Rn), z1 : : : zn 2 X . D. 6.36. ( B6.8) B , C B(X ). B , Qn ) Q P (X ), 'Qn ! 'Q . < , 'Qn ! 'Q , ' : X ! C fQng . ? ' = 'Q Q 2 P (X ) Qn ) Q. < N?]. % $ $ $ C (T X ) &$ %$+ % $ $ $ D(N0 1]q ) D(N0 1)q ), q > 1. 1
,
N0 1]q N0 1)q , (
, , t = 6 0
( f t , f (s) s ! t s 6= t , sk > tk , k = 1 : : : qP ). 1$ $ $ +; $ $ %$+ +, .
/
, , ( " " , , ( " " (, fx(t) = 1 x 1)(t) fy (t) = 1 y 1)(t)
C x ! y, D). / ! N?], N?].
/ ( + ( a ) ( a ) % & ; $. * Y1 Y2 : : : { ...
P (Y1(a) = ;a) = P (Y1(a) = a) = 1=2 a > 0: * 1() 2() : : : { ... , ! > 0, fYn(a)g f m()g
a . * P ( ) k a ;a k = j=1 j(), k = 1 2 : : : ( t = 0
). ? , k() P kj=1 Yj(a) ! P ) N k()) k(+1 ). ) , Xt(a ) = j6N (t) Yj(a), t > 0, N(t) = maxf k : k() 6 tg ( N(0) = 0, X0(a ) = 0). 8 , N() { . )
124
D. 6.37. ) .-.. X (a ) , a ! 0, ! 1 , a2 = 2 > 0 ( ( , ( ). 0 X (a )
DN0 1)? ' B { * + $$. J , ( " ") 1 2 : : : R (Rm
), 1 2 : : : , , ( ,
fW (t) t > 0g , ( !) L( 1 2 : : : ) = L( 1 2 : : : )
X
k ; W (t) = O(h(t)) .. t ! 1
k 6t
(6.45)
h { . & (6.45)
(
. . ! 2 D C (!) > 0,
j
X k 6t
k ; W (t)j 6 C (!)h(t)
t > t0(! C (!)). (6.45)
X
P
k 6t
k ; W (t) = o(g(t)) .. t ! 1
(6.46)
(, j k6t k ; W (t)j=g(t) ! 0 t ! 1 . . ! 2 D, g { . @ 86.38 (H$ ). 1 2 : : : { , E 1 = 0 E 12 = 1. (6.46), g (t) = (t log log t)1=2, t > e. B
! /, . 4, . ??. B ( (6.45), " ", ,, 3 ? N?], N?]. & , ( . < . N?], N?]. D. 6.39. / ( B6.38 C N0 1], ... 1 2 : : : p . 1 (k=n Sk = 2n log log n), k = 0 : : : n, n > 3, S0 = 0, Sk = 1 + : : : + k , k > 1. D. 6.40. ) ( , \ { : X1 X2 : : : { ... EX1 = 0, EX12 = 1. ? p 1
fSn= 2n log log n n > 3g N-1,1] ( Sn = X1 + : : : + Xn ). / ( . 6.39 ( h : C N0 1] ! R
125
D. 6.41. * fXk g fSk g { , 6.40. * f (t) { -
, L N0,1]. ? 1
R
X k Z 1 2 !1=2 F (u)du n Sk =
n 3 ; 1 = 2 lim sup(2n log log n) f n!1 k=1
0
F (u) = u1 f (t)dt u 2 N0 1]. & ( , ( . * X {
. * M (X ) {
- B(X ) ( .. M
), ,
f : 7! hf i =
Z
X
fd
f 2 CK+ { . D. 6.42. B , M (X ) {
. B ! , f1 f2 : : : {
, CK+ ,
( ) =
X k
2;k (jhf i ; hfk ij ^ 1) 2 M (X )
. * ! B(M (X )) f f 2 CK+ ( .. - , ! ), B : 7! (B ) B 2 X, 2 M (X ), X = fB 2 M : (@B ) = 0g, M {
X . C (D F P ) {
X 2 FjB(M (X )), X . $ , ( (
)
N (X ) M (X ), ( { . D. 6.43. * Y Y1 Y2 : : : { X {
. B , ( ! D Y n ! 1. 1. Yn ! D 2. hf Yni ! hf Y i f 2 CK+ (n ! 1). D (Y (B ) : : : Y (B )) B : : : B 2 X k 2 N, 3. (Yn(B1) : : : Yn(Bk )) ! 1 k 1 k Y XY = fB 2 M : Y (@B ) = 0 ..g. < , , , N?], N?]. 8 , ( . , , , N?].
126
7. '
% . ' fXt t 2 U g j B(R)- . 3 " 1 Rd. 3 " d- . . 4 d- . ! . $ , " @ @.
? . & 3 , ( ,
, $ , $ , ; -
&$ %$+ ; % . $ t $ $ , <+ , $ $ t. & -
, t ( ) t. C (t = 0 1 : : : ),
. J , , n , n. * , 6 , (. < 3 ( (), (. 0 , $ . ' ( . F $ . * X = fXt t 2 T Rg | , (D F P ) ( t
(X B). /.. X , t 2 T A 2 F6t = fXs : s 2 T \ (;1 t]g, B 2 F>t = fXs : s 2 T \ Nt 1)g
P (AB j Xt ) = P (A j Xt)P (B j Xt) .. (7.1) AB C A 2 F6t
t
A 1C
B 2 F>t
. 7.1
I (7.1) , 6% 7 6< 7 6 $ 7. ', P (A j A) := E(1 A j A) A 2 F - A F , E( j ) := E( j f g) .!.
. . . ? , -
(...)
. . - A F ( E( j A)). J , !
: D ! R, 1) A j B(R)-, 2) C 2 A E 1 C = E 1 C : / ( L{' , Ej j < 1,
(
! 127
( 1 2, P ( 1 = 2) = 1). B = ( 1 : : : n ): D ! Rn E( j A) = = (E( 1 j A) : : : E( n j A)) , (
. 7.1. X | $ , " t 2 T F G, F6t j B(R)- F>t jB(R)- ,
E(FG j Xt ) = E(F j Xt)E(G j Xt )
(7.2)
. .
2 < , (7.1) (7.2), F = 1 A , A 2 F6t G = 1 B , B 2 F>t. (7.2) (7.1). * ... (E( + j A) = = E( j A)+ E( j A) .. 2 R - A F , Ej j < 1, Ej j < 1) , (7.2) $ %
F=
M X i=1
ci1 Ai G =
N X j =1
dj 1 Bj
ci dj 2 R, Ai 2 F6t , Bj 2 F>t, i = 1 : : : M , j = 1 : : : N . 2 A jB(R)- h ( sup jh(!)j < H ) !2
P
2n ;1
$ % A j B(R)- hn = rn k 1 Dn k , k=0 rn k = ; H + kH 2;n+1 , Dn k = f! : rn k < h(!) 6 rn k + H 2;n+1 g 2 A, k = 0 : : : 2n ; 1. * ! sup jhn(!)j 6 H !2
sup jhn(!) ; h(!)j 6 H 2;n+1 n 2 N: !2
(7.3)
' E(hn j A) ! E(h j A) .. (jE(hn j A) ; E(h j A)j 6 E(jhn ; hj j A) 6 H 2;n+1 ). < , n ! .. ~n ! ~ .. n ! 1 n = ~n .. (n 2 N), = ~ .. 2 7.2. (D F P ) F j B- .. Xt, t 2 U , (X B ). " fXt t 2 U g jB (R)- h (sup jh(! )j < H ) . . ! L1 (D F P )
f1(Xs1 ) : : :fm (Xsm ). G fi | B jB(R)- # $ , "( si 2 U , i = 1 : : : m ( m 2 N, si # $ fi , i = 1 : : : m). 2 * An = fXt t 2 U g ,
7.1, 2P ;1 hn = rn k 1 Dn k , Dn k 2 A. ? (. k=0 4.3), A = fAg, A
L = f! : (Xs1 : : : Xsm ) 2 C g, q
(C1j : : : Cmj ) s1 : : : sm 2 U Cij 2 B i = 1 : : : m j = 1 : : : qP m q 2 N (7.4) 6 .
C=
128
j =1
B " > 0 D 2 A (.
4.2) L" 2 A: P (D 4 L") < ". *! n 2 N, k = 0 : : : 2n ; 1 "n 2 (0 1)
L n k 2 A, P (Dn k 4 Ln k ) < "n 2;n . n ;1 2P * ~hn = rn k 1 Ln k , k=0
Ejhn ; ~hnj 6
X
2n ;1
k=0
jrn k jEj1 Dn k ; 1 Ln k j = =
X
2n ;1
k=0 H 2;n+1
jrn k jP (Dn k 4 Ln k ) 6 H 2n "n2;n = H"n: (7.5)
/ , Ejh ; h~ nj 6 + H"n , n 2 N, (7.3) (7.5). *! ~hn ! h n ! 1
L1(D F P ). / , , , , , ( .. < , L 2 A, C (7.4), 1 L (!) =
q X j =1
1 C1j (Xs1 (!)) : : : 1 Cmj (Xsm (!)):
(7.6)
0 , $ % $+, $% ; ,
$ $, $ & ; H . B ! , ( , (
Ln k k = 0 : : : 2n ; 1 ;1 Ln j (
Lbn 0 = Ln 0 Lbn k = Ln k n kj=0 P n k = 0 : : : 2n ; 1, bhn = 2k=0;1 rn k 1 Lbn k , n 2 N ( ! "n
, "n2n ! 0, n ! 1). 2 $ 7.3. 3 $ X m n 2 N, " s1 < : : : < sm 6 t 6 t1 < : : : < tn ( T ) , (7.2) F = f1(Xs1 ) : : : fm(Xsm ) G = g1(Xt1 ) : : :gn (Xtn ) (7.7) fi gj | B jB (R)- # $ , (7.7)
fi = 1 Ci gj = 1 Dj Ci 2 B Dj 2 B i = 1 : : : m j = 1 : : : n: (7.8) 2 / ... 7.2 7.1. ,
, h h~ n , (
7.2, EjE(~hn j Xt ) ; E(h j Xt )j 6 E(Ej~hn ; hj j Xt) = Ej~hn ; hj: / , fnk g, E(~hnk j Xt) ! E(h j Xt ) .. k ! 1. 2 7.4. X | $ , " m n 2 N " s1 < : : : < sm 6 t 6 t1 < : : : < tn ( T ),
B jB (R)- # $ g1 : : : gn E(G j Xs1 : : : Xsm Xt) = E(G j Xt) . ., (7.9) G (7.7), (7.7) - # $ gj (7.8). 129
2 *
... (., , N?, . 2, x7]), A j B(R)- . ., . . , Ej j < 1, Ej j < 1, E( j A) = E( j A) .. ,
, Ej j < 1 - A1 A2 F , E(E( j A1) j A2) = E(E( j A2) j A1) = E( j A1) .. (7.10) * (7.9). ? F G, (7.7), E(FG j Xt) = E(E(FG j Xs1 : : : Xsm Xt) j Xt) = = E(F E(G j Xs1 : : : Xsm Xt) j Xt) = E(F E(G j Xt) j Xt) = = E(G j Xt)E(F j Xt) (
..., ,
1). * (7.2). /. . E(G j Xt ) fXtg j B(R)-. *!
(7.9) , E1 C E(G j Xt) = E1 C G C 2 fXs1 : : : Xsm Xtg. 8 , E(E( j A)) = E A F Ej j < 1, (7.2) E1 C G = E(E(1 C G j Xt )) = E(E(1 C j Xt)E(G j Xt )): / , E1 C E(G j Xt) = E(E(1 C E(G j Xt) j Xt)) = E(E(G j Xt )E(1 C j Xt )): 2 7.5. 3 $ X ( ) 7.4 n = 1. 2 G (7.7), n > 2. ? , (7.9) n = 1, E(G j Xs1 : : : Xsm Xt ) = = E(E(G j Xs1 : : : Xsm Xt Xt1 : : : Xtn;1 ) j Xs1 : : : Xsm Xt) = = E(g1(Xt1 ) : : : gn;1 (Xtn;1 )E(gn (Xtn ) j Xtn;1 ) j Xs1 : : : Xsm Xt) = = E(g1(Xt1 ) : : : g~n;1 (Xtn;1 ) j Xs1 : : : Xsm Xt) g~n;1 (Xtn;1 ) = gn;1 (Xtn;1 )E(gn (Xtn ) j Xtn;1 ). 3 (., , N?, . 236]) , E( j ) = '( ), ' | B jB(R)- ( ! , , ), , jE( j )j 6 H .., j j 6 H .. * , E(G j Xs1 : : : Xsm Xt) = E(~g1(Xt1 ) j Xs1 : : : Xsm Xt) = E(~g1(Xt1 ) j Xt) (7.11) g~1 (Xt1 ) = g1(Xt1 )E(~g2(Xt2 ) j Xtn ). ? E(G j Xt ) (
Xs1 : : : Xsm ), (7.11). 2
A + % $ &$ $ 7.6. X | $ , " m 2 N s1 < : : : < sm 6 t 6 u ( T ) " C 2 B P (Xu 2 C j Xs : : : Xsm Xt) = P (Xu 2 C j Xt) . . (7.12) 1
2 g = 1 C , (7.12) 7.5. B
-
,
7.1. 2
( $+ % $
130
@ 7.7. X = fXt t > 0g | $ ( , "( Rd (d > 1) , Xt F j B (Rd)- t > 0. X | $. 2 ) fXs : : : Xsm Xtg = fXs Xs ; Xs : : : Xt ; Xsm g, 1
1
2
1
. ' , 7.2, |
Rq Rl f : Rq Rl ! R ( . . B(Rq+l) jB(R)- ), Ejf ( )j < 1, E(f ( ) j = y) = Ef ( y) .. P
(7.13)
( E( j = y) , '(y), '( ) = E( j ). ? , 0 6 s1 < : : : < sm 6 t 6 u ( T ) B(Rd) j B(R)- g
mX+1 ! E(g(Xu ) j Xs : : : Xsm Xt ) = E g + i 1 : : : m+1 i=1 1
1 = Xs1 , 2 = Xs2 ; Xs1 , : : : , m = Xsm ; Xsm;1 , m+1 = Xt ; Xsm , = Xu ; Xt. / , .. ( 1 : : : m+1)
mX+1
E g +
i=1
mX +1
i j 1 = y1 : : : m+1 = ym+1 = Eg +
i=1
mX+1
yi = T
i=1
yi
T | . B
, h1(! z) = (!) h2(! z) = z, D R, , F B(R)jB(R)-. / , g( (!) + z) = g((h1 + h2)(! z)) ( ) F B(R)jB(R)- . F B(R) P Q (P Q {
F B(RR ), .. F B(R) ), ,
I (. N?, . 363]) g((h1 + h2)(! z))dP . ? ,
mX+1
E(g(Xu ) j Xs1 : : : Xsm Xt) = T
i=1
i
..
? ,
mX+1 mX+1 E(g(Xu ) j Xt ) = E g + i i = i=1 mX+1 i=1 mX+1 = E E g + i 1 : : : m+1 i = mX+1 i=1 mX+1 mX+1 i=1 =E T
i i = T
i .. 2 i=1 i=1 i=1
E & Rm W (t) = (W1(t) : : : Wm (t)), t > 0, m
Wi ( m .!. C N0 1)). 131
$ 7.8. @
Rm $. $ $.
@+ ; ;, $ $ ; ; + $ . I P (s x t B ), s 6 t (s t 2 T R), x 2 X , B 2 B, # $ ( ), 1) s x t P (s x t ) (X B), 2) s t B P (s t B ) B jB(R)-, 3) P (s x s B ) = x(B ) s 2 T , x 2 X , B 2 B, 4) s < u < t (s u t 2 T ), x 2 X , B 2 B / { P : P (s x t B ) =
Z
X
P (s x u dy)P (u y t B ):
(7.14)
3) (7.14) s 6 u 6 t. 9 , fXt t 2 T g % $ P (s x t B ), s 6 t (s t 2 T ), B 2 B
P (Xt 2 B j Xs ) = P (s Xs t B ) .., (7.15) , , P (s x t B ) = P (Xt 2 B j Xs = x) .. PXs . 8 -
, $ $ 6< 7 $ , $ $ (7.15). *
..., $+ & (7.14). (7.15), (7.10) (7.12) s 6 u 6 t, (s u t 2 T ) B 2 B P (s Xs t B ) = E(1 fXt 2 B gjXs) = E(E(1 fXt 2 B gjXs Xu )jXs ) = = E(E(1 fXt 2 B gjXu)jXs ) = E(P (u Xu t B )jXs): ) , PXs { x
P (s x t B ) = E(P (u Xu t B )jXs = x):
(7.16)
< (. N?, . 242]), (C ) = P (Xu 2 C jXs = x) ( s u x), g 2 BjB(R) E(g(Xu )jXs = x) =
Z
X
g(y) (dy) PXs ; ..
(7.17)
) , (7.16) (7.17) $ $ <+ % $ $ (7.14), PXs - x. 7.9. * W (t), t > 0, | m- . ? , (7.13), s < t
P (W (t) 2 B j W (s) = x) = E(1 fW (t);W (s)+W (s)2Bg j W (s) = x) = = E1 fW (t);W (s)+x2Bg = P (W (t) ; W (s) 2 B ; x) = Z ;kzk2 Z ;kz;xk2 1 1 = (2(t ; s))m=2 e 2(t;s) dz = (2(t ; s))m=2 e 2(t;s) dz: B ;x
132
B
* t = s ( (7.13) W (s) 0 2 Rd)
P (W (s) 2 B j W (s) = x) = P (W (s) + 0 2 B j W (s) = x) = P (x + 0 2 B ) = x(B ): ) , P (s x s B ) = x(B ), s < t 1
P (s x t B ) = (2(t ; s))m=2
Z B
e
;ky;xk2 2(t;s) dy
(7.18)
1){4), , ( (7.14) , ). 3 fXt t 2 T g, ( P (s x t B ), , x 2 X , B 2 B, h > 0 s s + h 2 T
P (s x s + h B ) = P (0 x h B ):
(7.19)
* (7.19) P (x h B ), !
B h, x. / (7.18), m- % & { . B fXt t > 0g 1) { 4) s t > 0, x 2 X , B 2 B ( : 1') x t P (x t ) B(X ), 2') t B P ( t B ) 2 BjB(R), 3') P (x 0 B ) = x(B ) x B , 4')
P (x s + t B ) =
Z
X
P (x s dy)P (y t B ):
(7.20)
@% (7.20) $ $ $ $+ $ $ . * ! ! .
A% $ &; .-.. . 7.10. X = fXt t 2 T Rg | $ (X B ), "( " # $ " P (s x t B ). " s 6 t (s t 2 T ) " B jB (R)- # $ g () E(g(Xt ) j Xs ) = '(Xs) . ., (7.21)
Z
'(x) = P (s x t dz)g(z) X
B jB (R)- # $ ( , g s t # $ ).
(7.22)
'
2 * g() = 1 B (), B 2 B. ? E(g(Xt) j Xs ) = P (Xt 2 B j Xs ) = P (s Xs t B ) .. 133
/ ,
Z X
P (s x t dz)1 B (z) = P (s x t B ):
PN
/ ,
g() = ck 1 Bk , Bk 2 B, k = 1 : : : N . k=1 C g B jB(R)- , sup jg(z)j < H , , z2X
7.1, $ % B jB(R)- gk , g ( ! sup jgk (z)j 6 H , k 2 N) z2X E(g(Xt) j Xs)= klim E( g ( X ) j X ) .., s !1 k t
Z
X
P (s x t dz)gk(z) !
R
Z
X
P (s x t dz)g(z) (k ! 1):
B j B(R)- P (s x t dz)gk (z) gk
X R
2) , B jB(R)- P (s x t dz)g(z) X
x s t 4.7 ( - ). 2
7.11. 9 7.10 " n 2 N, s 6 t1 6 : : : 6 tn T ) " B jB (R)- # $ g1 : : : gn
(
E(g1(Xt1 ) : : : gn (Xtn ) j Xs ) = T(Xs )
T(x) =
Z X
Z
Z
X
X
(7.23)
P (s x t1 dz1)g1(z1) P (t1 z1 t2 dz2)g(z2) : : : P (tn;1 zn;1 tn dzn)g(zn)
( 7.10 # $ ).
2 *
. B n = 1 (7.23) (
7.10. ? s t1 : : : tn ( tn+1 > tn
B j B(R)- gn+1 . ? , G = = g1 (Xt1 ) : : : g(Xtn ), (7.9), (7.10) 7.10 E(g1(Xt1 ) : : : gn(Xtn )gn+1 (Xtn+1 ) j Xs ) = E(E(Ggn+1 (Xtn+1 ) j Xs Xt1 : : : Xtn ) j Xs ) = = E(GE(gn+1(Xtn+1 ) j Xtn ) j Xs ) = E(G'n (Xtn ) j Xs )
'n (x) =
Z X
P (tn x tn+1 dzn+1)gn+1 (zn+1):
gn 'n gn (7.23) g1 : : : gn;1 gn'n , (7.23) g1 : : : gn+1 . 2 134
@ 7.12 (.-.. ). 7.10 " n 2 N, " s 6 t1 6 : : : 6 tn ( T ) B1 : : : Bn 2 B P (Xt1 2 B1 : : : Xtn 2 Bn) =
Z
X
Z
Z
B1
Bn
Qs(dx) P (s x t1 dz1) : : : P (tn;1 zn;1 tn dzn ) (7.24)
Qs = PXs . 2 B 7.11 gi = 1 Bi (7.23) (2.10). ? ET(Xs ) =
Z
X
T(x)Qs(dx): 2
? 7.13. I (7.24) (Xt : : : Xtn )
" " B1 : : : Bn, !
Bn = B : : : B. * 7.2 7.11,
( (7.24). C v 6 s (v s 2 T ), B 2 B 1
Z
Qs(B ) = Qv (dx)P (v x s B ):
(7.25)
X
2 Qs(B ) = P (Xs 2 B ) = E(E(1 fXs2Bg j Xv )) = Z = EP (v Xv s B ) = Qv (dx)P (v x s B ): 2 X
, T = N0 1), Qs s > 0 Q0 ( , 6 { ). ? , X = fXt t > 0g | $ $ (X B) ( t > 0), $ $
; ;, $ .-.. $+; ;$ + ( . . Q0(B ) = P (X0 2 B ), B 2 B) P (s x t B ). C
, , .
, B7.1.
. A $ (7.1) , $ & %% $+ . * , &, 7, $& 7.7 { 7.9, $ , ;$ ; , X = fXt t 2 T Rg { , -$ $$ $ $ (D F P ), ; & t $ $ $ (Xt Bt), t 2 T . B , 135
, t
(X B). * ! P (s x t B ), s 6 t (s t 2 T ), x 2 Xs, B 2 Bt, , $+ & $+ & $ x B (
s t). ', , { S s 6 u 6 t (s u t 2 T ), x 2 Xs, B 2 Bt
P (s x t B ) =
Z
Xu
P (s x u dy)P (u y t B )
(7.26)
(7.15) , B 2 Bt.
F $ ; + $ . * (Xt Bt)t2T {
. B n 2 N s0 6 t1 < : : : < tn ( T ), Bk 2 Btk , k = 1 : : : n & ( , ) C = B1 : : : Bn Qt1 ::: tn (C ) =
Z
Xs0
Qs0 (dx)
Z
Xt1
P (s0 x t1 dz1)1 B1 (z1)
Z
Xtn
P (tn;1 zn;1 tn dzn )1 Bn (zn)
(7.27) Qs0 { Bs0 . (7.27) , 7.10 . *, Qt1 ::: tn { B1 : : : Bn . C B1 : : : Bn = 1q=1B1(q) : : : Bn(q), 6 (Bk(q) 2 Btk , k = 1 : : : nP q 2 N), 1 B1 ::: Bn (z1 : : : zn) =
1 X q=1
1 B1(q) (z1) 1 Bn(q) (zn):
(7.28)
? 0. 2 (. N?, . 348]), (7.27) (7.28). *! & $ $+ Qt1 ::: tn & Bt1 : : : Btn . ; $ $ % $ $ 7.12 (
). @ 87.1. (Xt Bt) { - Bt t 2 T R. - Qt1 ::: tn , t1 : : : tn 2 Ts0 = T \Ns0 1), n 2 N ( s0 2 T ), " , (D F P ) ( $ X = (Xt t 2 Ts0 ), "( Qt1 ::: tn .-.., P (s x t B ) # $ X , Qs0 { Xs0 . 2 2.8 X .-.. Qt1 ::: tn $ $ $+ <+ 3 . ??. * 2 6 m 6 n ; 1 ( n > 3) ! (7.26), m = 1 m = n u 6 t (u t 2 Ts0 ) x 2 Xu Z P (u x t dz) = P (u x t Xt) = 1: Xt
136
F PXs s 2 Ts . n = 1 t1 = s (7.27). ? Z Z Z P (Xs 2 B ) = Qs (dx) P (s0 x s dz) = Qs (dx)P (s0 x s B ): 0
0
Xs0
Xs0
B
0
(7.29)
, s = s0 P (Xs0 2 B ) = Qs0 (B )
3) . 8 &, $ (7.15). /
2) , P (s Xs t B ) fXsgjB(R)- ( 1)) s t 2 Ts0 (s 6 t), B 2 Bt. *! A 2 fXsg , E1 fXt2Bg1 A = EP (s Xs t B )1 A : (7.30) 2 A 2 fXs g A = fXs 2 Dg D 2 Bs. * (7.27), E1 fXt2Bg1 A = P (Xs 2 D Xt 2 B ) =
Z
Z
Xs0
Qs0 (dx) P (s0 x s dz)P (s z t B ): (7.31) D
* (7.30), I ,
Z
EP (s Xs t B )1 fXs2Dg = =
Z Xs
Xs
P (s z t B )1 D (z)
=
Z Xs0
Z
Z
Xs0
P (s z t B )1 D (z)Qs(dz) = Qs0 (dx)P (s0 x s dz) =
Qs0 (dx) P (s0 x s dz)P (s z t B ):
(7.32)
D
) (7.31) (7.32) (7.15).
$+ (Xt t 2 Ts0 ). (7.12) , n > 2 t1 < : : : < tn ( Ts0 ), Bn 2 Btn P (Xtn 2 Bn jXt1 : : : Xtn;1 ) = P (Xtn 2 BnjXtn;1 ): B ! , A 2 fXt1 : : : Xtn;1 g E1fXtn 2Bng1 A = EP (Xtn 2 Bn jXtn;1 )1 A: (7.33) * 7.2, (7.6), 1 A = 1 B1 1 Bn , Bk 2 Btk , k = 1 : : : n ; 1. ? (7.33) P (Xt1 2 B1 : : : Xtn;1 2 Bn;1 Xtn 2 Bn ), (7.27). (7.15) (7.33) , EP (tn;1 Xtn;1 tn Bn)1 A. & , gn : Xtn ! R Btn jB(R)- , , ( , (7.27), ( E1 fXt1 2B1g 1 fXtn;1 2Bn;1gg(Xtn ) 1 Bn g(zn ). < ! , n ; 1 n, gn;1 (zn;1) = P (tn;1 zn;1 tn Bn )1 Bn;1 (zn;1 ): 2 137
$ 87.2. $ Ts = T \ Ns0 1), " T R, Qs (Xs Bs ), s 2 T , - (7.25) ( Xs ).
$ $ . * , -, (Xt Bt)t2T {
, T R P (s x t B ) { . s 2 T
Qs() = x(), x 2 Xs. * B7.1
(
X s x = fXts x t 2 Ts := Ns 1) \ T g, ( 0
, Xss x = x ..
B7.1, , ( 1.7), , X s x
(. (1.8))
(XTs BTs ), Qs x = L(X s x ). X s x D = XT
Yts x(!) := Xts x(T Ts!) t 2 Ts ! 2 XT
(7.34)
T Ts
T
Ts (. . 8). - F>s := T;1TsBTs Ps x = Qs xT;1Ts. D. 7.3. < 6 , Yts x, t 2 Ts { (D F>s Ps x). @ 87.4. 3 s 2 T " x 2 Xs $ Y s x = fYts x(!), t 2 Ts, ! 2 Dg (D F>s Ps x) # $ P (s x t B ). = , Yss x = x . . Ps x . 2 L
Yss x = x , Qs = x. * , u 6 t ( T ), B 2 Bt
Ps x(Yts x 2 B jYus x) = P (u Yus x t B ) Ps x ; ..
(7.35)
B , D 2 Bu
Ps x(Yts x 2 B Yus x
2 D) =
Z
fYus x 2Dg
P (u Yus x t B )dPs x:
* (7.34), (2.10), X s x
(XTs BTs Qs x),
Z
fYus x 2Dg
P (u Yus x t B )dPs x
=
Z
fXus x 2Dg
P (u Xus x t B )dQs x =
= Qs x(Xts x 2 B Xus x 2 D) = Ps x(Yts x 2 B Yus x 2 D):
3 Y s x (7.12) . * !
. 2 ? , & % (7.34) Y s x % <+ %$ (2.10)
. A ,
Yts x(!) = !(t) t 2 Ts ! 2 D: ) , Y (t !) = !(t), t 2 T , ! 2 D = XT , $ $ s T t > s 138
(t 2 T ). 1$$ , Ts (D F>s Ps x ) % $ , $ s $ $ x Ps x -.. (x 2 Xs). & , F>s . ??, - F>s
T t
XT t 2 Ts. B , F>s = fYt t > s t 2 Tsg. ? Y s x (Yt Ps x). * F>s BT s 2 T , Ps x F = BT
BT , F>s. ) , $ &$+ $ & (D F ), Ps x s 2 T , x 2 Xs, " $+" $ s $ x ( Ps x- 1), ,$ $ fYts x t 2 Tsg . * (7.35) u = s, $+ $ : Ps x (Yts x 2 B ) = P (s x t B ) Ps x ; .., s t 2 T (s 6 t) x 2 Xs B 2 Bt: 3 , P ( 2 B j ) = P ( 2 B ) .., = c .., c = const. & , T {
R, ,
B7.1 ? B1.2,
Xt t 2 R. <, ! , ,
Bs s 2 T . ) , $ & .-.. $ $+ ,
87.5. * X = R, B = B(R), T = N0 1). Q = 0 ( B, 0). B x 2 R, t > 0, B 2 B(R)
P (x t B ) = 1 B (x) Pe(x t B ) = 1 B (x + sign x) sign x = ;1 x < 0, sign 0 = 0, sign x = 1 x > 0. 2 2 , P Pe
1') { 4'), . ??, , , . * (7.24), , P Pe .-.. ,
. 2 * (
-
; &, $% $+ $, $ 7. D. 7.6. B , (7.1) , P (AjF6t) = P (AjXt) .. A 2 F>t t 2 T . ' . ?? . . , (D F P ), ( Ej j < 1, , ... E( jA), A { - - (A F ). * , ... %% ,
D. 7.7. *
H = L2(D A P ), A F , A { - . B 2 L2(D F P ) PrH H . B , PrH = E( jA) .. 139
D. 7.8. ( . 7.7). PrH { F P ). * L2(D F P ) L1(D F P ), PrH
L1(D F P ). B , E(jA)
L2(D
(..) ! . D. 7.9. * (Xt t 2 T ), T R { . < 6 , (Xt t 2 U ), U T , . , (Xt t > 0) { , ] > 0 (Xk k = 0 1 : : : ) .
? D. 7.10. * (Xt t > 0) { ( t)
(X B). * (Y E ) {
ht : X ! Y , ht 2 BjE , t > 0. B , ht t > 0 { - , (ht(Xt ) t > 0) . D. 7.11. * , (, ( ht,
. D. 7.12. * (Xt t > 0) { X R. * Yt = NXt], N] { . 3
, Yt { ? D. 7.13. ( . 7.10, 7.11). * W (t) = (W1(t) : : : Wm(t)) t > 0 1=2 P m 2 m- . * Xm (t) = , !
k=1 Wk (t) $ @. 0 (Xm (t) t > 0) ? , m = 1 X1(t) = jW1(t)j. D. 7.14. * fXt t > 0g fYt t > 0g {
. 0 fXt + Yt t > 0g fXtYt t > 0g ? S , Yt = c(t), c(t) { ? D. 7.15. * fXk k = 0 1 : : : g {
. * Xt = (t ; k)Xk + (k + 1 ; t)Xk+1 t 2 Nk k + 1), k = 0 1 : : : , .. N0 1) (k Xk ). 0 fXt t > 0g ? 0 Yt = X t] t > 0, N] { ? D. 7.16. * 1 2 : : : { ... , ( 1 ;1
1=2. * S0 = 0, Sn = 1 + : : : + n , n 2 N, Xn = max06k6n Sk . B , fXn n > 0g . D. 7.17. * 1 2 : : : { , 1 2 : : : N0 1], {
.. F(x). * Sn = 1, n 6 , Sn = ;1, n > (n = 1 2 : : : ). 0 fSn n > 1g ? D. 7.18. B ,
fXt t > 0g , 7.5 G(x) x, x 2 R. D. 7.19. B , $$ + fXt t 2 T g, T R % $ $ $ + $ , ;% t1 < t2 < t3 (t1 t2 t3 2 T ) $
r(t1 t3)r(t2 t2) = r(t1 t2)r(t2 t3) 140
(7.36)
r(s t) = cov(Xs Xt), s t 2 T . D. 7.20. B , < { 8 Xt = e;tW (e2t), t 2 R, W () { , . D. 7.21. * P (X0 = 1) = P (X0 = ;1) = 1=2 X0 fNt t > 0g > 0. * fXt t > 0g (! # ). 0 fXt t > 0g ? ' . D. 7.22. B x > 0 x = inf ft > 0 : W (t) = xg, W () { . B , f x x > 0g { (, , 7.7. D. 7.23. * fXt t > 0g { . B s > 0 Y = fXs;t t 2 N0 s]g. 0 Y ? < Y , X ? D. 7.24. B , { ! . D. 7.25. * X { N0 1), Y = fYt = (Xt t) t > 0g. B , X Y . B , Y { , . , X Y ? D. 7.26. * fXt t > 0g {
X . B , (
hs : X N0 1] ! X , s > 0 ( .. hs 2 B(X ) B(N0 1])jB(X ), s > 0) t s fXu 0 6 u 6 tg t s > 0 , Xt+s = hs (Xt t s) .. t s > 0. D. 7.27. * Rm P (x t B ) = P (x + y t B + y) x y 2 Rm, t > 0, B 2 B(Rm). B ,
(. A% $ %% $ . N?] , , .. ( , ). N?] " " (. . ??), ..
, (
, . 3 X = fXt t 2 T Rg, + $ $ - % Ft t 2 T (Fs Ft F s 6 t, s t 2 T ). J , (Xt Ft)t2T { ( t
(X B)), $ X (Ft)t2T ( .. Xt 2 Ft jB t 2 T ) s t 2 T , s 6 t B 2 B
P (Xt 2 B jFs) = P (Xt 2 B jXs) ..
(7.37)
2 ,
, .. Ft = F6t = fXs s 2 T \(;1 t 2 T
.
E & % $ $ $ . < 6 , (. N?, . 2]). 141
*
(D F P ) - A1 A2 E F 9 , E ( A1 A2, P (A1A2jE ) = P (A1jE )P (A2jE ) Ak 2 Ak k = 1 2: (7.38) ? , (7.1) , fXtg ( F6t F>t
t 2 T . D. 7.28. B , (7.38) , E(F1F2jE ) = E(F1jE )E(F2jE ) Fk 2 Mk , Mk {
L2(D Ak P ), k = 1 2. D. 7.29. C E ( A1 A2, E ( A1 _ E A2 _ E , A _ E - , ( A E . D. 7.30. (. . 7.6). 8 (7.38) , P (AjA1 _ E ) = P (AjE ) A 2 A2: D. 7.31. * - E A1 E ( A1 A2 ( 2 I ). ? \2I E ( A1 A2. D. 7.32. C (7.38), A1 \ A2 E . , (7.1) ( ( , fXtg F6t \ F>t. & , - A1 A2 , ( - E = f Dg. C (Xt t 2 T ) { ,
- A(U ) = fXt t 2 U g, U T . , T R, , ( ,
( " " " ( ". *! , - A(U1) A(U2) U1 U2 T - ( . / (7.1) ( , , , ( (., , N?]),
( C01(T ) (
), T Rn,
( ) , ( supp U ( U T ), " ", (7.1). ? , R
-
T , S 2 R (
: S1 = S , ;, S2 = T n NS ], NS ] = S @S , ; {
, ( @S ( S ). '
- A(U ), U T ( , A(U1 U2) = A(U1) _ A(U2)) - " R, A(;" ) ( A(S1) A(S2) " > 0 ( ;" "-
;). B , Zd, ( (. 1). ) ,
, &$ $+ (. . 7.25), $ $
$ $ , <; , , ; &;; $ $, $ $+ $+ . ?, N?, . 91] ,
, ( " " , , (
. 142
8. - ' .
" @ . % 3 . D " " . % 3 . . E .
*
X $,
.. , . ? xi 2 X
i, ! , i
f0 1 : : : rg, N f0g. * B
X . *
(X B) ,
, (x y) = 1 x 6= y (x x) = 0, x y 2 X . & , g : X ! R BjB(R)-. 3
$" : . L - A, D
Aj j 2 J , J {
(j2J Aj = D, Ai \ Aj = i 6= j ). B , ( Ej j < 1 8 E 1 A < j E( jA) = : P (Aj ) P (Aj ) 6= 0 (8.1) 0 P (Aj ) = 0 Aj ( E( jA) .. ). B X = (Xt t 2 T R) t1 : : : tn 2 T - fXt1 : : : Xtn g D fXt1 = j1 : : : Xtn = jn g, j1 : : : jn 2 X . *! (7.12) (8.1) X = (Xt t 2 T ) $ +; ( $ $ + $ , ;% m 2 N, s1 < : : : < sm < s 6 t ( T ) ;% i j i1 : : : im 2 X
$
P (Xt = j jXs1 = i1 : : : Xsm = im Xs = i) = P (Xt = j jXs = i)
(8.2)
P (Xs = i1 : : : Xsm = im Xs = i) 6= 0. & P (AjB ) = P (AB )=P (B ) 6 0 ( P (B ) = P (Aj = x) = '(x), P (Aj ) = '( )). * Xs = fi 2 X : P (Xs = i) = 6 0g, s 2 X . B s 6 t (s t 2 T ), i 2 Xs, j 2 X 1
pij (s t) = P (Xt = j jXs = i)
(8.3)
) (8.3) , pij (s t)
: 1) pP ij (s t) > 0 i 2 Xs , j 2 Xt, s 6 t, s t 2 T , 2) pij (s t) = 1 i 2 Xs s 6 t, s t 2 T , j 2Xt 3) pij (s s) = ij i j 2 Xs, j 2 Xt s 2 T , ij | ,, 4) i 2 Xs, j 2 Xt s 6 u 6 t (s u t 2 T )
pij (s t) =
X
k2Xu
pik (s u)pkj (u t):
(8.4) 143
/ (8.4) : . 8 1){3) , 4). 8 (8.2) (8.3), j Xs = i) = X P (Xt = j Xu = k Xs = i) = P (Xt = j j Xs = i) = P (XPt = (Xs = i) P (Xs = i) k =
X k : P (Xu =k Xs=i)6=0
=
X
k2Xu
k Xs = i) = P (Xt = j j Xu = k Xs = i) P (XPu = (X = i) s
P (Xt = j j Xu = k)P (Xu = k j Xs = i) =
X
k2Xu
pik (s u)pkj (u t): (8.5)
/ (8.4) { ! , { S. 3 (7.16), . ??, , "PXs -.." " i 2 Xs". 8 3 : i j s t , u k 2 Xu , i k, k j .
r r r k
6 i
s
u
q>7 j -
t
. 8.1
Xs X , s 2 T , X { . pi (0), i 2 X0, pi (0) > 0, pi(0) = 1. # $ pij (s t) s 6 t (s t 2 T ), i 2 Xs, j 2 Xt, i2X0 - 1){4). (D F P ) ( $ X = fXt t 2 T g Xt t 2 T ( .. Xt (! ) 2 Xt t ! 2 D), , pi(0) = P (X0 = i) pij (s t) = P (Xt = j j Xs = i) s 6 t (s t 2 T ), i 2 Xs, j 2 Xt.
P
@ 8.1. 0 2 T
N0 1),
2 C X (
, 0 6 t1 < : : : < tn, n 2 N, Bk Xtk , k = 1 : : : n X X X P (Xt1 2 B1 : : : Xtn 2 Bn) = pi (0) pij1 (0 t1) : : : pjn;1 jn (tn;1 tn): (8.6) i
j1 2B1
jn 2Bn
*! Pt1 ::: tn (B1 : : : Bn ) ( (8.6) (. 2.8) 3 . ??. ? (Xt t 2 T ), .-.. . 3
, B7.1 ( . 2 144
< ,
P (s i t B ) =
X j 2B
pij (s t)
(8.7)
s 6 t (s t 2 T0), i 2 Xs, B X . ??. *, $ -
$+; $ ( ) $$ $ &$ $ &$. ? , $ 8.1 % $ $ . ? 8.2. A$ + $ pij (s t), s 6 t (s t 2 T ), i 2 Xs, j 2 Xt, & $+ &, s 6 t (s t 2 T ) i j 2 X . B pij (s t) = 0 i 2 Xs, j 2= Xt pij (s t) = pi j (s t) i 2= Xs, j 2 X , i0 = i0(s) 2 Xs (Xs = 6 ). * ! ,
1) { 4)
($+ & $ $+ Xt = X t 2 T ). . @ 8.3. $ N = fNt t > 0g , N | $ : X = f0 1 2 : : : g, pi (0) = i0 0 6 s < t, i j 2 X 0
8 (m((s t]))j;i < ;m((s t]) j > i e pij (s t) = : (j ; i)! 0 j < i
m { - B(N0 1)). '
pij (s s) = ij
(8.8)
s > 0 i j 2 X .
2 * N | , .. N0 = 0 .., ( Nt ; Ns m((s t]) t > s > 0, . . ( * m((s t]). ? (
2.14. 7.7 N | , ! .. Nt = Nt ; N0 m((0 t]), t > 0P X = f0 1 : : : g. < , pi (0) = P (N0 = i) = i0, i 2 X . B 0 6 s < t j ; i Ns = i) = P (N ; N = j ; i) pij (s t) = P (Nt = j j Ns = i) = P (Nt ; NPs(= t s Ns = i) (8.8). * s > 0 i j 2 X pij (s s) = = P (Ns = j j Ns = i) = ij . A% $. * 8.1, fNt t > 0g
(8.8) , . < , N0 = 0 .., . . pi (0) = i0, i 2 X . B s < t k > 0 , (8.6) (8.8),
P (Nt ; Ns = k) =
1 X l=0
P (Nt ; Ns = k Ns = l) =
1 X l=0
P (Nt = k + l Ns = l) = 145
= =
1 X X l=0 i
pi(0)pil (0 s)pl k+l (s t) =
1 X (m((0 s]))l
l!
l=0
1 X l=0
p0l (0 s)pl k+l (s t) =
e;m((0 s]) (m((ks! t])) e;m((s t]) = (m((ks! t])) e;m((s t]): k
k
(8.9)
' , t > 0 Nt X = f0 1 : : : g. 1 P ) (8.9) , P (Nt ; Ns = k) = 1, ! P (Nt ; Ns = k) = 0 k < 0. k=0 ) , Nt ; Ns m((s t]), 0 6 s 6 t. B n 2 N 0 = t0 6 t1 < : : : < tn, 0 6 k1 : : : kn P (Nt1 = k1 Nt2 ; Nt1 = k2 : : : Ntn ; Ntn;1 = kn ) = = P (Nt1 = k1 Nt2 = k1 + k2 : : : Ntn = k1 + : : : + kn) = X = pi(0)pik1 (0 t1)pk1 k1+k2 (t1 t2) : : : pk1 +:::+kn;1 k1+:::+kn (tn;1 tn) = i
k1 k2 kn = (m((0k t!1])) e;m((0 t1]) (m((tk1 !t2])) e;m((t1 t2]) : : : (m((tn;k1! tn])) e;m((tn;1 tn]) =
=
Yn
m=1
1
n
2
P (Ntm ; Ntm;1 = km):
(8.10)
* (8.10) (8.6) (8.9). ' ( . 2 * m() { - B(N0 1)) m(N0 1)) = 1. < M (t) = m(N0 t)) t > 0 ( M ;1 (t) = inf fu > 0 : M (u) > tg t 2 N0 1): @ 8.4. fN (t) t > 0g { $ ( m(). f (t) = N (M ;1(t)) t > 0g, = 1. 1 , $ f (t) t > 0g = 1 - m() B (N0 1)) $ N (t) = (M (t)), t > 0, "( (" m(). 2 / , m((s t]) = M (t);M (s) 0 6 s < t M (M ;1 (t)) t t > 0. 2 ? , $ $ &$ %$+
$ $ +$ , + . @ 8.5 ( $ ).
1 : : : n : : :
| $ , . . "(
(
;x pi (x) = e x > 0 0 x < 0: 0 (! ) = 0, t > 0
t(!) = max k : 146
X i6k
(8.11)
i(!) 6 t
(8.12)
P = 0, . . (!) = 0, (!) > t. f t > 0g | t 1 t ?
$ " .
? 8.5 , , . . 8.2.
6 3 2 1 0
o
t(! )
-
o
o
|
{z
1 (! )
-}| 1 {z
-1
2 (! )
}|
{z
}
3 (! )
1
t
. 8.2
2 * 0(!) = 0. * t > 0. ? P (t = 0) = P ( 1 > t) =
Z1 t
e;xdx = e;x:
< Sk = 1 + : : : + k , k > 1. * k 2 N
8 (x)k;1 < ;x pSk (x) = : (k ; 1)! e x > 0 0 x < 0:
(8.13)
L
(8.13) ! . * (8.13), k > 1
P (t = k) = P (Sk 6 t Sk+1 > t) = P (Sk 6 t Sk + k+1 > t) = ZZ Zt (u)k;1 Z1 = pSk (u)pk+1 (v) du dv = (k ; 1)! e;u du e;v dv = u6t u+v>t
= e;t
Zt (u)k;1 0
t;u
0
k ;t
(t) e : du = (k ; 1)! k!
&, $ ft t > 0g $ t ; s t;s 0 6 s < t. 1
(
P (t1 = k1 t2 ; t1 = k2 : : : tn ; tn;1 = kn) =
Yn j =1
qkj ((tj ; tj;1))
)
(8.14) 147
6v t t
-u
. 8.3
n > 2, 0 = t0 6 t1 < : : : < tn k1 k2 : : : kn > 0, 8 k e; > > 0 k = 0 1 : : : < qk ( ) = > k0! < 0 k = 0 1 : : : : k = 0 k = 0 1 : : : : B
, (8.14),
P (t2 ; t1 = k2) =
1 X
k1 =0
P (t1 = k1 t2 ; t1 = k2) = = qk2 ((t2 ; t1))
1 X k1 =0
qk1 (t1) = qk2 ((t2 ; t1)):
) , (8.14). < A , ( (8.14). ? A = ft1 = k1 t2 = k1 + k2 : : : tn = k1 + : : : + kn g. C k1 = : : : kn = 0, P (A) = P ( 1 > tn) = e;tn = e;t1 e;(t2;t1 ) : : :e;(tn;tn;1) (8.14) . 8 + < $ +$ $ n. * 2 6 m 6 n k1 = 0, : : : , km;1 = 0, km > 1, kj > 0, m < j 6 n. ? A = ftm;1 < 1 6 tm Skm 6 tm Skm +1 > tm : : : Skm +:::+kn 6 tn Skm+:::+kn +1 > tng P (A) = E(E(1 A j 1)). * (7.13), E(1 A j 1 = x) = = E1 ftm;1 < x 6 tm x + 2 + : : : + km 6 tm x + 2 + : : : + km +1 > tm : : : x + 2 + : : : + km +:::+kn 6 tn x + 2 + : : : + km +:::+kn +1 > tng = = 1 ftm;1 < x 6 tmg P (Skm ;1 6 tm ; x Skm > tm ; x : : : Skm+:::+kn ;1 6 tn ; x Skm +:::+kn > tn ; x): (8.15) 3 , 2 3 : : : , 1 2 : : : ( , fS~k g, S~k = 2 + : : : + k+1 , , fSk g). B, , E(1 A j 1 = x) = 1 ftm;1<x6tm g 148
P (tm ;x = km ; 1 tm
+1
;x ; tm ;x = km+1 : : : tn ;x ; tm;1 ;x = kn ) =
= 1 ftm;1<x6tm gqkm;1((tm ; x))
Yn
j =m+1
qkj (N(tj ; x) ; (tj;1 ; x)]): (8.16)
(8.16) tm;x x > tm. < x (8.15) . *! , 1 ftm;1 <x6tm g tm;1 < x 6 tm. ? ,
P (A) = E1 ftm;1 <16tm gqkm ;1((tm ; 1)) =
Ztm tm;1
e;x ((t(mk ;;x))1)!
km ;1
;tm = (ke ; 1)! m
Ztm
tm;1
m
((tm
Yn
j =m+1 n
e;(tm;x)dx
; x))km;1dx
qkj ((tj ; tj;1)) =
Y
j =m+1
Yn
j =m+1
qkj ((tj ; tj;1)) =
qkj ((tj ; tj;1)) =
km = e;t1 e;(t2;t1 ) : : : e;(tm;tm;1 ) ((tm ; tm;1))
km !
Yn j =m+1
qkj ((tj ; tj;1)):
) , (8.14) k1 = : : : = km;1 = 0, km > 1, kj > 0, m < j 6 n (2 6 m 6 n). * k1 > 1. ? 0 6 x 6 t1 P (1 A j Sk1 = x) = = E1 f0 6 x 6 t1 x + k1 +1 > t1 x + k1 +1 + : : : + k1 +k2 6 t2 x + k1 +1 + : : : + k1 +k2 +1 > t2 : : : x + k1 +1 + : : : + k1 +:::+kn 6 tn x + k1 +1 + : : : + k1 +:::+kn +1 > tng = = E1 f0 6 x 6 t1 1 > t1 ; x Sk2 6 t2 ; x Sk2 +1 > t2 ; x : : : Sk2+:::+kn 6 tn ; x Sk2+:::+k1 +1 > tn ; xg = = 1 f0 6 x 6 t1gP (t1;x = 0 t2;x = k2 : : : tn;x = k2 + : : : + kn ) = = 1 f0 6 x 6 t1gP (t1;x = 0 t2;x ; t1;x = k2 : : : tn;x ; tn;1;x = kn ) =
Y = 1 f0 6 x 6 t1ge;(t1;x) qk n
j ((tj
j =2
; tj;1)):
(8.17)
* (8.17) , k1 = 0. ? , (8.13) E1 f06Sk1 6t1 ge;(t1;Sk1 ) =
Zt (x)k ;1 ;x e;(t ;x) dx = e (k1 ; 1)! 0 Zt (x)k ;1 1
1
1
1
= e;t1
0
dx = (t1) e;t1 : (k1 ; 1)! k1! 1
k1
) (8.17) (8.16) (8.14) k1 > 1. 2 B $ T = N0 1) ( T = f]k k = 0 1 : : : g, ] > 0), . . , 149
s t s + h t + h 2 T (0 6 s 6 t), x 2 X , B 2 B P (s x t B ) = P (s + h x t + h B ): (8.18) * ! t ; s, P (s x s + t B ), P (x t B ), t 2 T . B pij (t) = pij (s s + t), s t 2 T , P (t) | , ( pij (t). 2 , ! (8.4) P (s + t) = P (s)P (t) s t 2 T: (8.19) ) , + & $ " P (t), t 2 T , . . , i j 2 X t 2 T pij (t) > 0
X j
pij (t) = 1 pij (0) = ij
(8.20)
(
(8.20) , P (0) = I | ). & , &$ %$+ -
; $ < $ $ (., ., . 7.25). T = N0 1), T
! .
@ 8.6 (, $ ). j0 2 X
h > 0
pij0 (h) > 8i 2 X :
(8.21)
( ) "
lim p (t) = pj t!1 ij
t>0
-
i j ( (8.22)
jpij (t) ; pj j 6 (1 ; ) t=h]
(8.23)
N] | $ . ' ! , % +< -
$ , +;, % 6 % $7, $ $ $ . 2 < t 2 T mj (t) = infi pij (t) Mj (t) = sup pij (t): i
< , mj (t) 6 pij (t) 6 Mj (t) i j 2 X t 2 T . *, mj (t) % Mj (t) & t ! 1 Mj (t) ; mj (t) ! 0 t ! 1. ? (8.22)
. B s t 2 T , (8.19), (8.20),
mj (s + t) = infi
X
Mj (s + t) = sup i
150
pik (s)pkj (t) > mj (t) infi
k X k
X
i
X
k
pik (s)pkj (t) 6 Mj (t) sup
pik (s) = mj (t)
k
pik (s) = Mj (t):
B, t h t ; h 2 T , (8.19), Mj (t) ; mj (t) = sup pij (t) + sup(;prj (t)) = = sup(pij (t) ; prj (t)) = sup ir
= sup ir
X+ k
X
r
i
ir
k
(pik (h) ; prk (h))pkj (t ; h) =
(pik (h) ; prk (h))pkj (t ; h) +
6 sup Mj (t ; h)
X+
ir
k
X; k
(pik (h) ; prk (h))pkj (t ; h) 6
(pik (h) ; prk (h)) + mj (t ; h)
X; k
(pik (h) ; prk (h))
P P + k, pik (h) ; prk (h) > 0, ; | k, k k P P
pik (h) ; prk (h) < 0. * pik (h) = prk (h) = 1, X+ k
k
(pik (h) ; prk (h)) +
X; k
k
(pik (h) ; prk (h)) = 0:
*!
Mj (t) ; mj (t) 6 (Mj (t ; h) ; mj (t ; h)) sup ir
? , j0
X+ k
(pik (h) ; prk (h)) 6
X+
(pik (h) ; prk (h)):
P+, (8.22)
X+ k
pik (h) 6 1 ; pij0 (h) 6 1 ;
P j0 + , , (8.22), k X+ X+ / ,
k
(pik (h) ; prk (h)) 6
k
pik (h) ; prj0 (h) 6 1 ; :
(8.24) Mj (t) ; mj (t) 6 (1 ; )(Mj (t ; h) ; mj (t ; h)): B
Nt=h] , Mj (u) ; mj (u) 6 1, u = t ; Nt=h]h, (8.24) (8.23). 2 $ 8.7. " i j 2 X (8.22). j 2 X ( lim p (t) = pj (8.25) t!1 j pj (t) = P (Xt
= j ),
(8.21), jpj (t) ; pj j 6 (1 ; ) t=h]: P 2 * pj (t) = pi(0)pij (t). *! i 2 j 2 X
pj (t) ; pj =
X i
pi (0)(pij (t) ; pj ) ! 0 t ! 1: 151
C (8.21),
j
X X jpj (t) ; pj = pi(0)(pij (t) ; p) 6 (1 ; ) t=h] pi(0) = (1 ; ) t=h]: 2 j
i
i
$ 8.8. " i j 2 X t 2 T
X
pj =
i
i
pi (s)pij (t) > slim !1
P ) , pj > pi pij (t). B , i
pj >
-
X i
(8.26)
P (t) (
2 (8.25) j 2 X N 2 N
X
(8.22).
pi pij (t)
. . p $
pj = slim p (s + t) = slim !1 j !1
X i6N
pi(s)pij (t) =
X i6N
P (t)).
pi pij (t):
pi pij (t)
(8.27)
P
j t > 0. / (8.22) N i pj = j 6N P p (t) 6 1, ! = tlim ij !1 j 6N
X j
pj 6 1:
) , (8.27) ,
X j
pj >
XX j
i
pi pij (t) =
(8.28)
X X i
pi
j
pij (t) =
X i
pi :
* . 2
$ 8.9. " i j 2 X
(8.22).
P p = 1,
j j . . pj " , , pj = 0, . . pj = 0. j
P
2 C
P p 6= 0 ( (8.28)), p (0) = p= P p, j
i
j
i
j j pij (t),
i 2 X , 8.1. ?
8.8 t j 2 X P pp (t) i ij X pj i P P pj (t) = pi (0)pij (t) = (8.29) pj = pj = pj (0): i j
/ (8.25) pj = pj (0). / , 152
j
P p = 1. 2 j
j
? " " (
. * fXt t 2 T Rg $ ( $ $ ), n 2 N, t1 : : : tn 2 T h 2 R
, t1 + h : : : tn + h 2 T L(Xh+t1 : : : Xh+tn ) = L(Xt1 : : : Xtn ): ) , & $+ ( (
T P ! (
, , , T = Z). / @ 8.10. $ X = (Xt t > 0) $ fpj g. Y = (Yt t > 0) { ( ) $, "( , X , fpj g. Y { . 2 * Y (
8.1. ) (8.7), Y X , 0 6 t1 < : : : < tn, Bk X , k = 1 : : : n, n 2 N
P (Yt1 2 B1 : : : Ytn 2 Bn) =
X
j 1 2B 1
pj1 (t1)
X
j2 2B2
pj1 j2 (t1 t2) : : :
X
jn 2Bn
pjn;1 jn (tn;1 tn)
pj (t) = P (Yt = j ) = pj j 2 X , t > 0 (8.29). < , pij (s t) = pij (t ; s) = pij (s + h t + h) i j 2 X , 0 6 s 6 t, h > ;s. 2 *,
+ (8.22), (8.29), $..
$ + , $ $ ( .. P (Xt = j ) = P (X0 = j ) j 2 X t > 0). B ,
.
B . % % $ + , $ ; $ ; $ % $ $ ( ( 6, . 6.42). * (D F P ) k , k 2 N, , , .. ! 2 D0, P (D0) = 1, 0 < 1 < 2 < : : : n ! 1 n ! 1: (8.30) <
" "
(B !) =
1 X k=1
: B(R+) D ! N+ = f0 1 : : : g f1g,
k (!)(B ) B 2 B(R+) ! 2 D
(8.31)
R+ = N0 1), x { B. * ! 2 D0, , ( !) B(R+). 8 (8.30) , ( !) ! 2 D0 * , .. ,
R+, , , ( !) - B(R+) ! !. D. 8.1. < 6 , (B ) 2 FjA B 2 B(R+), A { -
N+, .. A
N+. 153
L $ , $ $ $ .
D
! = (t1 t2 : : : ), 0 < t1 < t2 < : : : tn ! 1 n ! 1. < fn n 2 Ng {
D, .. n(!) = tn ! = (t1 t2 : : : ). D - F = fn n 2 Ng (. 1.3)
(B !) =
1 X k=1
tk (B ) B 2 B(R+) ! 2 D:
(8.32)
! ( B ) L & ! 2 D. B
F , n n , (8.30). & , , ..
(ftg !) 6 1 t 2 R+ ! 2 D0:
$ Y = fY (t !) = ((0 t] !) t > 0 Y (0 !) = 0g: ?
Y (t !) =
1 X k=1
1 (0 t]( k (!)) t > 0:
(8.33)
< , fY (t) > ng = f n 6 tg t 2 R+ n 2 N+ = f0 1 : : : g, ! Y (t ) 2 FjA t 2 R+. D. 8.2. < m(B ) = E (B ) B 2 B(R+). 0 m() B(R+)? , ( N (t) t > 0, (m((s t]) = E(N (t);N (s)), 0 6 s < t < 1). D. 8.3. ( B6.10). B , (8.33) , Y (, Y R+ (. (2.35)). D. 8.4. * 7.7 ( , T = N) k = 1+: : :+ k , k 2 N 3 , . 0 (8.33), f k k 2 Ng { 3 ,
( (8.30)? D. 8.5. B , > 0 N (t)=t ! .. t ! 1. D. 8.6. N ] * ft t > 0g (8.12). ' t > 0 P +1 t t = j=1 jP; t, .. "" ( Sk = kj=1 j , k 2 N. L ! . * j { ! . / , , t? D. 8.7. ( 8.6). B , t > 0 t t +2 t+3 : : : ! . 154
? 8.4
$ ( $ = ). * R(t ! ) { (D F P ), ( ( ( ( t > 0, { ). *
= 1. , Z (t !) = (t !)(!). D. 8.8. 0 , ? < $ -
X (t !) =
1 X k=1
(t ; k (!)) t > 0
(8.34)
: R ! R , k (k 2 N)
(8.30) ( (8.34) t > 0
D0). D. 8.9. ' (8.34). B (8.34) t > 0 , 2 ? L $ ; $ . * k k 2 N,
( (8.30), (D F P )
k k 2 N. ',
k k , .. k (" ") k . <
(C P !) =
1 X k=1
( k(!) k(!))(C ) C 2 B(R+) B(R) ! 2 D:
/ (8.31) (B !) = (B RP !).
"(" # $ "
Y (t DP !) = ((0 t] DP !) t > 0 D 2 B(R): ?
0 Y (t DP !) = P1 1 6 t 2 Dg(!) k k=1 f k
! 2 fY (t) = 0g ! 2 fY (t) > 1g
Y (t) (8.33). B ,
0 Y (t D) = PY (t) 1 ( ) D k k=1
fY (t) = 0g fY (t) > 1g:
(8.35)
D. 8.10. ' Y (t D), t > 0, D 2 B(R). D. 8.11. * Y (t) t > 0 { ( m. * f k gk2N { ...
L( k ) = Q. D 2 B(R) , Q(D) > 0. B , " $" Y (t D), t > 0, (8.35), ( m()Q(D). 155
? % $ &; + & $ $, . /
fTt t > 0g,
(
B ( k k) ,
T0 = I Ts+t = TsTt s t > 0: *
,
(8.36)
sup kTtk 6 M < 1: t>0
(8.37)
B L : B ! B
kLk = supfkLf k=kf k f 2 B f 6= 0g
(8.38)
( ! B
, .. , 6 ). * (Tt)t>0 (, M = 1 (8.37), .. kTtf k 6 kf k f 2 B t > 0. D. 8.12. * P (x t B ) { (Xt t > 0), . (7.19). B , (Ttf )(x) =
Z
X
f (y)P (x t dy)
(8.39)
(
B (X P R),
. 6.15. & , (8.39) (Ttf )(x) = Exf (Xt)
(8.40)
Ex , L(X0) = x. B , (Xt t > 0) t = 0 x 2 X . * (Tt)t>0 , C0-, f 2 B
s ; tlim T f = f !0+ t
(8.41)
s ; lim B , .. kTtf ; f k ! 0 t ! 0+. D. 8.13. B , x 2 X lim P (x t B ) = x(B ) B 2 B
t!0+
(8.42)
x { B, (8.39) C0- B (X R). (8.39) , B (X R)
C0- (Tt)t>0,
Q(x t B ) := (Tt1 B )(x) x 2 X t > 0 B 2 B
(8.43)
( , , ..
1') { 4') . ??. 156
D. 8.14. * (Tt)t>0 (8.39) -
(7.18) m- . * , ! C0
B (Rm R). , P (x t V"(x)) = 1 ; o(t) t ! 0+ V" (x) { ( ) " x 2 Rm. D. 8.15. * (Tt)t>0 { ( B B0 f , Ttf , t = 0. * , Ttf N0 1) f 2 B0. * , B0
( ) TtB0 B0 t > 0. D. 8.16. (. N?, . ??]) * Q { Rn, .. . B , (Ttf )(x) =
Z
Rn
f (e;t x + (1 ; e;2t)1=2y)Q(dy) x 2 Rn t > 0
(8.44)
Lp(Rn B(Rn) Q) p > 1 ( C0- , ) - { I . J A ( ) (Tt )t>0 Tt f ; f Af = s ; tlim (8.45) !0+ t ! f , (
. 2 , DA A ( f1 f2 2 DA ,
f1 + f2 2 DA 2 R). ) ,
Af = ddt (Ttf )jt=0 f 2 DA : +
(8.46)
88.17. * A { ,
8.14. < Cb u(Rm) , Rm
. * @ 2f 2 C (Rm) k j = 1 : : : m: (8.47) @xk@xj b u B, f (, . 8.14 , ! f ) (8.48) Af = 21 ]f ] { 2. 2 < k k Rm. B t > 0 f
"
#
1 (T f (x) ; f (x)) = 1 1 Z f (y)e;ky;xk2=(2t)dy ; f (x) = t t t (2t)m=2 Rm
157
Z p 1 = t(2)m=2 (f (x + z t) ; f (x))e;kzk2 =2dz: Rm
* ?
m p m 2f (x + z pt) X X @f ( x ) 1 @ f (x + z t) = f (x) + zk t @x + 2 zk zj t @x @x (8.49) k k j k=1 k j =1 p R = (x z t) 2 R, jj 6 1. 8 , g(u)du = 0, g {
p
R
, jt;1(Ttf (x) ; f (x)) ; (1=2)]f (x)j 6 p @ 2f (x) Z m 2 X 1 @ f ( x + z 6 m (2)m=2 @x @x t) ; @x @x e;kzk2=2jzkjjzj jdz =: I (x t): (8.50) k j k j R k j =1 ) (8.47) , I (x t) (8.50) (
x 2 Rm, t > 0. L I (x t) Ir(x t), ( Br = fkzk 6 rg Jr (x t) {
fkzk > rg.
f " r(") >p 0 , p Jr (x t) < "=2
x 2 Rm t > 0. 8 (8.47) , kz tk 6 r t z 2 Br , Ir(x t) < "=2 x 2 Rm 0 < t < t0("). 2
F , $ $ % $ % $ $. C g t 2 R -
B , t ( (
) s ; hlim (g(t + h) ; g(t))=h: !0 B g : Na b] ! B , Na b] ( { , Na b], (
R b g( ), a t)dt, , L
, B . ?
. )
Z b Z g(t)dt 6 b kg(t)kdt a a
(8.51)
( , , ). & , g Na + h b + h],
Zb a
g(t + h)dt =
Z b+h a+h
g(t)dt:
(8.52)
C dg=dt { Na b] ( { ), Z b dg dt = g(b) ; g(a): (8.53) a dt 1 kg(u) ; g(v)k 6 supt2 v u] kdg=dtk(u ; v) a 6 v < u 6 b ( , ). C g { u 2 R,
(
Z u+h 1 s ; hlim g(t)dt = g(u): (8.54) !0+ h u 158
D. 8.18. B , g : Na b] ! B Na b],
L { B ,
Z b
L
a
Zb
g(t)dt =
a
Lg(t)dt:
(8.55)
D. 8.19. * (Tt)t>0 { ( C0- A. B , f 2 DA, Ttf t > 0 ( t = 0 ),
Tt f 2 DA
(8.56)
dTtf = AT f = T Af t t dt
(8.57)
Ttf ; f =
(8.58)
Zt 0
TsAfds:
D. 8.20. B , A C0- (Tt)t>0 .. # (
f(f Af ) f 2 DA g) B B . B , fn 2 DA fn ! f , Afn ! g n ! 1 ( B ), f 2 DA Af = g.
D. 8.21. B , B0 = NDA ], B0 8.14, A { ( (Tt)t>0, N]
B . , DA B ( C0- (Tt)t>0. D. 8.22. B , m > 1 2 M f : Rm ! R,
( (8.47). *! . 8.20 B8.17 , DA ! , M. D. 8.23. ' DA ,
. 8.14 m = 1. ) (., ., N?]), L, DL = B , L . & , B8.17 A
,
( (8.47),
. & ( . D. 8.24. 8 ;% $ L % $ $ B $ $ C0- , ; L $. J , , Tt = etL t > 0 (8.59) P (tL)n =n! (! etL := 1 n=0 P ktLkn=n! < 1) $ ; , t > 0, 1 n=0
kTtk 6 ejtj t > 0 (8.60) = kLk. & , . 8.19 8.24 ( . 159
D. 8.25. C A { ( ) (Tt)t>0,
f 2 DA Ttf
z(t) dz(t) = Az(t) dt
( kz(t)k 6 cet c > 0 t > 0, kz(t) ; f k ! 0 t ! 0+. I A { $ $ DA B (, , DA 6= B ), $ , $ $ $ C0- ( , C0- ) $ + &. ?+ & ;
+ $ $ +$.
* (Tt)t>0 { (
. /
Rg :=
Z1 0
e;tTtgdt g 2 B > 0
(8.61)
. ) (8.61) ( ) 0 u u ! 1. ) , Rg { ! 2 ( ) Ttg. ) (8.61) ,
kRk 6 1= > 0: D. 8.26. B , g 2 B s ; lim R g = g: !1 @ 88.27. (Tt)t>0 {
>0
"(
(8.62) (8.63)
A. -
R : B ! DA
(8.64)
(I ; A) : DA ! B
(8.65)
I { ,
R = (I ; A);1:
(8.66)
2 (8.64). * (8.55), (8.52) , , g 2 B > 0 1 (T ; I )R g = 1 (T ; I ) Z 1 e;sT gds = 1 Z 1 e;s(T g ; T g)ds = s t+s s t t t t t 0 0 Z1 Z1 1 t ; u ; s e Tugdu ; e Tsgds = =t e t 0 Z Z1 t 1 1 t ; u t = ;te e Tugdu + t (e ; 1) e;sTsgds = 0 0 Z t = ; 1t et e;uTu gdu + 1t (et ; 1)Rg: 0
160
* t ! 0+, (8.54), ARg = ;g + Rg: (8.67) ? , (8.64) $ . / (8.67) , g 2 B > 0 f ; Af = g (8.68)
$ < f = Rg. 8 $, $ (8.68) $ < f1 f2. ? v = f1 ; f2 2 DA v ; Av = 0. / (8.57)
dTtv = T Av = T v: t t dt *! ,
, d (e;tT v) = 0: (8.69) t dt ' ( 6 ), (8.69) , e;tTtv = x 2 B t > 0: t ! 0+, , x = v. 8 , (Tt)t>0 { ( ( (8.37) (8.60) > 0,
B8.27 > ), , 0 6 kvk = e;tkTtvk 6 e;tkvk t ! 1: / , v = 0 , , (8.68)
f = Rg > 0 g 2 B . ? , I ; A $+ $% & DA B (8.66). 2 $ 88.28. (Tt)t>0 (St)t>0 { C0- "( B . ! " , Tt = St t > 0. 2 ) B8.27 , (Tt)t>0 (St)t>0 (8.66). F 2 B . ? , (8.55), g 2 B > 0 Z1 e;tF (Ttg ; Stg)dt = 0: 0
*! Ttg = Stg t > 0, g 2 B . 3 , %
$+ - $% &, , N0 1) $$ + (., ., N?]). 2 1
, $ L &$ %$+ $ $ + (8.59). @ 88.29 (J { 2 ). B { A { " DA B . ) A "( C0- B
"( .
,
161
NDA] = B (DA B ). 2. 3 " g 2 B " > 0 f ; Af = g - f 2 DA . 3. 3 - f kf k 6 kg k=. 1.
162
9. # ) /. 0 1
/ . E " Q P (t), t > 0. ! " $ . QT . # % . 3 " , 1 . $ " .
/ $ + , $ $;$ $$ , N0 1), $ % & ,$ $ +$ $. < 3 fXt t > 0g ( P (t) = (pij (t))) , P (t) ! I t ! 0+, . . i j 2 X lim p (t) = ij : t!0+ ij
(9.1)
9.1. 3 $ fXt t > 0g i j 2 X " t h > 0 jpij (t + h) ; pij (t)j 6 1 ; pii (h): 9 , $ , i j 2 X # $ pij (t) N0 1). 2 ) pij (t + h) ; pij (t) =
X k
pik (h)pkj (t) ; pij (t) = pij (t)(pii(h) ; 1) + +
X k6=i
pik (h)pkj (t) 6
X k6=i
, , pij (t + h) ; pij (t) > pij (t)(pii (h) ; 1) > pii (h) ; 1. 2 @ 9.2. ! P (t), t > 0, | (
d+ P (t) = Q dt t=0
pik (h) = 1 ; pii(h)
,
(9.2)
. . ( " pij (t):
+ qij = d pdtij (t) : t=0
(9.3)
0 6 qij < 1
i 6= j
qi = ;qii 2 N0 1]:
(9.4) 163
3 Q
# .
2 0 + i 2 X . (9.1) > 0 , pii(h) > 0 h 2 N0 ]. B t > 0, (8.4), , pii (t) > (pii(t=n))n , n 2 N. n = n(t ) , t=n < , , pii (t) > 0 t > 0. *! N0 1) H (t) = ; log pii (t) ( -
). & ,
H (s + t) 6 H (s) + H (t) s t 2 N0 1)
(9.5)
pii(s + t) > pii (s)pii(t) (8.19) s t > 0. *
q = sup H (t)=t: t>0
(9.6)
< , q 2 N0 1]. L $ $ + . 1. $+ q < 1. ? " > 0 t0 = t0(") > 0, , H (t0)=t0 > q ; ". B h 2 (0 t0) t0 = nh + ], n = Nt0=h], 0 6 ] < h, N] { . * (9.5), H (]) : + (9.7) q ; " 6 H (t0)=t0 6 (nH (h) + H (]))=t0 = Hh(h) nh t0 t0 (9.1) H (]) ! 0 ] ! 0 (] ! 0, h ! 0+). *!
q ; " 6 lim inf H (h)=h: h!0+
(9.8)
) (9.6) , lim suph!0+ H (h)=h 6 q. / , (
limh!0+ H (h)=h = q. ? , H (h) = lim ; log(1 ; (1 ; pii (h))) = lim 1 ; pii (h) : q = hlim (9.9) !0+ h h!0+ h!0+ h h 2. $+ q = 1. ? M > 0 t0 = t0(M ) > 0 , H (t0)=t0 > M . B
,
(9.8), , lim inf H (h)=h > M: h!0+ ? , (
limh!0+ H (h)=h = 1. < (9.9). L $ $+ pij (t) i 6= j . < fij(k)(h) i i k h ( ), , ! j
mh, m = 1 : : : k. B n > 2
pij (nh) >
n;1 X k=1
fij(k)(h)pij ((n ; k)h):
(9.10)
B
, 8.2,
pij (nh) = P (Xnh = j jX0 = i) =
164
X
j1 ::: jn;1 2X
P (Xnh = j X(n;1)h = jn;1 : : : Xh = j1jX0 = i) >
>
n;1 X X
X
k=1 Jk jk+1 ::: jn;1 2X
P (Xnh = j X(n;1)h = jn;1 : : : X(k+1)h = jk+1jXkh = i)
P (Xkh = i X(k;1)h = jk;1 : : : Xh = j1jX0 = i) Jk = f(j1 : : : jk;1) : jm 2= fi j g m = 1 : : : k ; 1g. 3 ,
(8.2) 1 P (X = j X (n;1)h = jn;1 : : : Xkh = i : : : Xh = j1 X0 = i) = P (X0 = i) nh 1 P (X = j X = (n;1)h = jn;1 : : : X(k+1)h = jk+1 jXkh = i : : : X0 = i) P (X0 = i) nh P (Xkh = i : : : X0 = i) = = P (Xnh = j X(n;1)h = jn;1 : : : X(k+1)h = jk+1 jXkh = i)P (Xkh = i : : : Xh = j1jX0 = i): < ,
X
jk+1 ::: jn;1
P (Xnh = j X(n;1)h = jn;1 : : : X(k+1)h = jk+1 jXkh = i) = pij ((n ; k)h)
X
P (Xkh = i : : : Xh = j1jX0 = i) = fij(k)(h):
Jk (k) pij (h)
<
i j k P h ( ). ? , (9.10) k > 1 ( = 0)
pii(kh) = fij(k)(h) + < ,
k;1 X
m=1
p(ijm)(h)pji ((k ; m)h):
(9.11)
Pk;1 p(m)(h) 6 1, ! m=1 ij
fij(k) (h) > pii (kh) ; 16max p ((k ; m)h) m6k;1 ji (
). (9.1) " > 0 t0 = t0(" i j ) > 0 , pji (t) 6 " pii (t) > 1 ; " pjj (t) > 1 ; " t 2 N0 t0]: (9.12) ? fij(k)(h) > 1 ; 2" nh 6 t0 k = 1 : : : n ; 1: (9.13) &, pij ((n ; k)h) > pij (h)pjj ((n ; k ; 1)h) k = 1 : : : n ; 1 n > 2, (9.10) { (9.13)
pij (nh) > (1 ; 2")pij (h)
n;1 X k=1
pjj ((n ; k ; 1)h) > (1 ; 2")(1 ; ")(n ; 2)pij (h):
n = Nt0=h], pij (t) ( 9.1), 1 > pijt(t0) > (1 ; 2")(1 ; ") lim sup pijh(h) : 0 h!0+ 165
? , t0 , , lim inf pijt(t) > (1 ; 2")(1 ; ") lim sup pijh(h) : t!0+ h!0+ * " > 0 , , (
limt!0+ pij (t)=t. 2 P B N 2 N t > 0, , pij (t) = 1, j
X pij (t)
1 ; pii (t) : 6 t t
j 6N j 6=i
(9.3)
P q 6 q , , ij i
j 6=i j 6N
X j 6=i
qij 6 qiP
(9.14)
qi = 1
(9.14) . y , ! Q ! i
X j 6=i
qij = qi:
(9.15)
?, (8.8)
0 ; B B ; Q=B B ... @ 0
...
1 C C C A
0C
(9.15) .
@ 9.3 ( ).
$ : .
t>0
P 0(t) = QP (t) . . "
i j 2 X t > 0 ( " p0ij (t), p0ij (t) =
X k
qik pkj (t):
-
(9.16)
(9.17)
(9.2) pij (0) = ij , (9.17) t = 0, . 2 B t > 0, h > 0 i j 2 X X Lij (h t) = h1 pik (h)pkj (t): k6=i 166
?
pij (t + h) ; pij (t) = pii (h) ; 1 p (t) + L (h t): ij ij h h * N 2 N (9.3) X pik (h) X lim Lij (h t) > lim p qikpkj (t): kj (t) = h!0+ h!0+ k6=i h k6=i k6N
(9.18)
k 6N
/ , lim Lij (h t) >
h!0+
X k6=i
qik pkj (t):
(9.19)
P
* N > i, , pkj (t) 6 1 pik (h) = 1, i j 2 X t > 0, k h > 0 X X Lij (h t) 6 pikh(h) pkj (t) + h1 pik (h) = k6=i k>N X pik (h) X k6N 1 = h pkj (t) + h 1 ; pii (h) ; pik (h) : k6=i k6N
*!
lim L (h t) 6 h!0+ ij
/ , lim L (h t) 6 h!0+ ij
X k6=i
X k6=i k 6N
k6N k6=i
qik pkj (t) ; qii ;
qik pkj (t) + qi ;
X k6=i
X k6N k6=i
qik =
qik :
X k6=i
qik pkj (t)
(9.20)
(9.15). ) (9.19) (9.20) (9.17) . B t > 0 h < 0 (9.18), , h 2 N;t 0), ( ! ). 2
@ 9.4 (
). -
# $ P (t), t > 0, $ ( # $ Q qij , i 6= j
pij (h) = qij h + ij (h)
(9.21)
i
ij (h)=h ! 0 h ! 0 + : t > 0 ( P 0 (t) (( " p0ij (t)) P 0(t) = P (t)Q . . i j 2 X t > 0 X p0ij (t) = pik (t)qkj : k
(9.22) (9.23) (9.24) 167
2 * t > 0, h > 0 i j 2 X . ? pij (t + h) ; pij (t) = p (t) pjj (h) ; 1 + 1 X p (t)p (h): ij h h h k6=j ik kj
(9.25)
B " > 0 j (9.21) h0(" j ) > 0, jkj (h)j=h < " k 0 < h < h0(" j ), ! X (h) X pik (t) kj 6 " pik (t) 6 ": (9.26) h k6=j k6=j
P ) (9.21), (9.26) , h1 pik (t)pkj (h) 6 pij (th+ h) 6 h1 , , k6=j t > 0 i 2 X (9.24). / (9.25) h ! 0+, , (9.23) . / h < 0 (9.25) (
( ). 2 ? 9.5. I $ $ $ X $ P (t) $ $ , $ % $ / . B
, i j 2 X , t > 0 1 X p (t) = 1 ; pii ij t t j 6=i
, 9.2 . * ! (9.21), (9.22), , , .. i
X . ' Q (9.16) (9.23) ($% $%). $ 9.6. 9.4, i j 2 X (
lim p (t) = pj t!1 ij (
X k
(9.27) 8.6).
"
pk qkj = 0
j (9.28)
. . p | $ QT (QT " $ Q), "( ".
9.4, , 2 * , P j pk (t)qkj , pk (t) = P (Xt = k), k (
P
p0j (t) =
X k
pk (t)qkj :
(9.29)
P
C pj = 0, . . pj = 0, (9.28), , . C pj 6= 0, j j
pi(0) = pi , i 2 X (.
8.9). ? pj (t) = pj (8.29). / , p0j (t) = 0, (9.29) (9.28). 2 168
r r r r r r r r r r r h+o(h)
kh+o(h)
h+o(h)
nh+o(h)
? y
? ky k+1?
y 1?
k ;1
0
1;t+o(h)
n;1 n
1;(+k)h+o(h)
1;nh+o(h)
. 9.1
L $ $+ + ( 6 ), $ X = f0 1 : : : ng, $$ pij (h) h ! 0+ ;$ ,
. 9.1. ? ,
, P $ % 0 n
0 1 n n ; 1, ! ( | ). ? , $ + pij (h) = o(h) h ! 0+. ?
0 ; B B B B B B B Q=B B B B B B B @
;(+) 2
;(+2)
...
1 C C C C C C C C C C C C C A
0C
... k
...
;(+k)
...
...
(n;1) ;(+(n;1)) n ;n
0
*, 8.6. i = 0 : : : n
...
j0 = n. B
pin (nt) > pi i+1(t)pi+1 i+2(t) : : : pn;1 n (t)pnn (nt ; (n ; i)t) > > pi i+1(t)pi+1 i+2 (t) : : :pn;1 n (t)(pnn (t))i: ' t0 > 0, pk;1 k (t) = t + o(t) > 0, k = 1 : : : n (o() k) pnn (t) > 1=2 0 < t 6 t0. h = t0=n, , (8.21). 9.5 P P , , ! , , pj = 1, . . X ( pij (t) = 1 (9.27)). ) , j j p,
9.6. L QTp~ = 0:
0 ; B B B B B B B B B B B B @ 0
;(+)
...
2
...
...
;(+k) (k+1)
...
...
...
;(+(n;1))
1 0C C 0p 1 C C 0 C B C p1 C B C C = ~0: B C . C . @ A . C C pn C C n A
;n
, QT , : : : , k- | ( 169
( 2 6 k 6 n ; 1). * 0 ; 1 0 B C ; 2 B C B C . . B C . . ~p = ~0: . . B C B C @ ; n A 0 ;n ? , ;pk + (k + 1)pk+1 = 0 k = 0 : : : n ; 1: k / , pk+1 = pk =(k + 1), = = . < pk = k! p0, . .
Pn p k=0
k
= 1, p0 =
Pn
k=0
;1
k =k!
k =k! k = 0 1 : : : n: pk = P n j =j !
(9.30)
j =0
I (9.30) #
D .
@+ %:, $$ $ $ < + $$, . 9.1.
* $ % (/3<), ( n . * ! . 0 , , . . 1, 1 + 2, : : : , f k g , . . 8.2. . L $ %, . . , ($ %). ? , , . C , ( , ! ! (
, , ). C
, t | ! Xt . / . *, !
> 0. , , , ( , f k g, ( . A% (A % ;$ ajbjc, a k (-, k = 1 + : : : + k , k , k 2 N { ...
), b ( { ... , ( f k g), c {
. * ! ! M . ? , M jM jn. & GjGjc , ("G" { general). (, , , k .. (., ., N?]). 170
C ft t > 0g | , f k g, Am(t t + h) | (t t + h] m , )m e;h m = 0 1 : : : P (Am(t t + h)) = P (t+h ; t = m) = (h m! *! P (A0(t t + h)) = e;h = 1 ; h + o(h) h ! 0+P ;h = h + o(h) h ! 0+P P (A1(t t + h)) = he (9.31) S ; h ; h P Am(t t + h) = 1 ; e ; he = o(h) h ! 0 + : m>2
B
(
;x p (x) = e x > 0 0 x < 0
, x y > 0
P ( > x + y j > y) =
R1 p (z) dz
x+y
R1 p (z) dz y
;(x+y)
= e e;y = e;x = P ( > x):
(9.32)
(9.33)
? , ( ( x, , ! ( , ( , !
,
( (9.33)). & , ph = P ( < h) = 1 ; e;h = h + o(h) h ! 0 + : , . *! , k ( t l ! h, Ckl plh (1 ; ph)k;l ( 0). / , Bk l(t t + h) $ k ( l (t t + h]%, P (Bk 1(t t + h)) = k(1 ; e;h)e;h(k;1) = (9.34) = Sk( h + o(h))(1; (k ; 1) h + o(h)) = k h + o(h) h ! 0+P P Bk l(t t + h) = 1 ; (1 ; k h) ; k h + o(h) = o(h) h ! 0 + : l> 2
/ Cij (t t + h), ( , (
i j (t t + h], (
( (t t + h]). C 1 6 i 6 n ; 1, Ci i+r = ArBi 0 Ar+1Bi 1 : : : Ar+iBi i r = 1 : : : n ; i Ci i;r = A0Bi r A1Bi r+1 : : : Ai;r Bi i r = 1 : : : i: ' C0 k Cn k , k = 0 1 : : : n, fAqg fBm k g. ' , (9.31) (9.34), pij (h), ( . 9.1. 171
3 fXt t > 0g , Xt s Xs , t ; s
! , ( . *, <+ $$ , ; 1 , Xt - ! . & , ( : ( ( Q , ). < ,
! . ! % $ (9.30). B k = n
pn , . . , n , pn (
8.7 , pj (t) pj ! ). / N?], ( :
n = 2 = 03 n = 4 = 06 n = 6 = 09
9.2 p2 ' 00335: p4 ' 00030: p ' 00003: 6
1 ,
( (
( ). ' ## $ , ! (. (9.32)) E = 1= . *! . & , ( ) Pn kpk = (1 ; pn ). k=1 0. J. / , 1 % $ $ $+ ( , % & $% -& 1= , ! ! .
.
8 $ $ + $
Q = (qij )i j2X ($ $ ) X = fX (t) t > 0g. * qi := ;qii < 1 i 2 X : (9.35) ? ( 6 ) X N0 1) , B2.15, (
R N0 1)), . * s > 0 i P (X (s) = i) 6= 0. ? t > 0 ( X ) P (X (u) = i s 6 u 6 s + tj X (s) = i) = P (X (1s) = i) P (X (u) = i s 6 u 6 s + t) = 172
1 P (X (u) = i u = s + tk2;n k = 0 : : : 2n ) = !1 P (X (s) = i) nlim ;n ))2n : = nlim ( p ( t 2 ii !1
(9.36)
) qi (1 ; pii(h))=h = qi + i(h) i(h) ! 0 h ! 0 + : / , t > 0, i 2 X n 2 N (pii(t2;n ))2n = (1 ; qit2;n + o(t2;n ))2n = expf2n log(1 ; qit2;n + o(t2;n ))g: * log(1 + x) = x + (x)x2, j(x)j 6 1 jxj 6 1=2, , ;n ))2n = expf;qi tg t > 0: lim ( p ( t 2 ii n!1
(9.37)
< , (9.36) (9.37) t = 0. 2 (9.36) X i , t, , s i. / (9.36) (9.37), ,
s,
. *! X i, . ) , ( 8.5)
@ 89.1. $ X = fX (t) t > 0g " (9.35). $ i qi ( , # "( (9.36)). / i, 0 6 qi < 1, . < ( "( , qi = 0. C , ( (9.35) (9.36) qi = 0). / i , qi = 1. 1 6
D. 9.2. * qi = 1. B , X ! , .., ! , . ? 3 , ( , . ) , N?] , . P * , .. qi < 1 j6=i qij = qi i 2 X . ? qi 6= 0 qij =qi, j 6= i, & $$ $+, $$ $$ $ i $ j . ? , j 6= i, t > 0
Fij (t) := P (X (s + t) = j j X (s) = i X (s + t) 6= i): 2 ,
Fij (t) = pij (t)=(1 ; pii(t)) ! qij =qi t ! 0 + :
(9.38) 173
P
P
C 0 < qi < 1 j6=i qij < qi, 1 ; j6=i qij =qi
" ". B ! Q
, B . * X {
, X = fX (t) t > 0g Q qi 2 (0 1), i 2 X . 0 ( ) (D F P ) n X n R+ = N0 1). * 1 , t > 0, i 2 X P ( 1 > tj 1 = i) = e;qit (9.39) n > 1, i j i1 : : : in 2 X , x1 : : : xn 2 R+ P ( n+1 = j j 1 = x1 : : : n = xn 1 = i1 : : : n;1 = in;1 n = i) = qij =qi P ( n+1 ; n > tj 1 = x1 : : : n = xn 1 = i1 : : : n = in n+1 = j ) = e;qj t: (9.40) D. 9.3. B , (D F P ) (
n n,
( (9.39) (9.40). * ! 0 < 1(!) < 2(!) < : : : .. 3
, limn!1 n = 1 .. n ! 1? B , , supi qi < 1. D. 9.4. * 0(!) = 0 .. ) n , n , n 2 N, ( 9.3,
Y = fY (t) t > 0g, Y (t !) = n(!) n;1 (!) 6 t < n (!) n 2 N ( ! , n;1 (!) = n (!) n 2 N, Y (t !) = 0, t > 0). B , Y { , ( .-.., X , Q. < , , qi . 9.3 D. 9.5. B , supi qi < 1 , pii(t) ! 1 t ! 0+ i 2 X (, ! , pij (t) ! ij t ! 0+ i j 2 X ). D. 9.6. * , (9.1). * , (9.1), (9.35) . ) , 9.4 , , Q, , ( Q. B ! ( 8) { ! P (t), t > 0, . C P (t), t > 0, ( Q, 8.1 P (t) (
, . . ). D. 9.7. (. B9.10). *
X N ! , ( ) P (t) . B ,
P (t) = exp(tQ) t > 0 (9.41) 174
N N - Q
qij > 0 i 6= j
X j
qij = 0 i:
(9.42)
< . C Q (9.42), (9.41) . ? Q = (qij )i j2X
! ,
qij > 0 i 6= j qi = ;qii > 0
X j
qij 6 0 i 2 X :
(9.43)
/ , (
fP (t) t > 0g, Q { , ..
P 0(0) = Q
(9.44)
( ). 9.2 (9.14)
, (9.41). C (
P (t),
( (9.42), Q
, , . (9.16). ? , -
$ $$ < $ / . 1 ( . I N?] 2. 2 L N?] , Q " ". , N?] (. < , $+ < (9.42) % $ $ $ PP(t), t > 0, - (8.19), (8.20) j pij (t) = 1, t > 0, i 2 X P j pij (t) 6 1, t > 0, i 2 X . L P
Q-$, (. C , j pij (t) = 1 t > 0, i 2 X ,
P - ( ),
j pij (t) 6 1, t > 0, i 2 X , { $.
(
( , ) pij (t), t > 0, i j 2 X .
( ,
1. J , t>0 peij (t) = pij (t) i j 2 X pe1 j (t) = 1 j pei 1 (t) = 1 ;
X j
pij (t):
(9.45)
D. 9.8. B , (9.45) N0 1)
Xe = X f1g, ( Pe(t) = (peij (t))i j2Xe, t > 0. ) , (9.17) (9.24) : p0ij (t) t > 0 t > 0, - : pij (t) ( 2 ) t > 0. 175
D. 9.9. B , ,
( ) ! . B { ! (. N?]). @ 89.10. (9.41). ( Q-$, "( ,
= .
2 I $ $ $ , .. X = f0 : : : ng,
(9.16), .. !
P (t) = etQ t > 0: (9.46) * ! (8.19) P (t)
(9.23). B, (9.46) . < = min q C (t) = e;tP (t) t > 0: i2X ii
? C 0(t) = e;tP 0(t) ; e;tP (t) = e;tP (t)Q ; C (t) = C (t)Q ; C (t) = C (t)B B = Q ; I = (bij )nij=0. *
C (t) = etB
1 k X t
B k bij = qij ; ij > 0 i j = 0 : : : n k ! k=0
=
(C (t))ij > 0 , , pij (t) > 0 i j = 0 : : : n. ) (9.24)
X
(
j
X
pij (t))0 =
/ ,
j
p0ij (t) =
XX j
pik (t)qkj =
k
P pij (t) 6 P pij (0) = 1. j
X k
pik (t)
X j
qkj 6 0:
j
L $ $+ $ X , .. X = Z+. < "-
" , Pn0 (t) = QnPn (t) Pn (0) = In Pn(t) = ((Pn (t))ij ), Qn = (qij ), In = (ij ), i j = 0 : : : n, n 2 Z+. *
Pn (t) = etQn t > 0: (9.47) B An = (aij )nij=0 ( ) An = (aij )1i j=0, aij = 0 i > n j > n. ? &, $ & t > 0 9 nlim P (t) = P (t) (9.48) !1 n
( .. (
! Pn (t)) ,$$ $ ; $ ; ;. B i j 2 X , n > maxfi j g t > 0 (Pn+1 (t))0
ij
176
=
n+1 X
n X
k=0
k=0
(Pn+1 (t))ikqkj =
(Pn+1 (t))ik qkj + (Pn+1 (t))i n+1qn+1 j
C 0(t) = C (t)Qn + D(t) C (t) = ((Pn+1 (t))ij )nij=0, D(t) = ((Pn+1 (t))i n+1qn+1 j )nij=0. 3 , Pn (t) = etQn C 0(t) = C (t)Qn. *! , , C (0) = In, C (t) = Pn (t) +
Zt 0
Pn (t ; s)D(s)ds t > 0:
(9.49)
) (9.49), , Pn (u) D(s) ! u s > 0, n 2 N, , (Pn+1 (t))ij > (Pn (t))ij i j 2 X n > maxfi j g:
(9.50)
, , Pn (t), t > 0, (Pn (t))ij 6
X j
(Pn (t))ij 6 1 i j 2 X n 2 Z+:
(9.51)
* % ; $ +$+ $ () , , (9.48) . ' ! 89.11. ak (n) % ak < 1 n ! 1 k 2 Z+ ( .. ak (n) 6 ak (n + 1) k n 2 Z+ limn!1 ak (n) = ak k 2 Z+).
X k
ak = nlim !1
X k
ak (n):
(9.52)
) ! , (9.50) (9.51) , P (t) { $ $ $ & t > 0. , , s t > 0 i j n 2 Z+ (Pn (s + t))ij =
X k
(Pn (s))ik (Pn (t))kj
, ak (n) = (Pn (s))ik (Pn (t))kj ( s > 0, i j 2 Z+), B9.10 , $ P (t), t > 0 % ;$ , .. (8.19). 8 & $+, $ P (t), t > 0 $$ % $
/ ,
P (t) = I + P (t) = I +
Zt 0
Zt 0
QP (s)ds
(9.53)
P (s)Qds:
(9.54)
B (9.51), ..
QP (s) = P (s)Q s > 0:
(9.55) 177
B
, ( ) i j 2 X s > 0 (QnPn (s))ij = qii(Pn (s))ij +
1 X k=0
ak (n)
ak (n) = qik (Pn (s))kj (1 ; ik ), k n 2 Z+. * B9.10, lim Q P (s) = QP (s). J , Pn (s)Qn ! P (s)Q n ! 1. n!1 n n < , 1 k k+1 X QnPn (s) = s Qk!n = Pn (s)Qn n 2 Z+ s > 0: k=0 B " ( ) "
Pn (t) = I +
.. i j 2 X , t > 0 (Pn (t))ij = ij + qii
Zt 0
Zt 0
QnPn (s)ds
(Pn (s))ij ds +
Z tX 1 0 k=0
qik (Pn (s))kj (1 ; ik )ds:
/ n ! 1 0. 2 (9.53). 2 @ 89.12. $ Q, ( , " (9.41). = (9.16) $ - , "( Q-$, , (9.23).
2 / X , ! X = Z+. * , < (P (t) t > 0), $ $ +$ $ 89.10, % $ $ +$, .. (Pe(t) t > 0) { - (9.16), .. Pe0(t) = QPe(t) t > 0 Pe(0) = I
(Pe(t))ij > (P (t))ij i j 2 X t > 0:
(9.56)
I j 2 X (x(t))i = (Pe(t))ij , (x(0))i = ij , i 2 X . ? i 2 X , t>0 n X X 0 (x(t))i = qik (x(t))k + qik(x(t))k k=0
..
k>n
x0n(t) = Qn xn(t) + Rn (t) P - xn(t) = ((x(t))0 : : : (x(t))n) (Rn(t))i = k>n qik (x(t))k, i = 0 : : : n. J (9.49) xn(t) = Pn (t) + 178
Zt 0
Pn (t ; s)Rn (s)ds n 2 Z+ t > 0
.. i 6 n, j 6 n (xn(t))i = (Pn (t))ij +
Z tX n 0 k=0
(Pn (t ; s))ik (Rn (s))k ds:
8 , (Pn (u))ik > 0, (Rn(s))k > 0 i k = 0 : : : n s u > 0, (9.56).
$+ < % $ $ / $ $ $ $ $ %$. ? (P (t) t > 0) , . C (Pe(t) t > 0) P { - P , j peij (t) 6 1 i 2 X . ) (9.56) j pij (t) = 1, i 2 X , , peij (t) = pij (t) i j 2 X , t > 0. J , ,
( ) , , (
. 2 D. 9.13. * Q
(9.41) . B , (.
B9.11) (P (t) t > 0)
,
Q- . C (P (t) t > 0) {
, (
Q- ,
. D. 9.14. ( B8.29). * Q
(9.41) . ? ( (
Q- . 1. B > 0
(Q ; I )x = 0 (9.57) x = 0, .. x = (0 0 : : : ), , supi jxij < 1. 2. B > 0
(9.57) x = 0. D. 9.15. (. 9.5). B , supi qi < 1, ,
. ) , % $ $$ -
< / $ $ $ $ ,
@ 89.16 (. N?]). qi < 1. , -
= , , "( : X (s ! ) ! 1 s ! t , X (s ! ) ! 1 s ! t .
I$$+ $$+ < /
% $$+ & % . < (X (t) t > 0)
X = Z+ -
$ , Q
qi i;1 = i qii = ;(i + i) qi i+1 = i qij = 0 ji ; j j > 1 (9.58) 179
0 = 0, i > 0, i 2 N i > 0, i 2 Z+. C i = 0 i 2 Z+, $ , i = 0, i 2 Z+, { $ . D. 9.17. * , Pi ;i 1 < 1. B , ,
, { . D. 9.18. B , , Pi ;i 1 < 1,
, { ,
. S ( ,
. * n;1 0 = 1 n = 0 1 : :: : n 2 N (9.59) 1 2 : : n
R=
1 X
n 1 n;1 1 X X 1 X 1 T =X 1 ): S = ( i n n+ n n n=1 n n i=1 n=2 i=0 i i n=0
@ 89.19 (L $). Q {
"( "
(9.43).
1. ! R = 1, ( .
$ , -
Q-$,
-
2. ! R < 1 S = 1, ( Q-$ . , $ . 3. ! R < 1 S < 1, " T < 1, ( Q-$ , "( . * $ .
< , R = 1
Q- B N?]. B & $ $ $ ( , ..). B 3 ! , , N?], N?], N?]. C P (X (0) = i) 6= 0, t > 0
Gii (t) := P (X (t1) = 6 i X (t2) = i 0 < t1 < t2 6 tj X (0) = i)
.. ! , , i, ( ) ! t. / i , limt!1 Gii (t) = 1, . / i R ,
( 01 tdGii (t). D. 9.20. B , (
. 1. 2 , . 180
2. * ( ) ,
X n
(n n);1 = 1:
3. * ( ) ,
X n
X
n = 1
n
(n n);1 < 1:
n (9.57). D. 9.21. B , !
,
X n
n < 1
X n
(n n);1 = 1
{ ,
X n
n = 1
X n
(n n);1 = 1:
181
10. 3# #) .
. ! " - . E " , . " , @1 . = $ " .
A% & %:$ % $
( () $ & $ & . * ! . B L2- X = fX (t) t 2 Na b]g, (D F P ),
X (t) = m(t) +
n X k=1
k (t)zk
(10.1)
m(t) = EX (t), 1(t) : : : n(t) , z1 : : : zn { L2(D F P ). ? X (t), t 2 Na b], . * X ! ( . ?,
r(s t) = cov(X (t) X (s)) =
n X k=1
k k (s)k (t) s t 2 Na b]
k { ! zk , k = 1 : : : n. J (10.1), , ( . ', 3.6. * , L2(D F P ) { , ! z1 z2 : : : , X (t) 2 L2(D F P )
I ! 1(t) 2(t) : : : . < , ( ,
! ! "". <
, "" z1 z2 : : : , 1 2 : : : (,
L2Na b]). 0
, ( (10.1) . I ,. , . /. * . B $ $ $ + . * K |
R. * K - . 3 - A = fKg ( R), !
M = fB 2 A : (B ) < 1g - . * B 2 K Z (B ) 2 L2(D) = = L2(D F P ), . . Z (B ) = Z (B !) EjZ (B )j2 < 1, (Z (B ) Z (C )) = (B \ C ) 8 B C 2 K (10.2) 182
( ) = E Y 2 L2(D). ? Z () . ) (10.2) , B C 2 K B \ C = ?, Z (B ) ? Z (C ), . . (Z (B ) Z (C )) = (?) = 0. ) , Z , , Z
. @ 10.1. ) Z - 1 S L2 (D) # $ K, . . B = Bk , B B1 : : : 2 K, Bn \ Bm = ? k=1 n = 6 m,
Z (B ) =
1 X k=1
Z (Bk )
(10.3)
.
2 (10.2) ( , " ", .. (x y) = (x y) (x y) = (x y) x y 2 L2(D) 2 C )
kZ (B ) ;
n X k=1
Z (Bk )k2 = (Z (B ) ;
n X k=1
Z (Bk ) Z (B ) ;
n X k=1
Z (Bk )) =
= (B ) ;
n X k=1
(Bk ) ! 0 n ! 1: 2
& , Z (?) = 0 .., ! (10.3) , Z .. - K. * , &$ $ ! 2 D, $ (10.3), $ $ $$+ 1, $ $ $ B B1 : : : 2 K. 10.2. * R = N0 1), K = fNa b) 0 6 a 6 b < 1g (Na a) = ?). * Z (Na b)) = W (b) ; W (a), W | . 8 ( , , Z | K, ( 2 . ? & Z K M. C | - K, (
Rn 2 K, n 2 N, , 1 S R = Rn, Rn \ Rm = ? n 6= m (Rn ) < 1, n 2 N ( { n=1 , R = R1). < Kn = K \ Rn, . . Kn = fB 2 K : B Rng, An = fKng | - Rn. ?
A 2 A = fKg () A = !
(A) =
1
n=1
1 X n=1
An An 2 An
n (An)
(10.4)
(10.5)
n | jKn Kn An. ' , A R Rn 2 K, n 2 N, ! (10.5) . 183
E $ $+ , $ (R) < 1 ( . . $ % R Rn, n 2 N).
< A , ( m 6 (S
, ( K (R 2 K). B B = Bi, Bi 2 K, i = 1 : : : m, i=1 Bi \ Bj = ? (i 6= j ), &
Z (B ) =
m X i=1
Z (Bi):
(10.6)
Sr Dj , Dj 2K, j =1 : : : r, j =1 m P Pm Pr Z (B \ D ) Z (B ) =
1 $. B
, B =
Dj \ Dl = ? (j 6= l). ? (10.3) i i j i=1 i=1 j =1 r r P m P P Z (Dj ) = Z (Dj \ Bi), ; $ $+
j =1 $ ..j=1 i=1
' f : R ! C
,
f=
m X i=1
ci 1 Bi
(10.7)
Sm
ci 2 C , Bi 2 A, i = 1 : : : m, Bi = R Bi \ Bj = ? i 6= j , . .
Bi R. i=1 < (10.7)
Jf =
m X i=1
ciZ (Bi):
(10.8)
10.3. ) (10.8) .
Pr
2 * (10.7) f f = dj 1 Dj , j =1 Sr dj 2 C , Dj 2 A, j = 1 : : : rP Dj = R, Di \ Dj = ? (i 6= j ). ' , j =1 (Bi \ Dj ) 6= 0, ci = dj . / , , Z (?) = 0, r X j =1
dj Z (Dj ) = =
r m X X
dj
Z (Dj \ Bi) =
m X
X
dj Z (Bi \ Dj ) =
i=1 j : (Bi\Dj )6=0 m r m ciZ (Bi \ Dj ) = ciZ (Bi \ Dj ) = ciZ (Bi): i=1 j : (Bi \Dj )6=0 i=1 j =1 i=1
X m
j =1
X
i=1
XX
X
10.4. f g | # $ .
(Jf Jg) = hf gi h i | L2 (R) = L2(R A ). 184
2 (10.9)
Pr
2 * f (10.7), g = dj 1 Dj , D1 : : : Dr
j =1 R. ? (Jf Jg) = = =
X m
m X r X i=1 j =1
ZX m i=1
i=1
ciZ (Bi)
Z
r X j =1
X m X r
dj Z (Dj ) =
cidYj 1 Bi\Dj () (d) =
ci1 Bi
r
X j =1
dj 1 Dj (d) =
$ 10.5. J |
# $ L2(D).
Z
i=1 j =1
m X r X i=1 j =1
cidYj (Bi \ Dj ) =
Z
cidYj 1 Bi ()1 Dj () (d) =
f ()g() (d) = hf gi: 2
2 C f g | , , , f + g, 2 C , |
. 10.4 (J (f + g) ; J (f ) ; J (g) J (f + g) ; J (f ) ; J (g)) = = hf + g f + gi ; hf f + gi ; hg f + gi ; ; Yhf + g f i + Y hf f i + Yhg f i ; Yhf + g gi + Yhf gi + Yhg gi = 0: / , J (f + g) = Jf + Jg: 2 (10.10) 10.6. # $ L2(R). 2 * f 2 L2(R). B " > 0 H = H (") > 0 ,
Z
f 2()1 fjf ()j>H g (d) < ":
?
(7.3) f ()1 fjf ()j
f 1 fjf j
X
2n ;1
k=0
rn k 1 Dn k ;
X
2n ;1
k=0
rn k 1 Bn k 6 H
X
2n ;1
k=0
1 Dn k ; 1 Bn k = H
X
2n ;1
k=0
(Dn k 4 Bn k ) < ": 185
P
P
2n ;1
mn
< rn k 1 Bn k ci1 Bi , B1 : : : Bmn 2 A
i=1 k=0 R, . 2 ? J L2(R). B f 2 L2(R) & Jf = nlim Jf ( L2(D)), (10.11) !1 n () ffng | , fn L;! f n ! 1. 10.7. ) (10.11) . 2
() f 2 * 10.6 (
fn L;! 2
n ! 1. *
10.5 10.4, kJfn ; Jfmk = kJ (fn ; fm)k = fn ; fm ! 0 n m ! 1, . . ( ffng . * L2(D) , , (
Jfn L2(D). *, !
( L2 () . * gn ;! f , gn . ? gn ; fn 6 gn ; f + fn ; f ! 0 n ! 1. / , kJfn ; Jgn k = kJ (fn ; gn )k = fn ; gn ! 0 n ! 1: () () () ' Jfn L;! Jgn L;!
, Jfn ; Jgn L;! ; . ? , kJfn ; Jgnk! ! k ; k = 0, . . = .. 2 $ 10.8. 3 " f g 2 L2(R) 2
J
2
2
| L2 () ) hn ;! f .
(Jf Jg) = hf gi (10.12) L2(R) L2(D). hn 2 L2(R) (
( ) Jhn L;! Jf n ! 1: (10.13) (10.12) fn gn , n 2 N, 2 B
L2 () L2 ()
fn ;! f , gn ;! g (. 10.6), 10.4 . 2 J ,
10.5. * J (10.12), kJf ; Jhn k = kJ (f ; hn)k = f ; hn ! 0 n ! 1: 2 A% $ $+ $; Jf f 2 L2(R A ), | -2
.
L R Rn 2 K (Rn) < 1, n 2 N. ? fn := f jn 2 L2(Rn An n) n 2 N Z 1 Z X f 2 L2(R A ) () 2 f () (d) = fn2() n (d) < 1
186
n=1 n
(10.14)
An = fK \ Rn g, n | jKn Kn An. * ! (10.14) R Rn, n 2 N. B f 2 L2(R) &
Jf =
1 X n=1
Jn fn
(10.15)
L2(D), fn (10.14), Jn L2(Rn) . 10.9. ) (10.15) . 2 C f 2 L2(R), (10.14) fn 2 L2(Rn ) n 2 N. ? , Jnfn ? Jmfm n 6= m. C fn fm | , ! , . . (Z (B ) Z (D)) = 0 B 2 An D 2 Am (An | , Kn , n 2 N). B fn fm, ( , . / ,
X 2 MX_N M N X Jnfn ; Jnfn = kJn fnk2 ! 0 n=1 n=1 n=M ^N
N M ! 1
Jn (10.14). ? , Jf R. * R = 1 S = ;n , ;n 2 K, ;n \ ;m = ? (n 6= m), (;n ) < 1, n 2 N. < hn m = f jn \;m , n=1
1 P
PN
(n ) m n 2 N. ? fn = f jn f jn \;m , n 2 N. ? , hn m L;! fn m=1 m=1 N ! 1 , , (10.13)
Jn fn =
1 X
m=1
2
Jnhn m
L2(D). * Jn m h 2 L2(Rn \ ;m ) L2(Rn) Jn mh = Jn h, n m 2 N. 8 , 6
, 1 X n=1
Jn fn =
1 X 1 X
n=1 m=1
Jn m hn m:
J , gm = f j;m Jem { , L2(;m )
, Jn L2(Rn ), m n 2 N, 1 X e
m=1
Jmgm =
1 X 1 X
m=1 n=1
Jn mhn m :
(10.16)
< , Jn m hn m ? Jk lhk l, (n m) 6= (k l), , , 1 1 X X
n=1 m=1
kJn mhn m k = 2
1 1 X X
n=1 m=1
k
k
hn m 2L2(n\;m)
=
Z
f 2() (d) < 1: 2 187
, ,
J L2(R A )
Jf =
Z
f ()Z (d):
(10.17)
@ 10.10. ' (10.17) L2 (R A ) ( | - ) L2Z L2(D F P ). Z -$ M = fB 2 A : (B ) < 1g # Z (B ) = J 1 B :
(10.18)
S1
2 ? . * R = Rn, n=1 R1 R2 : : : | R (Rn) < 1, n 2 N. f g 2 L2(R). ? 1 1 P P f fn, g gn , fn = f jn , gn = gjn , L2(R). n=1
n=1
?
(Jf Jg) = = Nlim !1
X
N X
n
n=1
Jn f n
X m
Jmgm = Nlim !1
(Jnfn Jngn ) = Nlim !1
N X n=1
X N n=1
Jn fn
hfn gn i = Nlim !1
N X m=1 N
Jm gm =
X X N n=1
fn
m=1
gm = hf gi: (10.19)
3 , Jn fn ? Jmgm n 6= m (.
10.9) , hfn gniL2 (n) = hfn gniL2 (). ) , J | ! . < , L2(R) J
L2(D), L2Z . B B 2 K 1 P (B ) = (Bn ) < 1, Bn = B \ Rn, n 2 N. B, 1 B 2 L2(R A )
n=1 1 P 1 B 1 B , L2(R). / , (10.15) n=1
n
J1B =
1 X n=1
Jn 1 Bn 1 P
(10.8) Jn1 Bn = Z (Bn ), 10.1 Z (Bn) = Z (B ) ( n=1 L2(D)). / , B 2 K J 1 B = Z (B ), . . (10.18) J K M. , , B C 2 M
J (Z (B ) Z (C )) = (J 1 B J 1 C ) = h1 B 1 C i = (B \ C ): 2 ? 10.11. < Z $ , EZ (B ) = 0 B 2 M. * Z K M , EZ (B ) = 0 B 2 K, ! B 2 M. @ 10.12. | - $ K R ( A = fKg). ( (D F P ) $ Z , D -$ M = fB 2 A : (B ) < 1g, Z . 188
2 Z . * D = R, F = A. I B ) . 0 < (R) < 1 ( (R) = 0 ), P (B ) = ((R) p & Z (B !) = (R)1 B (!), B 2 A. < , Z (B ) 2 L2(D). B, B1 B2 2 A p 2Z (Z (B1) Z (B2)) = ( (R)) 1 B1 1 B2 dP = (R)P (B1 \ B2) = (B1 \ B2):
$+ $+ (R) = 1 R1 R2 : : : | R, (Rn ) < 1, n 2 N. B A 2 A & 1 X P (A) = (A \ Rn ) : n=1
(Rn )2n
? P | A (P (R) =
Z (B !) =
1 p X n n=1
1 P 2;n = 1). A B 2 M
n=1
2 (Rn )1 Bn (!)
Bn = B \ Rn, n 2 N, L2(D) ( ( , ! 1 p 1 1 X X X n) 2 n n = (B ) < 1): E( 2 (Rn )1 Bn ) = 2 (Rn )E1 Bn = 2n (Rn) (R(B)2 n n n=1 n=1 n=1 ? , (Z (B ) Z (C )) =
1 X n=1
2n (Rn)(1 Bn 1 Cn )L2( ) = =
1 X n=1
2n (Rn)P (Bn \ Cn ) =
1 X n=1
(Bn \ Cn) = (B \ C )
Bn = B \ Rn, Cn = C \ Rn, n 2 N. ' $ ; , ( . *, Z (B ) ; EZ (B ), B 2 M, ( ). *!
(D0 F 0 P 0)
, E0 = 0, E0( 2) = 1, E0 P 0. * Z { (D F P ) . (D^ F P ) = (D F P ) (D0 F 0 P 0) !
Ze(B !e) := Z (B !) (!0) B 2 M, !e = (! !0) 2 D D0. 2 , Ze ( ) . 2 ? .. X (t !), t 2 T , ! 2 D,
. . X (t !) = X1 (t !) + iX2(t !), X1(t ) X2(t ) F j B(R)-
t 2 T . ', X (t), ( EjX (t)j2 < < 1, t 2 T , $ # $ r(s t) = cov(X (s) X (t)) = E(X (s) ; EX (s))(X (t) ; EX (t)) s t 2 T: (10.20) 0 % $ $+ $ , . . EX (t) = 0 t 2 T ( X~ (t) = X (t) ; EX (t)). 189
@ 10.13 (/ ). $ # $ $ $ X (t ! ), t 2 T , (D F P ), , . .
r(s t) =
Z
f (s )f (t ) (d) s t 2 T
(10.21)
f (t ) 2 L2 (R) = L2 (R A ) t 2 T , | - . ( $ Z , -$ M = fA 2 A : (A) < 1g , ( , - (D^ F P ) = (D F P ) (D0 F 0 P 0), "( " , ,
X (t) =
Z
f (t )Z (d)
!
@ T() 2
L2(R
A ):
Z
t 2 T:
(10.22)
f (t )T() (d) = 0 8t 2 T
(10.23)
((10.23) , @ T: T ? f (t ) L2(R) t 2 T ), - (D F P ) Z . ? 10.14. C X (t), t 2 T , (10.22),
10.10 (10.21). 2 $+ (10.23). B t 2 T G : f (t )7!X (t ) (10.24) : n X
G(
k=1
ck f (tk )) =
n X k=1
ck X (tk )
(10.25)
ck 2 C n, tk 2 T , k = m1 : : : n. * $$+ ,$ , n P P P
. . ck f (tk ) = dl f (sl ), dl 2 C , sl 2 T , l = 1 : : : m, ck X (tk ) =
k=1 l=1 m P = d X (s ). B ! , l=1
l
X n k=1
l
ck f (tk )
=
XX n
m
k=1 l=1
m X l=1
X n X m
dl f (sl ) =
Z
ck dYl f (tk )f (sl ) (d) =
k=1 l=1 m ck dYlEX (tk )X (sl ) = k=1 l=1
XX ck dYlr(tk sl) = n
* (10.26), , n X k=1
190
k=1
ck f (tk ) ;
m X l=1
X n k=1
ck X (tk )
m X l=1
n m X X dlf (sl ) = ck X (tk ) ; dlX (sl ): k=1
l=1
dlX (sl ) : (10.26)
) , . , G L2Nf ], . . Pn L2(R) ck f (tk ). * ! (10.23) k=1 L2Nf ] = L2(R A ) G(L2Nf ]) = L2NX ], L2NX ] | X (t), . . L2(D F P ) c1X (t1) + : : : + cnX (tn ) (ci 2 C , ti 2 T , i = 1 : : : n). B B 2 M, , 1 B 2 L2(R),
Z (B ) = G1 B :
(10.27)
?
(Z (B ) Z (C )) = (G1 B G1 C ) = h1 B 1 C i = (B \ C ): / , Z | M . 3 Z , Z (B ) 2 L2NX ] B 2 M, E = 0 2 L2NX ], .. X . ? h 2 L2(R)
Jh =
Z
h()Z (d):
* 10.10 J | L2(R) L2Z . ) ,
G : L2(R)7!L2NX ] L2(D) J : L2(R)7!L2Z L2(D): * ! (10.27) (10.18)
G1 B = J 1 B B 2 M. ' L2(R) ( 10.6). / , G = J L2(R) ,
, L2NX ] = L2Z . ) (10.24) , Jf (t ) = X (t), . . (10.22)
. $+ $+ (10.23) . ? L2Nf ] 6 L2(R). L L2(R) ! L2Nf ], . . L2Nf ], - g(u ) 2 L2(R), u 2 T 0, T 0 \ T = ? (. N?]). (s t) =
Z
g(s )g(t ) (d) = (g(s ) g(t ))L2() s t 2 T 0:
< , ck 2 C , tk 2 T 0, k = 1 : : : n, n 2 N,
2 Z X n ck cYl(tk tl) = ck g(tk ) (d) > 0: k=1 k l=1 n X
3.3 (
fY (t) t 2 T 0g, (D0 F 0 P 0), cov(Y (s) Y (t)) = (s t) s t 2 T 0: (D^ F P ) = (D F P ) (D0 F 0 P 0). ? t 2 T , s 2 T 0 X (t) Y (s) !
( !~ = (! !0) 2 D~ X (t) = X (t !~ ) = X (t !), Y (s) = Y (s !~ ) = Y (s !0)). 191
(T T 0) D~ . . ?
(
(t !~ ) = X (t !0) t 2 T P0 Y (t ! ) t 2 T :
8 > (s t) s t 2 T P : 0 s 2 T t 2 T 0 ) , Z cov( (s) (t)) =
(
s 2 T 0 t 2 T:
h(s )h(t ) (d)
h(t ) = f (t ) t 2 T P0 g(t ) t 2 T : 3 , f (t ) ? g(s ) L2(R) t 2 T s 2 T 0. , , L2Nh] = L2(R),
. . @ T 2 L2(R): T ? h(t ) t 2 T T 0. / , (
Z M (D^ F P ), L2N ] = L2Z Z (t) = h(t )Z (d) t 2 T T 0:
' t 2 T (t !~ ) X (t !~ ). / , t 2 T
X (t !~ ) = X (t !) =
Z
h(t )Z (d) =
Z
f (t )Z (d):
y Z , , (10.23). 2
.
D. 10.1. B Z , K
R, . . ! 2 D, .. K? /(
Z
? D. 10.2. * , !
, (10.2) (. * B K
R Z (B ) 2 L2(D F P ) , (Z (B ) Z (C )) = 0, B C 2 K B \ C = ? (-, ( ) = E 2 L2(D F P )). * Z
(10.3). ? (B ) := EjZ (B )j2 K ! (10.2). D. 10.3. * Z 2 R { L2- (. . 28) , 1) EjZ ; Z j2 ! 0 2 R # , 2) ( , .. 1 < 2 < 3 E(Z1 ; Z2 )(Z3 ; Z2 ) = 0: ' K = f(a b] ;1 < a 6 b < 1g, (a a] = ?,
Z ((a b]) := Z (b);Z (a). B , Z { . 192
C Z, 2 R, (10.17) R Rf ()dZ . D. 10.4. * Z { B(R), EjZ (R)j2 < 1. * Z = Z ((;1 ]), 2 R. B , Z, 2 R {
1), 2), ( . D. 10.5. * Z, 2 R { (, ( , ). * t 2 T g(t ) : R ! C
Z
R
jg(t )j2 (d) < 1
{ ,
( Z , Z, 2 R (. 10.3). B ,
Y = fY (t) =
Z
R
g(t )dZ t 2 T g
. D. 10.6. * Z { , DB 2 M ( -
(R A) (, - B 2 M= fA 2 A : (A) < 1g). * h : R ! C , h 2 L2(R A ).
- Z N = fB 2 A : jh()j2 (d) < 1g B
Z
V (B ) :=
1 B ()h()Z (d) B 2 N :
(10.28)
< 6 , V ( , Z ) -
(B ) =
Z
B
jh()j2 (d) < 1 B 2 N :
B , g : R ! C , g 2 L2(R fNg ),
Z
g()V (d) =
Z
g()h()Z (d):
(10.29)
, 11 + % $ + & $ . L
L2Na b], ( , 2 Na b]. / !
(f g) =
Zb a
f (t)g(t)dt f g 2 L2Na b]:
(10.30)
/ , ( 10.13. 193
810.7. B, N0 2] -
1 X 1 ; eikt z 2 k=;1 ik k
W (t) = p1
(10.31)
zk { , t 2 N0 2] ( k = 0 (1;eikt)=ik = ;t). 2 B s u 2 N0 2] 1 X 1 0 s](u) = p1 ck (s)eiku: 2 k=;1
(10.32)
I p12 eiku , k 2 Z,
L2N0 2] s 2 N0 2] (10.32) !
, ! I
p
Z 2 1 ck (s) = p 1 0 s](u)e;ikudu = p1 (1 ; e;iks)=ik k 2 Z 2 0 2
(c0(s) = s= 2). L
* s t 2 N0 2]
1 X 1 (1 ; e;iks )(1 ; eikt) : minfs tg = (1 0 s] 1 0 t]) = 2 k2 k=;1
< 10.13, R = Z, f (t k) = p12 1;eik;ikt , t 2 N0 2], k 2 R ( R, .. (fkg) = 1, k 2 Z. 2 D. 10.8. B , (10.31) .. zk N (0 1), k 2 Z. ? , , zk , k 2 Z, (10.31) , ( minfs tg, s t 2 N0 2]. B %$ $ $&, $ $ $ +%$ $ $ . * ! D. 10.9. * X = fX (t) t 2 Na b]g { L2- , Na b]. B , !
, Na b] m(t) = EX (t) Na b] Na b] r(s t) = cov(X (s) X (t)). / Na b] Na b] r 0 ,
L2Na b]
( :
Af (s) =
Zb a
r(s t)f (t)dt f 2 L2Na b]
(10.33)
r ! A. D. 10.10. B ,
(10.33) A (
). 194
, A { 9 { h (. N?]),
Z bZ b a
a
jr(s t)j2dsdt < 1:
(10.34)
& , A , .. (. N?, . 531]) A ! r(t s), r(s t) = r(t s) s t. ' (. N?], . 4, x6) . @ 810.11 (! +%$ { H$). A { ( ) H . ( fn gn2J A, "( , , " h 2 H
h=
X n2J
cnn + u
(10.35)
cn 2 C , n 2 J , u 2 KerA, .. Au = 0. : J ( (10.35) k k = ( )1=2). , A n - " $ 0 , j1 j > j2 j > : : : . ! J , limn!1 n = 0.
< ,
A H
,
,
(
, , (n u) = 0, n 2 J . B A (10.33) H = L2Na b]
KerA fk gk2M , M { (
M J (, KerA = 0, M = ?). B fngn2J fgn2J M H , (10.35)
A: X ck k ck = (h k ) k 2 J M: (10.36) h= k2J M
C
J M , ! I L2Na b]. 10.9 10.10 r X = fX (t) t 2 Na b]g (10.36) t 2 Na b] X r(t ) = ck (t)k() (10.37) k2J M
&
ck (t) = !
Zb a
r(t s)k (s)ds = k k (t) = k k (t)
r(t ) =
X k2J
k k ()k(t):
(10.38) (10.39)
C J { , (10.36), (10.39) L2Na b] t 2 Na b]. 3 , k
, k = 0, k 2 M . 195
D. 10.12. B , k > 0 k 2 J -
r.
$< (10.39),
@ 810.13 ((). r { Na b] Na b] # $ . s t 2 Na b] r(s t) =
X k 2J
k k (s)k (t)
$ -
(10.40)
, J { , (10.40) "
Na b] Na b]. G, - k k " "( # $ (10.33).
@ 810.14 (/ { ,). X = fX (t) t 2 Na b]g { Na b] $ L2 -$, "( $ " # $ " r. t 2 Na b] X (t) =
Xp k 2J
k k (t)zk
. .,
(10.41)
k , k { , (10.40), fzk k 2 J g { $ . 3 J (10.41) .
p
2 * T = Na b], R = J f (t k) = k k (t) t 2 T , k 2 R. ? (10.40) ( (10.21), { ( R, .. (fkg) = 1 k 2 R. < (10.13). 2
D. 10.15. '
, EjX (t) ;
Xp k6n
k k (t)zk j2
B10.14, (10.41)
Zb 1 zk = p X (t)k (t)dt (10.42) k a ( (
) L. B ! 6 , Na b] Na b] r
k 2 C Na b], k 2 J . , , , (10.41) zk , k 2 J , t 2 Na b]. 810.16. ' , { 2! W (t), t 2 N0 1]. 2 <
,
( N0 1]N0 1] r(s t) = minfs tg, 196
s t 2 N0 1]. ? k > 0 k 2 J . 10.12, k 2 C N0 1] . 10.15, > 0 2 C N0 1]
Z1 0
..
r(s t)(t)dt = (s) s 2 N0 1]
Zs 0
t(t)dt + s
Z1 s
(t)dt = (s):
(10.43) (10.44)
* (10.44) s 2 N0 1] ( ), ,
Z1 s
(t)dt = 0(s) s 2 N0 1]:
/ s !
, 00(s) = ;(1=)(s): < ( (10.45) p p (s) = A cos(s= ) + B sin(s= ) A, B { . 8 , (0) = 0, 0(1) = 0, p p A = 0 (B= ) cos(1= ) = 0 k = ((k + 1=2));2 k 2 Z+ = f0 1 : : : g: 8
R 1 2 (s)ds = 1
0 k
(10.45)
(10.46)
p
k (t) = 2 sin((k + 1=2)s) s 2 N0 1] k 2 Z+: (10.47) 2 , k k , k 2 Z+,
(10.44). ? , 1 pX k + 1=2)t) z W (t) = 2 sin(( (10.48) k k=0 (k + 1=2)t
fzk k 2 Z+g { 10.15 , (10.48) N0 1]. 2 D. 10.17. B , X = fX (t) t 2 T g { , L2NX ] , .. n Y1 : : : Yn 2 L2NX ] (Y1 : : : Yn) . ? 810.18. *( , L2NW (t) t 2 N0 1]] { . / 10.15 fzk k 2 Z+g { . *
, 5.8, , (10.48) t 2 N0 1] , zk N (0 1), k 2 Z+, (D F P ), . 197
D. 10.19. 3
, (D F P ) zk N (0 1), k 2 Z+,
(10.48) N0 1]? D. 10.20. < 6 , I , B10.16, KerA = 0. * X = fX (t) t 2 Na b]g, (10.33) KerA 6= 0. D. 10.21. ' , { 2!
W0(t) = W (t) ; tW (1) t 2 N0 1]: W (t), t 2 N0 1], (10.48). D. 10.22. ' N0 c] , { 2! < { 8 , ( r(s t) = e;js;tj, > 0, s t 2 R. D. 10.23. * X = fX (t) t 2 Rg { r(s t) = R(s ; t). * X { , .. X (t + ) = X (t) .. t 2 R. ' , { 2! X . < , $ + + -
;$ $ (., ., N?], . VI, x4 N?], . 1, x15-18). * ( N?], N?], N?].
<
. , ,
,
( . ) ( x174 N?]. ! $
& , ; $$ % ; , $ .
* H {
, ( , T . I K : T T ! C (
H , 1. K (t ) 2 H t 2 T , 2. hf K (t )i = f (t) f 2 H , h i { H .
@ 810.24 (R< ). 3 , # $ K
( , $ .
,
2 $+ K { . B n 2 N, t1 : : : tn 2 T , z1 : : : zn 2 C 1 X
j r=1
zj zrhK (tj ) K (tr )i = k
kuk2 = hu ui, u 2 H . 198
n X j =1
zj K (tj )k2
A% $. $+ K { $ $ + T T . L Lin(K ) { K ( t), t 2 T . B f g 2 Lin(K ),
..
f=
n X j =1
aj K (tj ) g =
m X r=1
br K (sr )
(10.49)
aj br 2 C , tj sr 2 T (j = 1 : : : n, r = 1 : : : m),
hf gi :=
n X m X j =1 r=1
aj br K (tj sr ):
(10.50)
D. 10.25. * , (10.50) , ..
, ( ! aj , br tj , sr f g 2 Lin(K ). ) (10.50) , br = ir , i r 2 f1 : : : mg, hf K (si )i =
n X j =1
aj K (tj si) = f (si):
* si { ,
2. K h i,
(10.50), Lin(K )Lin(K )
( ). B f 2 Lin(K ) kf k2 = hf f i. ? Lin(K ) k k. C fn { Lin(K ), t 2 T ,
, { 0 { h ,
jfN (t) ; fM (t)j = jhfN ; fM K (t )ij 6 kfN ; fM k(K (t t))1=2: (10.51) 3 , kK (t )k2 = hK (t ) K (t )i = K (t t). ? , ffN g
f T . < H Lin(K ). B f g 2 H h i . B
D. 10.26. * , H -
, ( K ( . 2 / 3.3 . *! ;% ; ; ; r(s t), s t 2 T & $ $+ $ +%$ $ $ H h i ( B10.24 , H ). / ,
r(s t) = hr(s ) r(t )i:
@ 810.27 ( ). F
f
(10.52)
2 g
X = X (t) t T $ 2 L -$ (D P ) $ # $ r(s t), s t T . H { ( r . (D P ) ( $ L2-$ Y = Y (h) h H ,
h i
F
2
f
X (t) = Y (r(t ))
. .
t 2 T
2 g
(10.53) 199
(Y (h) Y (g)) = hh gi " h g 2 H (10.54) ( ) = E 2 L2 (D F P ). L2 NX ] H , L2 NX ] { Lin(X ) ( .. X ). P 2 B h = nk=1 ck r(tk ) 2 H , ck 2 C , tk 2 T , k = 1 : : : n, n 2 N, n X
Y (h) :=
k=1
ck X (tk ):
2 , ! , H
. 2
(& $ 810.27 10.13 $ $ +, $ D. 10.28. B , r -
(10.21),
H ( r
h(t) =
Z
g()f (t ) (d) g 2 L2Nf ]
(10.55)
L2Nf ] { L2(R A ) f (t ), t 2 T ( , g t, , f (t )). / H
Z
hh1 h2i = g1()g2() (d)
(10.56)
hk gk , k = 1 2 (10.55). * Z { B 2 M(D F P ), ( (. 10.10), L2Nf ] = L2(R A ). < 6 , X = fX (t) t 2 T g (10.22), Y , ( B10.27,
Y (h) =
Z
g()Z (d)
(10.57)
h g (10.55). D. 10.29. B , (
)
H
( r(s t) = minfs tg, s t 2 R+ ( .. r { ),
h(t) =
Zt 0
g()d t 2 R+
Z1 0
g2()d < 1
(10.58)
d { 2 . < 6 , L2NW ], .. L2(D F P ) W (t), t 2 R+, (10.57), Z { B 2 M(D F P ), (. 10.2), g
(10.58). ? 810.30. h h (. B4.11 q = 1) { !
H N0 1] ( r(s t) = minfs tg, s t 2 N0 1]. '
- q > 1. 200
11. #) # # B . = ? . 4" . = 2 {0 . " . % " L2()). "1 . " . . < . , . , > " . , , 1 ". $ $ .
1$ , , $ + $ ;
$ ( ) , $ % $ , $ $ / . B ! . , g(t) t 2 T , T ( { ), $ , R(s t) = g(s ; t), s t 2 T , .. (3.6). , L2 { fX (t) t 2 T g ( .. EjX (t)j2 < 1, t 2 T ), T { , $ - , EX (t) = a t 2 T
(11.1)
r(s t) = cov(X (s) X (t)) = r(s ; t 0) =: R(s ; t) s t 2 T
(11.2)
) 3.3 , $ $ + fR(t) t 2 T g, T { , $ $ ( ) fX (t) t 2 T g. * fX (t) t 2 T g, T { , $ , n 2 N u t1 : : : tn 2 T (X (t1 + u) : : : X (tn + u)) =D (X (t1) : : : X (tn))
.. . 2 , L2- . B ! .
A $ $ + Z $ ; $ . @ 11.1 (! $). 0 $ R(n) n 2 N, $
,
R(n) =
Z
;
einQ(d) n 2 Z
Q { $ B (N; ]). G N; ].
(11.3)
R
;
201
2 < , (11.3) , n 2 N, t1 : : : tn 2 Z z1 : : : zn 2 C n X
k q=1
Z X n 2 zk zq R(tk ; tq ) = zk eitk Q(d) > 0: ; k=1
(11.4)
A% $. B N > 1 2 N; ]
( R()) N X N 1 X fN () = 2N R(k ; q)e;ikeiq = 21 k=1 q=1
X
jmj
(1 ; jmj=N )R(m)e;im :
(11.5)
3 , N ;jmj (k q), k;q = m ( k q 2 f1 : : : N g, jmj < N ). < B(N; ]) QN fN 2 , .. Z QN (B ) = fN ()d B 2 B(N; ]): B
? , (2.10), N > 1
Z
;
einQ
N (d) =
Z
;
einf
(1 ; jnj=N )R(n) jnj < N N ()d = jnj > N:
0
(11.6)
$ ( 4), K = N; ] , QN (N; ]) = R(0) < 1 N ( n = 0
(11.6)), fNk g N , QNk ) Q, Q { N; ]. ? , (11.6), n 2 Z
Z
;
Z
einQ(d) = klim einQNk (d) = R(n): 2 !1 ;
,
( . @ 11.2. fX (t) t 2 Zg { $ $ - $, (D F P ). ( Z (), B (N; ]), ,
X (t) =
Z
;
eitZ (d) t 2 Z:
2 * 9 s t 2 Z r(s t) = cov(X (s) X (t)) =
Z
;
ei(s;t)Q(d) =
(11.7)
Z ;
eiseitQ(d)
(11.8)
Q { B(N; ]). & , , ( 10.13), f (t ) = eit, 2 N; ], t 2 Z. * ! 202
(10.23). B
, L2 = L2(N; ] B(N; ]) Q) L2 , (
; , I. 2 3 Q, ( (11.3), ( ), "" Q(f;g) ; , "" Q(f;g)+Q(fg). * ! (11.3) , e;in = ein n 2 Z. D $ <+ $, $% $ $+ $ & $ (; ] $ &$. C , Z (f;g) = 0 .. 10.12, ! (11.7) (; ]. L , N; ). 3 $ & $ + $ (11.7) $ $ % +< .
@ 11.3.
;1 1 NX L2 ( ) X k ;! Z (f0g) N k=0
11.2.
N ! 1:
(11.9)
2 (11.7).
Z 1 NX;1 Z ;1 1 NX ik N k=0 Xk = N k=0 e Z (d) = TN ()Z (d)
;
;
8 1 (1 ; eiN) < 6= 0 TN () = : N (1 ; ei) 1 = 0:
/ , 10.10
1 NX;1 Z 2 N k=0 Xk ; Z f0gL ( ) = jN ()j G(d) 2
;
(11.10)
N () = TN () ; 1 f0g(), G() | ,
( Z (). 8 1 NP;1 ik
, N e 6 1, 2 R, , N (0) = 0 6= 0, 2 (; ] k=0
jN ()j 6 N j1 ;2 eij ! 0 N ! 1 2 (11.10) (11.9). 2 ? 11.4. C fX (t) t 2 Zg | EXt = a, t 2 Z,
Xt ; a =
Z
;
eitZ (d) 203
NP ;1 NP ;1 2 ( ) N1 (Xk ; a) = N1 Xk ; a L;! Z (f0g). / , k=0
k=0
;1 1 NX L2 ( ) X k ;! a N ! 1 N k=0
(11.11)
$ $ + $ , EjZ (f0g)j2 = 0, $ + ; G(f0g) = 0,
. . ( ) . /
(11.11) " L2(D). L (D F P ) L2- fXt t 2 Rg. * X () t 2 R,
kX (s) ; X (t)k ! 0 s ! t (11.12) k k = (Ej j2)1=2, 2 L2(D F P ). '
-
!
.
@ 11.5. ' $ - $ fX (t) t 2 Rg , $ # $ R(), (11.2), . 2 * X (t) X (t) ; a, a = const, , ( ) - t 2 R. *! -
( ( r(s t) = EX (s)X (t) = R(s ; t) s t 2 R. I fX (t) t 2 Rg $ R, s t u v 2 R
jr(u v) ; r(s t)j 6 kX (u) ; X (s)kkX (v)k + kX (s)kkX (v) ; X (t)k ! 0 u ! s v ! t. 3
, { 0 { ( ) h , X (v) L;! X (t), kX (v)k ! kX (t)k v ! t. < , R(t) ; R(0) = r(t 0) ; r(0 0). A% $. kX (s) ; X (t)k2 = ;(R(t ; s) ; R(0)) ; (R(s ; t) ; R(0)) ! 0 s ! t, R() 0. 2 $ 11.6. 0 $ R(), $ R,
R , . 2 * R() R . * 2
3.3 ( ) fX (t) t 2 Rg r(s t) = EX (s)X (t) = R(s ; t), s t 2 R. 11.5 ! R. ?
jR(s) ; R(t)j = jEX (t)X (0) ; EX (s)X (0)j 6 kX (t) ; X (s)kkX (0)k ! 0 s ! t. ? , R() . 2 A $ ( ) , $ R, &$ 204
@ 11.7 (E{J ). R(t), t 2 R, | $ # $ . " t 2 R R(t) =
Z1
;1
eitG(d)
(11.13)
G | $ B(R). < , (11.13) . / (11.5) , . ? ,
$ 11.7 $ + $ $ , %. B
0 { \ ( 4. ? 11.8. C G, ( (11.13), X (t), t 2 R, f () 2 , . . t 2 R
R(t) =
Z1
;1
eitf () d
f " . )
11.7, 4, , R() 2 L1(R), (
, (??). @ 11.9 (/ { / ). (D F P )
$ $ - $ fX (t) t 2 Rg ( ). ( Z (), B (R),
X (t) =
Z1
;1
eitZ (d) t 2 R:
(11.14)
2 11.5 R(t) = EX (t)X (0) ( R)
3.3 . * 0{\ (11.13). / , r(s t) = cov(X (s) X (t)) = R(s ; t) =
Z1
;1
ei(s;t)G(d) =
Z1
;1
eiseitG(d)
G() | - B(R). ? , , ( 10.13), f (t ) = eit (t 2 R) (10.23). , L2 = L2(R B(R) G) f ()1 fjj6ag, L2 , ( in a a ;a, e , n = 0 1 : : : . 2 ? % $ ; $ + $$ $ . 1 6 , , , , ( . < . 205
y ( L2(D)) " = f"(n) n 2 Zg -. 2 , ! , ( f () = 1=(2), 2 N; ]. ) , " ", " ( ) .
@ 11.10. ' $
- $ $
X = fX (t) t 2 Zg
"
, 1 fck k 2 Zg 2 l2 ( . . jck j2 < 1) - k=;1 2 ^ " = f"(n) n 2 Zg, L (D F P ) ( - (D F P )), ,
Xt =
1 X
k=;1
P
ck "t;k t 2 Z:
(11.15)
* , ( (11.15), , $ ( . 9 , X ( # , ". I # ( , ck = 0 k < 0. 1
, X t "k k 6 t ( .. X (t) t " (" ( "). 2 $+ (11.15), fck g f"k g . L L2(D^ F P ) ( "k , k 2 Z, !
), . . 2 ( ) PN ct;k "k ; Pn ct;k "k L;! PN 0 N n ! 1 M m ! ;1. & , E ct;k "k = 0, k=M
k=m
k =M
L2 ( )
N M 2 Z(M 6 N ). B, M N ;! , E M N = 0, jE ;E M N j 6 Ej ; M N j 6 6 k ; M N k ! 0 (N ! 1, M ! ;1), , E = 0. ) , EXt = 0 t 2 Z. *
* r(s t) =
X k
cs;k "k
X l
X
ct;l"l =
k
cs;k cYt;k =
X j
cj cY;j;(s;t) = R(s ; t) s t 2 Z (11.16)
. . fX (t) t 2 Zg | . 1 X 1 X() = p c eik: (11.17) 2 k=;1 ;k L (11.17) L2N; ] = L2(N; ] BN; ] mes), mes { 2 , fck g 2 l2
Z
;
(
eiseitd = 2 s = t 0 s 6= t (s t 2 Z):
?
L2N; ] 1 1 X X 1 1 i ( k + s ) is cs;j eij: c;k e =p X()e = p 2 k=;1 2 j=;1 206
(11.18)
(11.19)
*
* (11.19) (11.18) (11.16)
Z
;
ei(s;t)jX()j2 d =
Z
;
X()eisX()eitd =
X j
cs;j cYt;j = r(s t):
(11.20)
' X() 2 L2N; ], . . jX()j2 2 L1N; ]. ) , (11.20) , f () = jX()j2, 2 N; ], fX (t) t 2 Zg. A% $. * f () | fX (t) t 2 Zg. ? f () 2 L1NR; ] f () > 0 . . 2 p ( G G(B ) = f () d). < X() = f () (X() = 0, f () < 0). ? B X() 2 L2N; ] , , , f(2);1=2eis 2 N; ]g, s 2 Zg, , 1 X c;k eik X() = p1 2 k=;1
(11.21)
fck g 2 l2 (11.21) L2N; ]. ? ,
* ,
r(s t) = R(s ; t) =
Z
;
ei(s;t)f () d
=
Z
;
eisX()eitX()d =
X k
cs;k cYt;k : (11.22)
R = Z, | ( - B
Z( . . - , (fkg) = 1, k 2 Z). * f (t ) = ct;, t 2 Z, fck g 2 L2(R B ). ? (11.22) (10.21), , 10.13 , ,
(D^ F P ) (
Z (),
Xt =
Z
f (t )Z (d) =
1 X
k=;1
ct;k "k
(11.23)
"k = Z (fkg), (11.23) L2(D). * ! k , k 2 Z, , , . . Ej"k j2 = (fkg) = 1, k 2 Z. 2 ? 11.11. )
11.10 ,
(
, ,
( (11.15):
1 X 2 1 ik f () = p c e 2 N; ]: 2 k=;1 ;k
? $ $$ $ + $$. * fX (t) t 2 Zg | , R(n) = EXn+k X k , k n 2 Z. I (11.6) , fN (),
(11.5),
$ + $$ f () ( (
). 207
I$$ & (11.5) fN $ $+ $ R(m) $ $$ ; , $ ; % ; X0 : : : XN ;1.
8 m;1 > 1 N ;X > > < N ; m k=0 Xm+k X k 0 6 m 6 N ; 1 RbN (m) = > Rb(m) ; (N ; 1) 6 m < 0 > > : 0 m 2 Z
(11.24)
X b mj : RN (m)e;im 1 ; jN fbN () = 21 jmj
*
N X;1 ;ik 2 1 ^ fN () = 2N Xk e : k=0
1
.
(11.25)
2 ,
EfbN () = fN ()
(11.26)
ERbN (m) = R(m) m, jmj 6 N ; 1. ? , fN ( : NX ;1 NX ;1 N X;1 NX;1 Z i(k;l) 1 1 ; i ( k ; l ) fN () = 2N R(k ; l)e = 2N e f ( ) de;i(k;l) = k=0 l=0 k=0 l=0
= =
Z
;
Z
;
;
;1 NX ;1 1 NX i( ;)k e;i( ;)l f ( ) d = 2N k=0 l=0 e
Z ;1 2 1 NX i ( ; ) k f ( ) d = XN ( ; )f ( ) d 2N k=0 e
(11.27)
;
I
N X;1 ik2 1 sin(N=2) 2 1 XN () = 2N e = 2N sin(=2) : k=0
8 , f () 2 L1N; ], . . ( 2 )
Z
XN ( ; )f ( ) d ! f () N ! 1:
(11.28)
) (11.26) (11.28) , $ $$ $ + $$. < , . . EjfbN () ; f ()j2, N , , 208
. *! fbNW , . .
b
fNW () =
Z
WN ( ; )fbN ( ) d
;
WN (), n 2 N, , a) WN () 0P R b) WN () d = 1P ; c) Ejf^NW () ; f ()j2 ! 0 N ! 1 2 (; ]. ', E $ $ ( WN () = aN B (aN ),
sin 2 2 1 BN () = 2 BN (0) = 21 2
aN % 1, aN =N ! 0 (N ! 1). & ! ( . * L2- X = fX (t) t 2 T Rg. ? X (t) kk
L2(D), X s < t. C ,
, - Fs = fX (u) u 6 s u 2 T g,
, inf fkX (t) ; yk : y 2 L2(D Fs P )g = kX (t) ; E(X (t)jFs)k: <, - Fs , , . I $ $+, $ &$ $+ $ + % Xu u 6 s, u 2 T ( $ $ $ L2(D)), $ ( $ . * ! !
, .
Hs (X ) s 2 T , X (u), u 6 s, u 2 T ,
H;1 (X ) := \s2T Hs (X ) H (X ) = L2NX ] L2NX ] { L2(D) Xu , u 2 T . < , H (X ) {
L2(D), ( Hs (X ), s 2 T . ) , X (t) < $ s $ $, < % & X (t) (
L2(D)) , $ $ $ Hs (X ). )- ](s t) := inf fkX (t) ; gk : g 2 Hs (X )g:
(11.29)
' ( ), L {
H ( k k, ) x 2 H , inf fkx ; gk : g 2 Lg
! h = PL x, PL { H
L. ? , Ps X (t), Ps { H (X ) Hs (X ). 209
* X , H;1 (X ) = H (X ),
.. Hs (X ) = Ht (X ) s t 2 T . * X , H;1 (X ) = 0. &, ,$
$ $+; $ , &$ T = R T = Z. 3; % $ $+ X $ . ? EY = 0 Y 2 H (X ), Y -
L2(D) ! X , ( . *, (11.30) ](s t) = ](s + u t + u) s t u 2 T: B , y = c1X (s1)+: : :+cn X (sn) 2 Hs (X ), ck 2 C , sk 2 T , sk 6 s z = c1X (s1+u)+: : :+cn X (sn +u) 2 Hs+u (X ) kX (t);yk2 = kX (t+u);zk2 (
). I (11.30) (t) := ](s s + t) t 2 T: < ( (11.29), , (t) = 0 t 6 0 (t 2 T ) (v) 6 (u) v 6 u (v u 2 T ): (11.31) '
(11.31) , , (11.29) ,
. @ 11.12. ' $ $ X 1) , (t0) = 0 t0 > 0, t0 2 T ( (t) = 0 t 2 T ), 2)
,
(t) ! R(0) t ! 1 (11.32) R { $ # $ X . 2 1) I X , X (t) 2 Ht(X ) = Hs (X ) , , ](s t) = 0 s t 2 T . A% $. * (t0) = 0 t0 > 0 (t0 2 T ). ? X (s+t0) 2 Hs (X ) s 2 T . (11.31) X (t) 2 Hs (X ) t 6 s + t0. < , Ht(X ) = Hs (X ) t 6 s + t0. s , X . 2) $+ (11.32). B s t 2 T R(0) = kX (t)k2 = kPs X (t)k2 + kX (t) ; Ps X (t)k2 = kPs X (t)k2 + (t ; s): (11.33) *! kPs X (t)k ! 0 s ! ;1 t 2 T . ? , L1 L2 {
H L1 L2, PL1 x 6 PL2 x x 2 H . ? , kP;1 X (t)k 6 kPsX (t)k s t 2 T . / , P;1 X (t) = 0, .. X (t)?H;1 (X ) t 2 T , H (X )?H;1 (X ). * H;1 (X ) H (X ), , H;1 = 0. A% $. * H;1 = 0. 8 (11.33), ,
11.13. Ht, t 2 T { "( H , .. Hs Ht s 6 t, s t 2 T . ! \s2T Hs = 0, Psx ! 0 s ! ;1 x 2 H , Ps { H Hs . 210
2 * kPs xk 6 kPtxk x 2 H s 6 t, , kPtk xk ! 0 k ! 1, tk+1 < tk , k 2 Z+ limk!;1 tk = ;1. < Lk := Htk ! Htk+1 , .. Htk+1 " Lk = Htk , k 2 N. ? x 2 H m > k m X;1 Ptk x = Ptm x + Qj x (Qj { H Lj ). <
j =k
kPtk xk = kPtm xk 2
2+
X
m;1 j =k
P kQj xk2 6 kPt xk2 6 kxk2. / , 1 j =0 m X;1 2 2 0
kPtk x ; Ptm xk =
j =k
kQj xk2:
? , m > k ,
kQj xk ! 0 k m ! 1
( , k > m). H (
! y = klim P x. !1 tk * Ptk x 2 Htm k > m, y 2 Htm m 2 N. *, y 2 \1m=0 Htm , \1m=0Htm = \t2T Ht
Ht . *! y = 0. 2 ? , 11.12 . 2
@ 11.14 (3 +). K $ ( - ) X = fX (t) t 2 T g, T = R T = Z
X (t) = M (t) + N (t) t 2 T (11.34) M = fM (t) t 2 T g { , N = fN (t) t 2 T g { $, M ?N , .. M (s)?N (t) s t 2 T . = , $ M N -
$
$ $ . I $
M (t) 2 Ht (X ) (
t 2 T .
X
, N (t) 2 Ht(X ))
2 * M (t) = P;1 X (t), N (t) = X (t) ; M (t), t 2 T ,
(11.34). * M N M (t) 2 H;1 (X ) N (t)?H;1 (X ) t 2 T . / , M ?N . ) (11.34) , X (t) 2 Ht (M ) " Ht (N ). *! Ht(X ) Ht(M ) " Ht(N ). / , M (t) 2 H;1 (X ) Ht(X ). < Ht(M ) Ht(X ) (11.34) Ht(N ) Ht(X ) t 2 T . ? , Ht(M ) " Ht (N ) Ht (X ) Ht(X ) = Ht(M ) " Ht(N ) t 2 T: (11.35) ? , Ht(N )?H;1 (X ) t 2 T , H;1 (N )?H;1 (X ).
Ht (N ) Ht(X ), ! H;1 (N ) H;1 (X ). ) , H;1 (N ) = 0,
.. N { . 3 , Ht (M ) H;1 (X ), t 2 T . ) (11.35) , H;1 (X ) Ht (M ) " Ht(N ). 8 , Ht (N )?H;1 (X ), H;1 (X ) Ht (M ), t 2 T . / , Ht(M ) = H;1 (X ), t 2 T , M { .
8 &, $ M N { $ . B !
H (X ) u 2 T ,
Su ,
Su(c1X (t1) + : : : + cnX (tn )) = c1X (t1 + u) + : : : + cnX (tn + u) 211
ckP2 C , tk 2 T , k = 1 : : : n, n 2 N. 1 . B
, P n k=1 ck X (tk ) = mj=1 dj X (sj ), dj 2 C , sj 2 T , j = 1 : : : m, m 2 N, X u 2 T 0=k
n X k=1
ck X (tk ) ;
m X j =1
dj X (sj )k = k 2
n X k=1
ck X (tk + u) ;
m X j =1
dj X (sj + u)k2:
J , u 2 T (Sux Suy) = (x y) x y 2 Lin(Xt t 2 T )
.. Su { Xt t 2 T , kSuxk = kxk x 2 Lin(X (t) t 2 T ), u 2 T . / , Su H (X ), Su. 2 ( Lin(X (t) t 2 T )), (Su )u2T { H (X ), .. Su+v = SuSv u v 2 T S0 = I {
. ) , $ ( ) $ (Su )u2T H (X ). 11.15. 3 " u v 2 T (11.36) SuHv (X ) = Hu+v (X ) Su H;1 (X ) = H;1 (X )
Pu+v Su = SuPv P;1 Su = Su P;1 (11.37) Pt { H (X ) Ht (X ), t 2 T f;1g. 2 B u v 2 T , , Su (Lin(X (t) t 6 v t 2 T )) 2 Hu+v (X ), ! SuHv (X ) Hu+v (X ). 8 , Su S;u = I , Hu+v (X ) = SuS;u Hu+v (X ) SuHv (X ). *
(11.36) . C x 2 H;1 (X ), x 2 Ht (X ) t 2 T . * Su x 2 Ht+u (X ) t 2 T . / , Sux 2 H;1 (X ),
.. Su H;1 (X ) H;1 (X ). *! H;1 (X ) = SuS;u H;1 (X ) SuH;1 (X ),
(11.36). Su, Pv , u v 2 T ,
X (t), t 2 T , , (
(11.37). B u v t 2 T Pu+v Su X (t) = Pu+v X (t + u). / , X (t) = yv (t) + zv (t), yv (t) = Pv X (t) 2 Hv (X ) zv (t)?Hv (X ). B, X (t + u) = SuX (t) = Su yv (t) + Su zv (t), Su yv (t) 2 Hu+v (X ) h 2 Hu+v (X )
(11.36) h = Sug, g 2 Hv (X ). *! (h Suzv (t)) = (g zv (t)) = 0: C
H = L " L? (L? {
L), ! x 2 H
x = y + z, y 2 L, z 2 L? ( y = PLx, PL { H L). *! , SuPv X (t) = Suyv (t) = Pu+v X (t + u). *
(11.37) . J , P;1 SuX (t) = P;1 X (t + u), X (t) = y;1 (t) + z;1 (t), y;1 2 H;1 (X ) z;1(t)?H;1 (X ). ? X (t + u) = Su X (t) = Su y;1(t) + Su z;1(t). * Su y;1(t) 2 H;1 (X ) Suz;1(t)?H;1 (X ) (
(11.36)) Su P1 X (t) = Su y;1(t) = P;1 X (t + u). 2 212
& $ +$ $ 11.14. * M (t) 2 H;1 (X ) H (X ),
EM (t) = 0, , , EN (t) = 0, t 2 T . ? , M N { $ . B t u v 2 T , 11.15
Su,
(M (v + u) M (t + u)) = (P;1 X (v + u) P;1 X (t + u)) = (P;1 Su X (v) P;1 SuX (t)) = = (Su P;1 X (v) SuP;1 X (t)) = (P;1 X (v) P;1 X (t)) = (M (v) M (t)): J , (N (v + u) N (t + u)) = (X (v + u) ; M (v + u) X (t + u) ; M (t + u)) = = (X (v) X (t)) ; (P;1 Su X (v) SuX (t)) ; (SuX (v) P;1 SuX (t)) + (M (v) M (t)) = = (X (v) X (t)) ; (M (v) X (t)) ; (X (v) M (t)) + (M (v) M (t)) = = (X (v) ; M (v) X (t) ; M (t)) = (N (v) N (t)): 8 & $$+ & (11.34). J , X (t) = U (t)+V (t), U = fU (t) t 2 T g, V = fV (t) t 2 T g {
, U ?V U (t) V (t) 2 Ht(X ), t 2 T . ?
Ht(X ) = Ht (U ) + Ht(V ) = H (U ) " Ht (V ) t 2 T: / ,
H;1 (X ) = \t2T (H (U ) " Ht(V )) = H (U ) " \t2T Ht (V ) = H (U ) + 0 = H (U ): ? ,
M (t) = PH;1(X )X (t) = PH (U )X (t) = PH (U )U (t)+PH (U )V (t) = PH (U )U (t)+0 = U (t) t 2 T: ? . 2
? 11.16. L (11.34) . , , L2- ( ) X ,
T R, , , { . 8 (. 11.14)
. 1 ,
11.14. < , 11.15 (11.30): ](s + u t + u) = kX (t + u) ; Ps+u X (t + u)k = kSu X (t) ; Ps+u Su X (t)k = = kSuX (t) ; SuPs X (t)k == kX (t) ; Ps X (t)k = ](s t): 0 " = f"n n 2 Zg "( $ L2- X = fXn n 2 Zg, Hn (X ) = Hn (") n 2 Z. <
213
@ 11.17. 3 , $ $ $ X = fXn n 2 Zg , , - "( $ " = f"n n 2 Zg 2 fcn g1 n=0 2 l , , . . Xn =
1 X k=0
ck "n;k n 2 Z
(11.38)
.
2 $+ X { . 2 , m 2 Z
Hm (X ) L2(D) ! hm;1 + cX (m), hm 2 Hm;1 (X ), c 2 C . * X (m) = gm;1 + zm, gm;1 2 Hm;1 (X ) zm?Hm;1 (X ), Hm;1 L2(D) hm;1 + czm;1, , ! , hm;1 2 Hm;1 (X ), c 2 C ( , ). & , zm 6= 0 (0 { L2(D), 0 ..). ) Hm;1(X ) = Hm (X ) (11.36) Hn;1 (X ) = Hn (X ) n 2 Z, . n = zn=kznk ,
11.13. J , n 2 Z tk = n ; k, k 2 Z+. * !
Lk := Htk ! Htk+1 n;k (k 2 Z+). * kPtk X (n)k ! 0, k ! 1,
(11.38), 1 X k=0
jck j2 6 kXn k2 < 1:
? , (11.38) n 2 Z, ! ! ck , k 2 Z, ( , n. B , , 1 X Xn = c(kn) n;k k=0
= (Xn n;k ), n k 2 Z. *( H (X ). J , "k = Sk 0, k 2 Z (-, Sk { H (X )). 8 (11.36), , "k 2 Hk (X ) "k ?Hk;1 (X ). ,
, kvarepsilonkk = kxi0k = 1, k 2 Z. *
Hk (X ) ! Hk;1 (X ) k , , , "k = k k , k 2 C , k 2 Z f"k g { ( X . L2(D) c(kn)
X0 =
1 X k=0
c(0) k 0;k "0;k
=
1 X k=0
ck ";k :
Sn (n 2 Z), ( ,
Xn = Sn X0 =
1 X k=0
ck Sn";k =
1 X k=0
ck Sn S;k 0 =
1 X k=0
ck Sn;k 0 =
1 X k=0
ck "n;k :
A% $. B " = f"n n 2 Zg { ( .. -
( X ), (11.38) , X { , Hn (X ) Hn ("), n 2 Z. < H;1 (X ) Hn " 214
n 2 Z. ' "n+1?Hn ("), n 2 Z, ! zn?H;1 (") n 2 Z. "n { H (X ). / , H;1 = 0. 2 ? , (11.38) $, $ $ -
$ $ $ +$ $ $ % < . B X = fXn n 2 Ng
Xn = Mn + Nn = Mn +
1 X k=0
ck "n;k n 2 Z
(11.39)
Mn { (, f"n n 2 Ng { ( Nn, (n 2 Z), fck g1k=1 2 l2. 8 , PHm(X )Mn+m = Mn+m Hm (X ) = Hm (") m n 2 Z,
n2 = EjXn+m ; PHm(X )Xn+m j2 = EjNn+m ; PHm (X )Nn+m j2 = = EjNn+m ; PHm (")Nn+m j2 == Ej
n;1 X k=0
ck "n+m;k j2 =
n;1 X k=0
jck j2: (11.40)
I (11.40) ( (. (11.32)), X = fXn n 2 Ng, .. ( fNn n 2 Ng, n ! 1
n2
!
1 X k=0
jck j2 = EjN0j2 = EjX0j2 = R(0)
R { X . ) 11.17 11.15 , , .. (11.38),
1 2 X 1 ; ik f () = p ck e 2 N; ]: 2 k=0 ) (
@ 11.18 (/ ). ' $ $ X = fXn n 2 Zg
f () "
$
,
Z ;
log f ()d > ;1:
B
!
\ H2 (. N?, . ]).
.
$ % $+ , $ < . ' .
215
D. 11.1. * X = fX (t) = ei(t+) t 2 Rg,
, N0 2]
( ), E 2 < 1. 8 , X (11.13), G = E 2F , F { . D. 11.2. ( Y = fY (t) = cos( t + ) t 2 Rg: * , Y { . C X = fX (t) t 2 Rg { ,
, fRe X (t) t 2 Rg fIm X (t) t 2 Rg { ? D. 11.3. N ] * fNt t > 0g { , ( > 0. *
fNt t > 0g P ( = ;1) = P ( = 1) = 1=2. , # , (11.41) X (t !) = (!)(;1)Nt(!) t > 0 ! 2 D: ' ! . ' EX (t) cov(X (s) X (t)), s t > 0 (
3.19). D. 11.4. ( ( 11.3). * 0 1 : : : { ...
, ( fNt t > 0g, ( 1 1 < 2 < : : : (. 8.5). * X (t !) = k (!) k (!) 6 t < k+1(!), k = 0 1 : : : ( 0 = 0). C f k (!)g1k=0 ( (
0), X (t !) = 0 t > 0. / E 02 < 1, fX (t) t > 0g. D. 11.5. N ] ) 11.4, : R ! R $ -
X (t) =
1 X k=0
k (t ; k ) t 2 R
(11.42)
.. ( ). ' r ! , lims!1 r(s s + t), , " " , ! s. ' ( (11.42)? * (11.42) t > 1 k > 0, k 2 Z+ ( N0 1) () > 0, (0) = 1, ( . k , k 2 N ! . * ! k k ,
. / ! . ? X (t) { t. < ( (11.42), - ,
(., ., N?], N?]). D. 11.6. ' < { 8 (. . 3.19). 216
D. 11.7. * ( 11.8) R(t), t 2 R, , t = 0. D. 11.8. * h1(t) : : : hn (t) { , P
T . B , r(s t) = nj=1 hj (s)hj (t), s t 2 T , -
. 8 , ( ! . ) ! . 10.20. D. 11.9. B , Pn (z1 : : : zm) { n- ( m ) ! rk (s t), k = 1 : : : m, s t 2 T { , Pn (r1(s t) : : : rm(s t)) . B fX (t) t 2 T g ! . D. 11.10. * fXn n 2 Zg { a R(n), n 2 Z. B , N ! 1 ;1 ;1 1 NX 1 NX L2 ( ) X ;! a , (11.43) N k=0 k N k=0 R(k) ! 0: D. 11.11. * ( N X N 1 X 1 X R(k)(1 ; jkj=N ) 6 cN ; R ( k ; j ) (11.44) N 2 k=1 m=1 N jkj6N ;1
c > 0 N 2 N. B , ;1 1 NX (11.45) N k=0 Xk ! 0 .. N ! 1: * , (11.44) , R(n) = O(n; ) n ! 1. @ 811.12 (! <). fX (n) n 2 Zg { $ $ a = 0 $ # $ R(n), n 2 N ( R(0) = 1).
(11.45) , G(f0g) = 0
Z
lim Z (d) = 0 . ., n!1 0<jj62;n Z { , # "( (11.12), G { . B
! N?] k;1 kj=1 Xj 2n 6 k < 2n+1 , n = 0 1 : : : ) jj62;n Z (d) + k , limk!1 k = 0 .. L2(D).
R
P
D. 11.13. * fXn n 2 Ng {
R(n), n 2 Z. B , ;1 1 NX 2 (11.46) N k=0 R (k) ! 0 N ! 1 , RbN (m),
(11.24) , .. m 2 Z EjRbN (m) ; R(m)j2 ! 0 N ! 1:
217
D. 11.14. * fXn n 2 ZgP{1 ( (11.15). B , k=;1 jck j < 1 "k , k 2 Z, ... , ( E"40 < 1. * fbN () { (11.25). ' lim cov(fbN () fbN ( )) 2 N; ]. N !1 D. 11.15. * Z { ,
(
< { 8 > 0 (. 3.19). B , Z 1 eit ; 1 i + W (t) = Z (d) t > 0 (11.47) ;1 i ( (eit ; 1)=i = t = 0)
( fW (t) t > 0g ( , B2.22). ) , (11.47) , < { 8 , 3.19 , < { 8 . 13 2 , ( ( ) ! . * 10.6, (11.47) ( B10.7) Z 1 eit ; 1 W ( t) = V (d) t > 0 (11.48) ;1 i V K, ( 10.6, (10.28) h() = (i + )=, 2 R. ? 811.16. / 4.1 .. t 2 R+. 2 ( ), t 2 R+ (
W (t)
. ? , + t
(11.48), " -"
W_ (t) =
Z1
;1
eitV (d):
(11.49)
1 , eitP (11.49) , V 2 . ? ! " " ( , . N?],
! N?].
L +$ $, & 11, $ ;$ $ , $ & . 3 C (s t) = (Cjk (s t))mjk=1, s t 2 T ! $ (
n 2 N z1 : : : zn 2 C m , t1 : : : tn 2 T
)
zjC (tj tk)zk > 0
(11.50)
n X
j k=1
218
# ! . X = fX (t) t 2 T g C n L2- , EkX (t)k2 < 1 t 2 T . & ( ,
)
kzk = (z z)1=2 (z w) =
m X k=1
zk wk z w 2 C m :
(11.51)
< " # $ " " $ " ( ) # $ " L2- X , a(t) = EX (t) 2 C m r(s t) = E(X (s) ; a(s))(X (t) ; a(t)): (11.52) D. 11.17. B , R(s t), s t 2 T ( ! ) L2- X = fX (t) t 2 T g , . * T {
, L2- X = fX (t) t 2 T g
C m $ - , a(t) = a 2 C m r(s t) = R(s ; t) s t 2 T: (11.53) 3 C (t), t 2 T , $ , r(s t) := C (s ; t), s t 2 T . 3; % $ $+ $ ( ). 9 , "(n), n 2 Z, C m - ( ! ,
, ), (11.53) a = 0 R(0) = I R(n) = 0 n 6= 0 (11.54) I { m- . ( ) $ (
X (n) =
1 X
k=;1
Ak "(n ; k) n 2 Z
(11.55)
f"(n) n 2 Zg { , Ak , k 2 Z { ( ), ( C m C m . L (11.55) , ( . D. 11.18. B , L2(D F P ) (11.55) 1=2 P P 1 m 2 1 = 2 2
, k=;1 jAk j < 1, jAj := (TrAA ) = j q=1 jajq j A = (ajq )mjq=1. ' ( . C T {
, L2- X = fX (t) t 2 T g C m t 2 T , EkX (s) ; X (t)k2 ! 0 (s t) ! 0: (11.56) ' T (11.56) t 2 T . B , 219
@ 811.19. 0 $ R(t), t 2 Rd,
$ # $ , X = fX (t) t 2 Rdg ,
R(t) =
Z
i(t )G(d) t 2 Rd e d
R
G { $
.
(11.57)
B(Rd).
G
J 11.1. / X , ( B11.19, Z G
(8&0S) . 1 ( .I. 9. @ 811.20 (! <). X = fX (t) t 2 Rng { $ $ # $ R(t), t 2 Rn ( , R(0) = 1). !
Z
Yn
(
0<jj61 k=1
log log(3 + jk j;1))2G(d) < 1
G { , T1 ! 1 : : : Tn ! 1 (T1 : : : Tn);1
ZT
1
0
Z Tn 0
X (t1 : : : tn)dt1 dtn ! (!)
. .
(11.58)
G(f0g) = 0. 8 B11.20 , N?]. < 8&0S . N?], N?]. , X = fX (t) t 2 Rdg , r(s t) t ks ; tk. C ( , r(s t) = R(ks ; tk) s t 2 Rd. ? (11.57) ( ,
@ 811.21 (. N?], . 1, . 262). 3 , # $ R(u), u 2 R+,
D " . .
,
$ # $ , X = fX (t) t 2 Rdg, ,
m Z 1 I(m;2)=2(u)
(11.59) (u)(m;2)=2 Q(d) u 2 R+ I (x) { # $ , Q { $
B(R+), Q(R+) = G(Rm) = R(0). 8 X , Rd D ( Rd. * (11.53)
(, ..
R(u) = 2(m;2)2;
2
0
(S ;1X (St1) : : : S ;1X (Stn )) =D (X (t1) : : : X (tn)) 220
( S Rd t1 : : : tn 2 Rd (n 2 N). ? , ( $ " ( , .. (X (t1 +h);X (t0 +h) : : : X (tn +h);X (tn;1 +h)) =D (X (t1);X (t0) : : : X (tn );X (tn;1)) n 2 N t0 : : : tn h 2 Rd, (., ., N?]). ( $ ; $- $ ; ; &$ G(B ) = (Gjk (B ))mjk=1, B 2 K (K {
R) $ , G(B ) B 2 K. B X = fX (t) t 2 Rdg C m 2 C m
Y = fY (t) = (X (t) ) t 2 Rdg. < ( ). * ! , , @ 811.22. 3 , # $ R(t), t 2 Rd $ # $ X = fX (t) t 2 Rdg C m , (11.57), R , m m # $ , G { m m - # $ B(Rd).
< ( 0 { \
4. 9 ( C m ) Z K
R L2- fZ (B ) B 2 Kg, , EZ (B )Z (C ) = G(B \ C )
(11.60)
G { - K, ( ) . /$ $ $ +
%% $ $ . ) 10.13 B11.22 @ 811.23. X , # "(
X (t) =
Z Rd
311.22, -
ei(t )Z (d) t 2 Rd
(11.61)
Z { B (Rd) G, "( " $ # $ . L2 NX ], .. L2 (D F P ) Xt , t 2 Rd, L2 ( ) = L2 (R B (R) ), = TrG, , 1) X (t) $ ei(t ), 2) Yk $ gk (), Yk 2 L2 NX ], gk 2 L2 ( ), k = 1 2,
Yk = EY1Y = 2
Z
Z
gk ()Z (d)
(11.62)
g1 ()g2()G(d):
(11.63)
Rd
Rd
221
* ,
@ 811.24 (/$ + { H). , X = fX (t) t 2 Rdg , ..
N;u1 u1] : : : N;ud ud] .
X (t) =
X
Yd sin(uk tk ; nk ) n1 nd uk tk ; nk X u1 : : : ud
n=(n1 ::: nd )2Zd k=1
(11.64)
t = (t1 : : : td ) 2 Rd . 1 , ( ) t 2 Rd - .
) ( N?], N?], N?].
A + & & $ % . *
(" "), X = fX (t) t 2 T g Y = fY (t) t 2 S g. 0 + , Y = AX , (
. C , A {
, , ( ) X A,
A ! ( ). 0 , $ A X $ & $ t 2 S ; .. Y (t). 8
,
( , ). ) ; $ . * X 0(t) ( ) L2- X , t 2 R, , X (t) ( )
B = L2(D F P ) (. . ??), .. EjX 0(t) ; (X (t + h) ; X (t))=hj2 ! 0 h ! 0:
(11.65)
? ( ' { 2 (8.53)). C (X (t + h) ; X (t))=h h ! 0,
, ( . D. 11.25. B , fNt t > 0g > 0 t 2 R+ .. ) , % $ $+ $ , ! . , , k- t ( (
) X (k;1)(t) ! . 222
D. 11.26. * X = fX (t) t 2 Na b]g { ! L2- r(s t). B , X 0(t) (
t 2 (a b)
e2
, (
( @ r(s t) (t t) ( @s@t
( ! ), @e2r(s t) := lim (r(s + h t + u) ; r(s + h t) ; r(s t + u) + r(s t))=hu: (11.66) h u!0 @s@t J
, . B ! ; ; + $ ; , $ . 811.27. I # $ f (t), t 2 Na b], "( H ( ) ( t0 2 Na b] , ( lims t!t0 (f (s) f (t)). D. 11.28. * (
X 0(t). ? (
dEX (t)=dt EX 0(t) = dEX (t)=dt: , ,
(s t) EX 0(s)X 0(t) = @e2r(s t) EX 0(s)X (t) = @r@s @s@t
(11.67)
(
(
, (
, . 2r(s t) D. 11.29. B , (
(s t) @ @s@t e2r(s t)
(
! @ @s@t
,
2 . * L - X = fX (t) t 2 Rg, ( u 2 R, ( v 2 R. 3 X ? D. 11.30. B , X = fX (t) t 2 Rg X (k)(t) (
t ,
Z1
;1
2k G(d) < 1
(11.68)
G { X . 0 X (k) ? C , ? ' ( ..
A(X + V ) = AX + AV 2 C X V 2 DomA) $ $ . B L2- X = fX (t) t 2 Na b]g R b
a X (t)dt ( (
), L Na b] X ( )
223
L2(D) = L2(D F P ), . . ??. B , a = x(0n) < : : : < x(knn) = b si 2 Nx(i;n)1 x(in)], i = 1 : : : kn (n kn 2 N)
Z b 2 kn X ( n ) ( n ) E X (t)dt ; X (si )]xi ! 0 a i=1
]n ! 0
(11.69)
]x(in) = x(in) ; x(i;n)1, i = 1 : : : kn ]n = max16i6kn ]x(in). D. 11.31. B
Na b]
x(in) i = 0 : : : kn t(in), i = 1 : : : kn ]n ! 0 n ! 1. B , L2- X = fX (t) t 2 Na b]g, ( r, Na b], .. (11.69), ,
9 n!1limm!1
kn X km X i=1 j =1
r(s(in) s(jm))]x(in)]x(jm):
(11.70)
< , L2- X = fX (t) t 2 Na b]g, ( r, (11.70) , (
L,
..
9
Z bZ b a
a
r(s t)dsdt:
(11.71)
& , N?] X = fX (t) t 2 Na b]g, Na b], (11.71), . N?, . 200]. ,
B ( (
( ). , X = fX (t) t 2 Na b]g Lp- p > 1, .. EjXtjp < 1 R b t 2 Na b], (Lp) a X (t)dt, L
Lp(D). R C
(Lp) ab X (t)dt $ L % , % $ $ $ ( ! ). B
N?], . 2, x2.1, , , . 47 , . 348 349 D. 11.32. * fX (t) t 2 Na b]g
Lp(D) p > 1. ? (Lp)
Zb a
X (t)dt)(!) =
Zb a
X (t !)dt
(11.72)
2 , . 8 X (t)
(11.72) ! (, . & , B2.15 Na b] Lp(D) . 224
* D. 11.33. * X = fX (t) t 2 T g T = Na b] Nc d]. ? E
Zb
cov(
a
Zb
cov(
a
Zb
X (t)dt =
a
Zb
X (s)ds X (t)) =
X (s)ds
Zd c
a
EX (t)dt
Zb a
X (t)dt) =
r { .
r(s t)ds s 2 T
Z bZ d a
c
r(s t)dsdt
(11.73) (11.74) (11.75)
F%$ $ $ $ $ % , ( (
) $$$ ; $ $ Na b], a ! ;1 b ! 1. ) , L2- X = fX (t) t 2 Na b]g r,
T h, T R,
Y (t) =
Z1 ;1
h(t s)X (s)ds t 2 T:
(11.76)
, Y (t) (11.76) (
,
L
Z 1Z 1
;1 ;1
h(t s)r(s u)h(t u)dsdu:
(11.77)
I h, ( (11.76), # $ . B
, + (11.76) h B s, Y (t) = h(t s). B , h(s t) t - , s. T = R ,
,
Y (t) =
Z1
;1
h(t ; s)X (s)ds t 2 R
..
# . 8 ## $
H (i) =
Z1
;1
h(s)e;isds 2 R:
(11.78)
(11.79)
C h 2 L1(R), .. 2 , H , , (
. S ( . I h 2 L1(R), $ & 2 R X (s) = eis, s 2 R % $ %$ % (11.78), $ ; %$ ; H (i). 1 ( . 225
D. 11.34. * (11.78) X = fX (t) t 2 Rg, ( (11.9) -
Z , G. B , H (i) 2 L2(R B(R)G), Y = fY (t) t 2 Rg
R(t) =
Z1
;1
Y (t) =
eitjH (i)j2G(d) t 2 R
Z1 ;1
eitH (i)Z (d) t 2 R:
(11.80) (11.81)
! , , jH (i)j2
$ + $ , $ $ ; X & +$. D. 11.35. ,
# -
, .. , ( ( Na b] (! , H (i) = 1 a b]())?
) $ + $ -
<; +
d
Pn dt Y = X (11.82) X { , Pn (z) = a0zn + a1zn;1 + : : : + an { ! , Y = fY (t) t 2 Rg n t 2 R. L
(11.82) .. t 2 R. (11.9) X . 0 Y , X ( ! Y , X ). 11.34 11.30,
Z1
;1
eitP
n (i)H (i)Z (d) =
Z1
;1
eitZ (d) t 2 R
( ), Pn (i)H (i) = 1 . . G X . 0 , , X W_ (11.49). , , X (11.82) Qm(d=dt)X , Qm(z) = b0zm + b1zm;1 + : : : + bm { m ! ,
Pn (i)H (i) = Qn (i)
(11.83)
H (i) = Qn(i)=Pn (i), 2 R, Pn (z) . ! , Qm(z)=Pn (z) , (. N?, . 282{284])
, Y X . J Pn (d=dt)Y = Qm(d=dt)X , , 226
(t 2 T Z) { ( . < ! . N?]. ( , ( ( . L L2- X = fX (t) t 2 T Rg
Ht(X ), t 2 T , ( X (s), s 6 t (s t 2 T ). / , "( $ Y = fY (t) t 2 T g, .. (,
Ht(X ) = Ht(Y ) t 2 T:
(11.84)
\ N?] X = fX (t) t > 0g, (11.84) . , N?] , fGn 1 6 n 6 N g B(R), G1 % G2 % : : : ), (
Zn , 1 6 n 6 N , (, EjZn()j2 = Fn(), 2 R, Fn { Gn , (
X = fX (t) t 2 Rg, N X
H t (X ) = 1
X (t) =
n=1
Ht (Zn ) t 2 R:
(11.85)
= :
N Z t X
n=1 ;1
gn(t )dZn () t 2 R
(11.86)
gn , n = 1 : : : N
N Z t X
n=1 ;1
jgn (t )j2dFn () < 1 t 2 R:
(11.87)
* N 2 N f1g " $ X . /
(11.86) (. N?]).
227
12. 3# 3#
4 @ . . " Z " , . E E, Z . > E . . $ E . , L2. E . , ( , ( , ( , , , L, 2 , L { / , 2 { / , B (. J.'. , ") " N?]). ) ( 0 *
N?]. , , ( " ". ?, 10 .
. + ,$ { $ $ 2$ $+ $ . $+ % ; $ $, $% $+ $ $+ $ + , & , $ + $ . &
. ' (. ! ). *- , '.
It(f ) =
Z
(0 t]
f (s)dWs t > 0
f (s), s > 0, " " (. N?] N?]). J , "
" d(fW ) = fdW + Wdf ,
R
It(f ) = f (t)W (t) ;
Zt 0
f 0(s)Ws ds
(12.1)
0t f 0(s)Wsds , ( .. ! 2 D) L N0 t] f 0(s)Ws(!), s > 0. &R , (0 t] f (s)dWs 2 { / ! , 4.1 . 1944 ,.) N?] (
" ", , ( ( !
. * , + % % . J , (Ft)t>0 { (D F P ). * fW (t !) t > 0 ! 2 Dg, (D F P ) ! , , .. , 06s
? 4.3 ,
. 1
, , - ,
d ( ) , W (t) { . & ,
W (t), t > 0,
Fs Fs = f W (u) u 2 N0 s]g, s > 0. L (Ft)t>0,
- Pred
(0 1) F ,
K = f(s t] A A 2 Fs 0 6 s 6 t < 1g (12.3) (s t] = ? s > t. 12.1. K $. 2 * B = (s t] A, C = (u v] D 2 K. ? BC = f(s _ u t ^ v] ADg 2 K A D 2 Fs_u AD 2 Fs_u. C B C , (s t] (u v] A D. ) C n B = f(s u] Dg f(u v] (D n A)g f(v t] Dg (12.4)
.. (12.4) 6 (
K ( , D 2 Fs Fu, D n A 2 Fu . 2
A + K ; (12.5) Z ((s t] A !) := (W (t !) ; W (s !))1 A(!): < , EjZ (B )j2 < 1 B = (s t] A 2 K, (12.2), ,
EZ (B ) = E(E(W (t) ; W (s))1 A jFs) = E1 A E(W (t) ; W (s)) = 0:
12.2. 0 $ Z (B !) $ K, "( " = mes P , mes { +
B((0 1)). 2 * B = (s t] A, C = (u v] D 2 K. ? EZ (B )Z (C ) = E1 AD (W (t) ; W (s))(W (v) ; W (u)) = = E1 AD E((W (t) ; W (s))(W (v) ; W (u))jFs_u)): 2 , E((W (t) ; W (s))(W (v) ; W (u))jFs_u)) =
0
(s t] \ (u v] = ? t ^ v ; s _ u (s t] \ (u v] 6= ?:
*!
EZ (B )Z (C ) = P (AD) mes((s t] \ (u v]) = (BC ): 2 , 10.10, Z - M, (
B 2 Pred, (B ) < 1. 229
I f : R ! R, R = (0 1) D, , f 2 PredjB(R). * f R Re f Im f . ) , f 2 L2(R Pred )
Jf =
Z
f ()Z (d) = (t !) 2 R
(12.6)
f g 2 L2(R Pred ) EJf = 0 (Jf Jg) = hf gi (12.7) ( ) { L2(D F P ), h i { L2(R Pred ), ..
hf gi = E
Z1
f (t !)g(t !)dt
0
(12.8)
( I (12.8) ). ' (12.6)
If =
Z1
f (t !)dWt:
0
(12.9)
1 1 . ' , f () Z (d) (12.6) !. < ! . S
, # $
f (t !) =
X
m;1 k=0
fk (!)1 (tk tk+1](t) t 2 (0 1) ! 2 D
(12.10)
f : D ! R fk 2 Ftk jB(R), 0 = t0 < t1 < : : : < tm < 1.
12.3. = # $
.
2
B 2 B(R) t > 0. ?
f(t !) 2 (0 1) D : f (t !) 2 B g =
m;1 k=0
f(tk tk+1] f! : fk (!) 2 B gg 2 Pred
(12.11)
.. f! : fk (!) 2 B g 2 Ftk (k = 0 : : : m ; 1), (12.11) 6
K. 2 & , f (12.10) L2(R Pred )
, fk 2 L2(D Ftk P ). 1 ,
Z1 0
j
j
f (t !) 2dt =
X
m;1 k=0
jfk (!)j2(tk+1 ; tk ):
@ 12.4. 3 # $ f If = 230
X
m;1 k=0
(12.10)
(12.12)
fk (!)(W (tk+1 !) ; W (tk !)):
(12.13)
2 2 , fk , L2(D Ftk P )
Nn X
h(kn)(!) =
j =0
c(knj )1 A(kn) (!) c(knj ) 2 R A(knj ) 2 Ftk j = 0 : : : Nn n 2 N j
(12.14)
.. Ejf ; h(kn)j2 ! 0 n ! 1 (k = 0 : : : m ; 1). ? ,
f (n)
:=
X
m;1 k=0
() h(kn)(!)1 (tk tk+1](t) L;! f n ! 1: 2
( ) / , If (n) L;! If n ! 1. * I { (.
10.5), 2
If (n)
XX
= I(
m;1 Nn
k=0 j =0
c(knj )1 A(kn) (!)1 (tk tk+1](t)) = j
X
m; 1 k=0
h(kn)(!)(W (tk+1 !) ; W (tk !)):
< , Ej
X
m;1 k=0
fk (W (tk+1 !) ; W (tk !)) ;
X
m;1 k=0
h(kn)(W (tk+1 !) ; W (tk !))j2 = =
X
m;1 k=0
Ejfk ; h(kn)j2(tk+1 ; tk ) ! 0 n ! 1:
3 , k 2 Ftk jB(R) Ej k j2 < 1, (k = 0 : : : m ; 1), 0 l rkl := E k l (W (tk+1) ; W (tk ))(W (tl+1) ; W (tl)) = Ej j2(t ; t ) kk 6= = l: k k+1 k (12.15) 3 (12.15) , .. Ej k j2(W (tk+1) ; W (tk ))2 = Ej k j2E(W (tk+1) ; W (tk ))2 < 1 ( k 2 Ftk jB(R) W (tk+1) ; W (tk )??Ftk ), 2 L2(D), 2 L1(D)
, { 0 { h . , , E( jA) = E , ??A Ej j < 1, ! rkk = Ej k j2E(W (tk+1) ; W (tk ))2jFtk ) = Ej k j2(tk+1 ; tk )
rkl = E( k l(W (tk+1) ; W (tk ))E(W (tl+1) ; W (tl)jFtl )) = 0 k < l (12.16) rkl = 0 k > l, (k l = 0 : : : m ; 1). 2 ) , , (12.10) ) (12.13). , , , !
(12.7). ? , $ &$+ $+ $ 2$ $ (12.10) (12.13), $ $ $+ % < . 231
& , $ $ J & % $$+ $ $ & , $ + $ + &$ fNs t)A A 2 Fs 0 6 s 6 t < 1g. * !
f : N0 1)D ! R,
,
f (t !) =
X
m;1 k=0
f (tk !)1 tk tk+1)(t)
(12.17)
0 = t0 < t1 < : : : < tm < 1,
. . f (tk ) 2 Ftk jB(R), k = 0 : : : m;1. !
, (Ft)t>0, - Prog
N0 1) D:
C 2 Prog , C \ fN0 t] Dg 2 Bt Ft t > 0 - Bt = B(N0 t]). I f : N0 1)D ! R ) , t > 0 B 2 B(R)
, f
(12.18)
2 ProgjB(R).
f(s !) 2 N0 t] D : f (s !) 2 B g 2 Bt Ft: (12.19) *! $+ f $, f N0 t] D Bt FtjB(R) ; .!. t > 0 (12.20) 12.5. # $ (12.17) . 2 B 2 B(R) t > 0. C t 2 N0 t1], f(s !) 2 N0 t] D : f (s !) 2 B g = N0 t] f! : f (0 !) 2 B g 2 Bt Ft: C t > t1, N = maxfk : tk 6 tg, f(s !) 2 N0 t] D : f (s !) 2 B g = Nk=0;1fNtk tk+1) f! : f (tk !) 2 B g NtN t] f! : f (tN !) 2 B g 2 Bt Ft: 3 , f! : f (tk !) 2 B g 2 Ftk Ft tk 6 t. 2 * L2 = L2(N0 1) D Prog ), = mes P , .. h 2 L2 E
Z1 0
jh(s !)j2ds < 1:
(12.21)
C h { , , Re h Im h {
. & , f (12.17) L2 , f (tk ) 2 L2(D Ftk P ), k = 0 : : : m, (12.12). 8 $ f ( .. (12.17)) fW (t) t > 0g (
( (12.2) (Ft)t>0) $ 2$ (12.13)
If = 232
X
m;1 k=0
f (tk !)(W (tk+1) ; W (tk )):
(12.22)
1 . C (12.17)
f (t !) =
r ;1 X j =0
g(sj !)1 sj sj+1 )(t) mes P { ..,
P
;1 g (s ! )(W (s ) ; W (s )) P { .. 0 = s0 < : : : < sr < 1, rj=0 j j +1 j (12.13). B
,
f (tk !)(W (tk+1) ; W (tk )) = f (tk !)(W (u) ; W (tk )) + f (tk !)(W (tk+1) ; W (u)) tk < u < tk+1 ( ( W (t) W (t !)). *! , N0 1). ? ! P { .. ? 12.6. (12.22) W (t) ( ). < (
, , ""
, . 12.7. 3 # $ f g 2 L2 (12.7). P 2 * f (12.17), g(t !) = mj=0;1 g(tj !)1 tj tj+1)(t). , 6 , ,
, ( N0 1). )
XX
m;1 m;1
(If Ig) =
k=0 j =0
Evkj
(12.23)
vkj = f (tk !)g(tj !)(W (tk+1) ; W (tk ))(W (tj+1) ; W (tj )). Evkj (12.15). ?, EIf = 0 f 2 L2, (12.2). 2 12.8. # $ (12.17) L2. 2 8 u 2 L2N0 1) = L2(N0 1) B(N0 1)) mes) > 0
1 X
u (t) =
k=0
u(k)1 k (t)
(12.24)
]k = ]k () = Nk (k + 1)), k = 0 1 : : :
u(0)
= 0
u(k)
= ;1
Z
k;1
u(t) dt k > 1:
(12.25)
*
,{0 {h m > 1
Z
m
ju j
(t) 2 dt =
j
j 6
u(m) 2
Z
m;1
ju(t)j2dt
(12.26) 233
!
Z1 0
ju(t)j2dt =
u () 2 L2N0 1). B, *
u(t) =
X
m;1 j =0
1 Z X k=1 k
ju (t)j2dt 6
1 Z X
Z1
k=1k;1
ju(t)j2dt = ju(t)j2dt 0
u () ! u() L2N0 1) ! 0.
(12.27)
(12.28)
rj 1 tj tj+1 )(t) 0 = t0 < : : : < tm < 1P rj 2 C j = 0 : : : m ; 1:
(12.29) ? ]k;1 ]k Ntj tj+1), u (t) = u(k) = rj = u(t) t 2 ]k . ? , u u e j = ]ij ]ij +1 tj , j = 0 : : : m. 8 , ( ) ] tj , (12.28). ? , (12.26)
Z
Sm e j
ju(t) ; u j
(t) 2dt 6 2
j=0
Z
Sm e j
j
j
u(t) 2dt + 2
j=0
Z
Sm e j
ju j
j=0
(t) 2dt 6 4
Z
Sm 0i
ju(t)j2dt
j=0
1 ]ij ]ij +1 (];1 = ?), u 2 L2 N0 1) R ju(]t)0ij2dt= !]i0j ; mes B ! 0. B * u | L2N0 1). B " > 0 v(") (12.24), ku ; v(")k < ", k k |
L2N0 1). ? ku ; u k 6 ku ; v(")k + kv(") ; v(")k + kv(") ; uk 6 2" + kv(") ; v(")k < 3" . 3 , v(") ; u = (v(") ; u) , (12.27). 3 + $+ h(t !) 2 L2 R ! 2 D = 1=n (12.24), (12.25) h1=n(t !). C h(t !) dt (
, k . ) (12.21) I ,
Z1 0
jh(t !)j2dt < 1 ..
R
/ , . . ! (
h(t !) dt k = 0 1 2 : : : . , k R
, I h(s !) ds Fk jB(C )- . 3 k;1
(12.20). * (12.27) (12.21), , h1=n(t !) { L2 ( , ). (12.27)
Z1 0
234
Z1
jh(t !) ; h1=n(t !)j2dt 6 4 jh(t !)j2dt .. 0
/ (12.28)
Z1 0
jh(t !) ; h1=n(t !)j2dt ! 0 .. n ! 1.
/ , 2
Z1
E jh(t !) ; h1=n(t !)j2dt ! 0 n ! 1: 0
< , (12.24)
u (t N ) =
N X k=0
u(k)1 k (t)
(12.28) > 0 u( N ) ! u() L2N0 1) N ! 1: /
, h1=n(t !) n 2 N ,
Z1
jh1=n(t !) ; h1=n(t !P N )j2dt ! 0 N ! 1: 2 0 @+ $+ I $ $ L2. & E
If = l:i:m:Ifn
(12.30)
L fn ;! f n ! 1, (12.30), , 2 L (D). fn 12.8, , (12.30) ( , ( 12.7. * ! (. (12.7)) (If Ig) = hf gi E(If ) = 0 f g 2 L2: (12.31) ) ( : 2
If =
Z1 0
f (t !) dWt f 2 L2 :
(12.32)
B 0 6 t1 < t2 6 1 f 2 L2
Zt
2
t1
f (t !)dWt =
Z1 0
f (t !)1 t1 t2)(t) dt:
(12.33)
B f f 1 t1 t2) . ( L2, ( mes P -.. 4.7. t1 = t2 $ $ (12.33) ;. * (12.32), , 0 6 v < s < u 6 1
Zu v
f (t !) dWt =
Zs v
Zu
f (t !) dWt + f (t !) dWt .. s
(12.34) 235
? 12.9. B T > 0 (..)
R f (t !) dW , (12.33) T
0
t
) (12.17), ( tn = T ( , f (t !) = ftn;1 (!) t 2 Ntn;1 T ]). (12.18) ! t 2 N0 T ]. @ 12.10. # $ f (t !), 0 6 t 6 T < 1, .
ZT
f (t !) dWt = ln:i!1 :m:
n;1 X
f (t(kn) !)(W (t(kn+1) ) ; W (t(kn)))
k=0 0 0 = t(0n) < : : : : : : < t(nn) = T , . .
n = 06max (t(n) ; t(kn)) ! 0 (n ! 1): k6n;1 k+1 n;1 2 I f (t(kn) !)1 t(kn) t(kn))(t) { L2 k=0
(12.35)
P
Ifn =
n;1 X k=0
f (t(kn) !)(W (t(kn+1) ) ; W (t(kn))):
L2 < , fn ;! f N0 T ]. B
,
ZT 0
Ejf (t !) ; fn(t !)j2dt 6 T sup Ejf (t !) ; f (s !)j2 ! 0 jt;sj6n
n ! 0 , , , ! (
). 2 * ! , ,
ZT 0
Wt dWt = l:ni:!m0:
n;1 X
2 W (t(kn))(W (t(kn+1) ) ; W (t(kn))) = W2T ; T2 : k=0
*, % (12.35) f & $ Nt(kn) t(kn+1) ) $ + ; +. ', n;1 2 X l:ni:!m0: W (t(kn+1) )(W (t(kn+1) ) ; W (t(kn))) = W2T + T2 : k=0 2 I f 2 L
Yt =
Zt 0
f (s !) dWs t > 0
R
(12.36)
(12.33) 01 f (s !)1 0 t)(s)dWs (Y0 = 0). B (12.36) It(f ) (f B )t, B = fB (s) s > 0g { . 236
@ 12.11.
t > 0 " Yt , " # (12.36), ( ), "( : 1) (Yt Ft)t>0 | 7 2) . . ! Yt(!) N0 1)7 3) .#. Yt(!), 0 6 t < 1, . 2 8 & 1). * f 2 L2 f (t !) = 0 .. 0 6 t < s. ? ,
E(If j Fs) = 0 .. (12.37) C f | (12.17) L2 s > 0, s = t1. ? E(If j Ft1 ) =
n;1 X k=1
E(E(f (tk )(Wtk+1 ; Wtk ) j Ftk ) j Ft1 ) =
=
n;1 X k=1
& ,
E(If j F0) =
E(f (tk )E(Wtk+1 ; Wtk j Ftk ) j Ft1 ) =
n;1 X k=0
n;1 X k=1
E(f (tk ) 0 j Ft1 ) = 0:
E(E(f (tk )(Wtk+1 ; Wtk ) j Ftk ) j F0) = 0 ..
(12.38)
L f n ! 1. < , g = * 12.8 fn ;! n 2 L 2 = fn1 s 1) 2 L . . f (t) = 0 .. 0 6 t < s, gn ;! f . / , 2 ( ) 2 ( ) Ign L;! If n ! 1. ' E(Ign j Fs) L;! E(If j Fs). B
, L2 ( ) L2 ( ) n ;! , E( n j A) ;! E( j A), A F . *
), EjE( j A) ; E( n j A)j2 = EjE( ; n j A)j2 6 E(Ej ; n j2 j A) = Ej ; n j2: 8 , E(Ign j Fs) = 0 .., (12.37) ( . (12.34) (12.37), s 6 t 2
Z1
E(Yt j Fs) = Ys + E
Rt
0
f (u !)1 s t)(u) dWu j Fs = Ys ..
(12.39)
3 , f (s !) dWs
f Ft jB(R)- 0
(12.18), (12.22), (12.30) 4.7. ) ,
1) . * ! , Yt , ! ( - ). 8 & 2). C f | , (12.17), 8 f (t0)(Wt ; Wt0 ) = f (0)Wt t 2 N0 t1) > > m ; 1 > X Zt < f (tk )(Wtk+1 ; Wtk ) + f (tm)(Wt ; Wtm ) t 2 Ntm tm+1) f (s !) dWs = > k=0 m = 1 : : : n 0
> > > :
n X k=0
f (tk )(Wtk+1 ; Wtk )
t > tn:
(12.40) 237
L B f 2 L2 fn ;! f n ! 1. 8 Rt fn(s !) dWs 0 (. (12.40)), , ! ! N0 1). ) fn fnk , 2
Z1
E jfnk+1 (s !) ; fnk (s !)j2ds 6 2;k k 2 N:
(12.41)
0
* , ) | , L2, T > 0
5.17
1)
" > 0, n m 2 N
Zt
ZT 2 Zt ; 2 P sup fn (s !) dWs ; fm(s !) dWs > " 6 " E (fn ; fm)dWs : 06t6T
0
0
(12.42)
0
) (12.41), (12.42) (12.31) ,
Zt 1 X 1 P sup (fnk ; fnk )dWs > k2 6 k42;k < 1: 06t6T k=1 k=1
1 X
+1
0
*! 0{, . . ! k > K(!)
Zt sup (fnk ; fnk )dWs 6 k;2: 06t6T +1
0
C N0 T ] .. , ! .. N0 T ] . ? , t > 0
Zt 0
fn1 (s !)dWs +
XZ
N ;1 t k=1 0
(fnk+1 ; fnk )dWs =
Zt 0
L2 ( )
fnN (s !) dWs ;!
Zt 0
f (s !) dWs
( ) :: N ! 1. ' t 2 N0 T ] k (t) ;! (t) k (t) L;!
(t), L2 ( ) (t) = (t) .. t 2 N0 T ] ( . . k (t) ;! (t), mj (t) ! (t) .., ! (t) = (t) ..). ? , N0 T ] Yt. 1 N0 n], n 2 N. ? , Xt = Zt .. t 2 N0 T ] ! .. N0 T ], P (! : Xt(!) = Zt(!) t 2 N0 T ]) = 1. B
, N0 T ]
MT . ? Xt Zt " > 0 P (! : sup jXt (!) ; Zt(!)j > ") = P (! : sup jXt(!) ; Zt(!)j > ") = 0: 2
t2 0 T ]
t2MT
) , 6
, , Yt .. N0 1). /
2) . 238
8 & 3). *
1) , Yt Ft j B(R)- t > 0 ( -
). , , N0 1) . . ! Yt
,
12.12. $ X = fX (t !) t 2 N0 T ] ! 2 Dg , . . N0 T ). ! Xt 2 Ft jB (R) t 2 N0 T ], (Ft)t2 0 T ] { # $ , $ ( # $ . 2 * m > 2 X (t !) =
X
m;1 k=0
X (tk !)1 k (t) t 2 N0 T ] ! 2 D
(12.43)
0 = t0 < : : : < tm+1 = u, ]0 = N0 t1], ]k = (tk tk+1], k = 1 : : : m ; 1. ? , , . < ( . J , ! 2 D0, P (D0) = 1, X N0 T ),
Xn (t !) =
X (q (t) !) n
X (T !)
qn (t) := (N2n t] + 1)2;n 6 T qn (t) > T
t 2 N0 T ], ! 2 D, n 2 N, N] . ? Xn (12.43). *
Vn (t !) = Xn (t !)1 0 (!) t 2 N0 T ] ! 2 D: 2 , Vn t 2 N0 T ], ! 2 D lim V (t !) = X (t !)1 0 (!): n!1 n < 4.6. 2 ? 12.11 . 2 *c f 2 L2 Yt, ( 12.11 sup EjYtj 6 sup(EjYtj t>0
t>0
R
2 )1=2
= sup(E t>0
Zt 0
j
j
f (s !) 2ds)1=2
6 (E
Z1 0
jf (s !)j2ds)1=2
, Y1 = 01 f (s !)dWs , , (12.39) t = 1, 5.14 , (Yt Ft)t2R+ f1g { $ , F1 = _t>0Ft.
12.8 ,
f 2 L2T = L2(N0 T ] D Prog mes P ) T < 1,
12.11 t 2 N0 T ].
. 239
3 - % Pred Prog & $ ; $ Xt , t 2 T R, $ & $ t 2 T " $ $ % "
(t !). * " (
". & , N0 1) ( (0 1))
T R ( ! N?], x6.2). J , Pred
, - ,
(T \ (t 1)) B , t 2 T , B 2 Ft. 3
A T D Prog,
A \ ((;1 t] D) 2 Bt Ft t 2 T Bt = B(T \ (;1 t]). < , ) ( (0 1), ) N0 1). * ! -
, ..
. N?], x6.2, - Pred Prog
T D (T R), , , . 153
@ 812.1. +"
# $ " # $ - (Ft)t2T R ( Pred Ad, Prog Ad, " A T D - Ad T D, f! : (t !) 2 Ag Ft t 2 T ). ! T - t0 , " # $ Xt (t 2 T ) , Xt0 () = const.
D. 12.2. B , T { , ,
Pred Prog.
D. 12.3. * , -
.
D. 12.4. * fXt t 2 T
Rg {
-
{ ( (Ft)t2T R). B , X (!)(!)
f! : (!) < 1g - F B(R). A< $ $ , $ 2$ (0 1). * (Ft)t>0 { (D F P ), fW (t) t > 0g { . L
H, (
f , R (t !) ( .. f 2 B(0 1)FjB(R)) , f (t ) 2 FtjB(R) t > 0 E 01 f 2(t !)dt < 1. ? (
(., ., N?, I, . 45]). @ 812.5 (8 %). H L2((0 1) D A ), A { - Pred = mes P , mes { + (0 1). ) 12.8 (12.10), , , 12.5. D. 12.6. B , f 2 L2((0 1) D A ), h 2 H , f (t !) = h(t !) .. . 240
D. 12.7. * { , (!) 6 T ! 2 D
(T { ). < It(f ) (12.36). B , I (f ) = IT (f 1 (0 ]), I (f ) := I (!)(f ). D. 12.8. B , f : N0 1) ! R
/ DN0 1), , f { ! cadlag- , .. ( t > 0 N0 1), It(f ), t > 0 { . ' . / " " . / X = fXt t > 0g Rm , ( ), a > 0
b > 0,
Law(Xat t > 0) = Law(bXt t > 0): (12.44) ) , < (t ! at) $ $ & +$ $ , $ < (x ! bx). C -
(12.44) a > 0 b = aH , X $ 8 H . D = 1=H # " X . ' (. (3.31)), fBH (t) t > 0g 0 < H 6 1 cov(BH (s) BH (t)) = s2H + t2H ; js ; tj2H s t > 0: D. 12.9. < 6 , fBH (t) t > 0g { \ . * BH J.'. , 1940 . N?], 9 . ? ( , "fractional")
0. 3 k. ' 1968 . N?], ( .
@ 812.10 (( +%$, 3 F). 3 0 < H < 1 t > 0
Z0
BH (t) = cH f
;1
N(t ; s)H ;1=2 ; (;s)H ;1=2]dW
s+
Zt 0
(t ; s)H ;1=2dWs g
(12.45)
2 (1) = 1 (., ., L?, "( cH , EBH . 281]), fWs s > 0g fBs = W;s g s 6 0 { $.
I ( )
, , (. N?], N?]). / BH (t) 0 < H 6 1
, H = 1=2 ( .. ) ! ( .. , , . N?, . 4]). C (12.45) H Ht ( .. jHt ; Hs j 6 cjt ; sj, > 0) (0 1), , # . 1
N?]. 241
D. 12.11. (. N?]). B , BH n ! 1
X Hbn := ln(n;1 jBH (k=n) ; BH ((k ; 1)=n)j)= ln(1=n) ! H n
k=1
..
0 , .. N0 1) Xt ## $ , 1 t > 0
Zt
Zt
0
0
Xt = X0 + f (s !) dWs + g(s !) ds
(12.46)
f 2 L2(N0 1)), g : N0 1) D ! R ,
P
Z1 0
jg(s !)j ds < 1 = 1
(12.47)
( fWt t > 0g (Ft)t>0,
( (12.2)). f g (12.46) .. . / (12.46)
dXt = f (t !) dWt + g(t !) dt: (12.48) B F F (t), Ft. & , , t 2 Nu v], 0 6 u < v < 1. ) , ( ), . ) . & & < $ ; +$ $,
. @ 812.12 ( 2$). $ Xt, t > 0, ## $ ( # $ f g, "( - ). # $ h : N0 1) R ! R , ( " @h=@t, @ 2h=@x2, sup j@h(s x)=@xj 6 M0 < 1
(12.49)
= h(t Xt), t > 0, @h (t X )dX + 1 @ 2h (t X )(dX )2 dYt = @h ( t X ) dt + t @t @x t t 2 @x2 t t (" (12.48), :
(12.50)
dt dt = dt dWt = dWt dt = 0 dWt dWt = dt:
(12.51)
s>0 x2R
$ Yt
(dXt)2
1 , . .
242
z>0
Zz
h(z Xz ) = h(0 X0 ) + f (s Xs ) @h @x (s Xs )dWs + 0
+
Zz @h
@h (s X ) + 1 f 2 @ 2h (s X ) ds: (12.52) ( s X ) + g s s s @t @x 2 @x2
0
3 ! , . 812.13. ?? # $ f g , . .
f (s !) =
X
m;1 j =0
fj (!)1 tj tj+1)(s) g(s !) =
0 = t0 < t1 < : : : < tm = z , fj ,
X
m;1 j =0
gj (!)1 tj tj+1)(s)
(12.53)
= f (tj !) gj = g(tj !) " Ftj j B(R)-
Efj2 < 1 j = 0 : : : m:
-
(12.54)
(12.52). 2 2 )
(12.52) , #
Ztj @h 2h @h 1 @ 2 h(tj+1 Xtj ) ; h(tj Xtj ) = @t (s Xs ) + gj @x (s Xs ) + 2 fj @x2 (s Xs ) ds + tj Ztj @h +1
+1
+1
+
tj
fj @x (s Xs) dWs j = 0 : : : m ; 1: (12.55)
j 2 f0 : : : m ; 1g ( u = tj , v = tj+1. ) (12.46) (12.53)
Xt = Xs + (Wt ; Wu)fj + (t ; u)gj t 2 Nu v]:
(12.56)
L Nu v] u = s(0n) <: : :< s(nn) = v, (n) (n) ; s n = i=0max ( s i +1 i ) ! 0 n ! 1: ::: n;1
(12.57)
Xs (!)) .. * aj (s !) = fj (!) @h(s@x Nu v] (Xs | .. , . (12.56), @h=@x ). B
, (12.49) (12.54) jaj (s !)j 6 M0jfj j 2 L2(D). , , aj (s !) Nu v] . / , 12.10 ( u v)
Zv @h n;1 X fj @x dWs = l:i:!m0: fj @h @x (si Xsi )(Wsi ; Wsi ) n u
i=0
+1
(12.58) 243
( si s(in). ) , , n;1 X i=0
h(si+1 Xsi+1 ) ; h(si Xsi ) ; fj @h @x (si Xsi )(Wsi+1 ; Wsi ) ! Zv @h 2h @h 1 @ 2 ! @t (s Xs ) + gj @x (s Xs) + 2 fj @x2 (s Xs ) ds .. (12.59) u
( , . . fnk g, nk ! 0 k ! 1. ? ,
h(si+1 Xsi+1 ) ; h(si Xsi ) = (2) = (h(si+1 Xsi+1 ) ; h(si Xsi+1 )) + (h(si Xsi+1 ) ; h(si Xsi )) = ](1) i + ]i : (12.60) (2) * ](1) i ? , ]i | , (2) @h si Xsi+1 )(si+1 ; si) + @h ](1) i + ]i = @t (e @x (si Xsi )(Xsi+1 ; Xsi ) + 2 + 21 @@xh2 (si Xesi )(Xsi+1 ; Xsi )2 i = 0 : : : n ; 1 (12.61) esi = sei(!) 2 (si si+1), Xesi (!) | Xsi (!) Xsi+1 (!). < ,
, sei Xesi , @h s (!) X (!)) | .
](1) i si+1 i | , @t (e 2h @ J , 2 (si Xesi ) | ( Xsi (!) = Xsi+1 (!), @x Xesi (!) = Xsi (!) (12.61)). * (12.56), (12.61), , (12.59)
n;1 X @h
2h @ @h 1 (esi Xsi+1 )(si+1 ; si) + @x (si Xsi )gj (si+1 ; si) + 2 @x2 (si Xesi ) @t i=0 2 2 2 2 Nfj (Wsi+1 ; Wsi ) + 2fj gj (Wsi+1 ; Wsi )(si+1 ; si) + gj (si+1 ; si) ] :
(12.62)
B
12.13 812.14. $ Xs . . Nu v], # $ G N0 1) R.
M (GP ) = supfjG(s Xs ) ; G(r Xq )j : s r q 2 Nu v] js ; rj < jr ; qj < g ! 0
. .
! 0: (12.63)
2 I Xs (!) . . ! Nu v]. / , ! 2 D0 (P (D0 ) = 1) t 2 Nu v] jXt(!)j 6 L(!) < 1. B L > 0 G(s x) Nu v] N;L L]. *! " > 0 = (" L) > 0, jG(s x) ; G(r y)j 6 ", 244
js ; rj 6 jx ; yj 6 . < , > 0 ! 2 D0 (
= ( !) > 0,
sup
s q2 u v] js;qj<( !)
jXs(!) ; Xq (!)j 6 : 2
(12.64)
*
12.13. * 12.14 @h=@t
X n;1 n;1 X @h @h (esi Xsi )(si+1 ; si) ; (si Xsi )(si+1 ; si) 6 @t @t i=0 i=0 @h +1
6 M @t P n (v ; u) ! 0 .. n ! 0: (12.65)
8 @h=@t .. Xt, , n;1 X @h
Zv @h
i=0
u
@t (si Xsi )(si+1 ; si) !
@t (s Xs ) ds .. n ! 0:
(12.66)
@h=@x , n;1 X @h
(si Xsi )gj (si+1 ; si) ! @x i=0
Zv @h u
@x (s Xs )gj ds .. n ! 0:
(12.67)
0 Nu v], !
X n;1 2 @ h fj gj e 2 (si Xsi )(Wsi ; Wsi )(si+1 ; si ) 6 @x i=0 +1
6 C (!)jfj gj j(v ; u) sup jWy ; Wr j ! 0 .. n ! 0 (12.68) y r2 u v] jy;rj6n
C (!) = maxfj@ 2h(t x)=@x2j : t 2 Nu v] jxj 6 tmax jX (!)jg: 2 u v] t J ,
X n;1 gj2 @ 2h2 (si Xesi )(si+1 ; si)2 6 gj2C (!)(v ; u)n ! 0 i=0 @x
..
(12.69)
(12.70)
? , ( fnkg Nu v] n;1 2 X @h
e 2 (si Xsi )(Wsi+1 ; Wsi ! @x i=0 )2
Zv @ 2h u
@x2 (s Xs) ds ..
(12.71)
* 12.14 245
X n;1 2 n ;1 2 X @ h @ h 2 2 e 2 (si Xsi )(Wsi ; Wsi ) ; 2 (si Xsi )(Wsi ; Wsi ) 6 @x @x i=0 @ 2h i=0X n;1 +1
+1
6 M @x2 P n
i=0
(Wsi+1 ; Wsi )2 ! 0 .. nk ! 0 (12.72)
, n;1 X i=0
( ) (Wsi+1 ; Wsi )2 L;! v ; u n ! 0: 2
(12.73)
B
, ( ,
X 2 X n;1 n;1 2 E (Wsi ; Wsi ) ; (si+1 ; si) = D(Wsi ; Wsi )2 = i=0 i=0 n;1 X +1
+1
=2
i=0
(si+1 ; si)2 6 2(v ; u)n: (12.74)
/ , (
fnk g, (12.73)
.. B,
X n ;1 n;1 @ 2h(si 2Xsi ) (Wsi ; Wsi )2 ; X @ 2h(si2Xsi ) (si+1 ; si) 6 @x @x i=0 i=0 n;1 X 2 +1
6 C (!)
i=0
j(Wsi ; Wsi ) ; (si+1 ; si )j: (12.75) +1
C (!) (12.69), (12.74),
X n;1
E
i=0
2
j(Wsi ; Wsi ; (si+1 ; si)j 6 2(v ; u)n: +1
)2
*! (
fn0k g fnk g, (12.75) 0 .. k ! 1 ( nk n0k ). ' , @ 2h=@x2 n;1 2 X @h
2 (si Xsi )(si+1 ; si ) ! @x i=0
Zv @h u
@x (s Xs) ds .. n ! 0:
(12.76)
? , .. :: :: fnk g k ! 1. < , nk ;! , nk ;!
nk = nk .., = .. 2 12.13 . 2 & $ +$ $ 12.12. * 12.8 f (n) ! f n ! 1
L2(N0 1)). J 12.8, (12.47), g(n), ( . . ! g
L1(N0 1)). 246
* t > 0
Zt
Zt
0
0
Xt(n) = X0 + f (n)(s !) dWs + g(n)(s !) ds: * 12.13 .. z > 0 n 2 N
h(z Xz(n) ) ; h(0 X0 ) = +
Zz @h 0
(12.77)
Zz @h 0
(n) (n) @x (s Xs )f (s !) dWs +
(n) ) + g (n) (s ! ) @h (s X (n) ) + ( s X s s @t @x
1 (f (n)(s !))2 @ 2h (s X (n)) ds: (12.78) s 2 @x2
8 &, $ $ (12.78) $ $ +$ fnk g $ $$ $$$ $ (12.52). * 12.11
(5.29), " z
P ( sup jXs(n) ; Xs j > ") 6 06s6z
Zs " Zs " ( n ) ( n ) 6 P sup (f ; f )dWs > 2 + P sup (g ; g)ds > 2 6 06s6z 06s6z 0 0 2 2 Zz Zz "
6 " E (f (n) ; f )2 ds + P 0
0
jg(n) ; gjds > 2 ! 0 n ! 1:
(12.79)
h , z > 0 P h(z Xz(n) ) ; h(0 X0 ) ! h(z Xz ) ; h(0 X0 ):
(12.80)
/ (12.31), (12.49) Xt Xt(n) (n 2 N),
2 Zz @h Zz @h ( n ) ( n ) E @x (s Xs )f (s !)dWs ; @x (s Xs)f (s !)dWs = 0 0 2 Zz @h @h E @x (s Xs(n) )f (n)(s !) ; @x (s Xs )f (s !) ds 6
=
0
2 Zz @h ( n ) 6 2 E @x (s Xs ) (f (n)(s !) ; f (s !))2ds + 0 @h 2 Zz @h
+ 2 E(f (s !))2 @x (s Xs(n)) ; @x (s Xs ) ds 6 0
6 2M02
Zz 0
E(f (n)
;
f )2ds + 2
Zz
E]n ds
(12.81)
0
247
@h
2 @h ]n ; @x (s Xs) 6 4M02f (s !)2: (12.82) @h , (12.79) s > 0 , , @x @h (s X (n)) ! P @h (12.83) s @x @x (s Xs ) ,
(s !) = (f (s !))2
(n) ) ( s X s @x
P ]n(s !) ! 0 n ! 1:
(12.84)
) (12.84) (12.82) , E]n(s !) ! 0, n ! 1, s > 0, E]n(s !) 6 4M 2E(f (s !))2. ? , 2
Zt 0
E](s !) ds ! 0 n ! 1:
(12.85)
? , 2 , ( (12.78),
( (12.52), ( Nu v]. 8 , f (n) ! f L2(N0 1)), fnk g ,
Zz 0
(f (nk )(s !) ; f (s !))2ds ! 0 .. k ! 1:
(12.86)
* (12.79), fnk g ( , fn0k g fnk g),
X k
P ( sup jXs(nk ) ; Xs j > 2;k ) < 1: 06s6z
(12.87)
? Xs(nk ) Xs .. N0 z]. *! 1 @h @h ( n ) (12.88) "1(nk !) := sup @x (s Xs k ) ; @x (s Xs ) ! 0 k ! 1: s2 0 z] * ! ,
B12.14, @h=@x N0 z] N;L L]. / ,
Zz @h @h ( n ) (s Xs k ) ; (s Xs) ds 6 z"1(nk !) ! 0 @x @x 0
.. k ! 1:
(12.89)
* fnk g ,
"2(nk !) :=
Zz 0
248
jg(nk )(s !) ; g(s !)jds ! 0 .., k ! 1
(12.90)
( fn00k g fn0k g, - nk ). ? (12.88) (12.90)
Zz @h @h g(nk )(s !) (s Xs(nk )) ; g(s !) (s Xs) ds 6 @x @x 0 z @h Z 6 "1(nk !) jg(s !)jds + sup @x (s Xs(nk )) "2(nk !): s2 0 z] 0
(12.91)
8 (12.47), , (12.91) .. k ! 1, (12.88) , @h @h ( n ) sup (s Xs k ) ! sup @x (s Xs) < 1 .. (12.92) s2 0 z] s2 0 z] @x / ( (12.86) , (@h @ 2 h (12.92)) , k ! 1 @x @x2
Zz 2h 2h @ @ (f (nk )(s !))2 2 (s Xs(nk )) ; (f (s !))2 2 (s Xs) ds ! 0 @x @x
..
(12.93)
0
? B12.12 . 2 ? 812.15. ) ( ) . ', X { Rm, { Rm ( , { ). 3 15.19, 15.21 N?]. & , ( (12.49) ( (13.84) ). , ( ) (12.50) 13. 812.16.
ZT
Ws dWs:
(12.94)
0
2 B12.12, Xt = Wt h(t x) = 12 x2. < ,
! . ? Yt = h(t Wt) = 21 Wt2
@h dW + 1 @ 2h (dW )2 = 0 + W dW + 1 (dW )2 = W dW + 1 dt: dYt = @h dt + t t t t @t @x t 2 @x2 t 2 t 2 (12.95) / , 1 d 2 Wt2 = Wt dWt + 12 dt:
? ,
(. (12.46)) . . W0 = 0, 1 W 2 = Z W dW + 1 t t > 0 s s 2 t 2 t
(12.96)
0
249
Zt 0
Ws dWs = 21 Wt2 ; 21 t: 2
(12.97)
D. 12.17. / ( ) , f = f (t) C 1 (12.1). * f = f (t !), (12.1) . D. 12.18. * Xt = expft + Wtg, t > 0, fWt t > 0g { , 2 R. ' dXt . , E (W )t = eWt ;t=2 dE (W )t = E (W )t dWt . ' 0 1 jfs 2 N0 t] : W (!) 2 (;" ")gj Lt(!) := "lim (12.98) s !0+ 2" j j 2 , (12.98) ( , (
) L2(D F P ). @ 812.19 (@ ). ' jWtj =
Zt 0
#
sgn(Ws)dWs + Lt t > 0
(12.99)
sgnx { x. B
! ) h(x) = jxj, ., ., N?, . 42], N?, ?]. * ( ) h 2= C 1 2 . N?], N?]. < $ %% $ 2$ (. 13). ) J1. 9 , f 2 J1, f : (0 1) D ! R,
P
Z t 0
f 2(s !)ds <
1 = 1 t > 0:
(12.100)
) , L2 fn , n 2 N, It(fn) ,
Zt 0
P (f (s !) ; fn (s !))2ds ! 0 n ! 1:
(12.101)
< , fIt(fn )gn2N . / , (
,R It(f ), P
, It(fn) ! It(f ), n ! 1. * It(f ) (0 t] f (s !)dWs , (f W )t. S fBt t > 0g, (f B )t. D. 12.20. B , (
ffn n 2 Ng, (12.101). B , f 2 J1, (
.. It(f ), t > 0. < , f 2 J1 It(f ) % %$+ $ , $ , .. (
, n " 1 .. (n ! 1) n " " It n (f ) := It^ n (f ), t > 0, . 250
D. 12.21. L f 2 J1
Zt
Zt 1 Zt = expf f (s !)dWs ; 2 f 2(s !)dsg t > 0 (12.102) 0 0 " ". B , dZt = Ztf (t !)dWt. J
J1(N0 T ]) f : N0 T ] D ! R,
( (12.100) t 2 T . / ( (. B12.5), , $ $ % . @ 812.22 (/ ). X = X (!) FT jB(R)- T > 0, (Ft )t>0 { # $ . "( . 1. ! EX 2 < 1, $
f = (f (s !))s2 0 T ] 2 L2(N0 T ]) ,
X = EX +
ZT 0
f (s !)dWs
. .
(12.103)
2. ! EjX j < 1, (12.103) $ f 2 J1 (N0 T ]). 3. ! X ( .. P (X > 0) = 1) EX < 1, $ f 2 J1N0 T ] , X = ZT EX , ZT (12.102) .
D. 12.23. < 6 , X , ( B?? , .. X (!) = g(W (s !) 0 6 s 6 T ), g : C N0 T ] ! R g 2 B(C N0 T ])jB(R). ) 12.22 ( +$ $ $ $ % $ . @ 812.24 (/ ). M = (Mt Ft)t2 0 T ] { , # $ (Ft )t>0 { , 312.22.
1. < $ f = (f (s ! ))s2 0 T ] 2 L2 (N0 T ]) , Mt = M0 +
Zt 0
f (s !)dWs t 2 N0 T ]:
(12.104)
2. ! M , (12.104) $ f 2 J1 (N0 T ]). 3. ! M , $ f 2 J1(N0 T ]) , Mt = M0 Zt , Zt { , t 2 N0 T ].
B
B12.22, B12.24 , N?], ) N?], B N?], , ., ., N?], N?, ?], N?]. / ( ! . /
251
fW (t) t 2 N0 1]g , V = inf EjW ; 0max W j2 :
6s61 s
3 , (, ) ( , ! , .. ) ( . & , W p ( max06s61 Ws , EW = 0 E max06s61 Ws = 2= (
). *!
Ve = a2infR EjW + a ; 0max W j2: 6s61 s
2 ( ! ),
Ve = V ; 2=:
*
St = 0max W t 2 N0 1]: 6s6t s
@ 812.25 (H). *-
- #-
p
= inf ft 2 N0 1] : St ; Wt = z 1 ; tg
z
4X(z) ; 2z(z) ; 3 = 0 X { # $ . z = 1:12 : : : , V = 2X(z ) ; 1 = 0:73 : : : .
,
! max Ws = a +
06s61
a = const ( ! )
Z1 0
f (s !)dWs
p
f (s !) = 2f1 ; X((St ; Wt)= 1 ; t)g s 2 N0 1] ! 2 D:
A%% ; $ 2$ N?]. * ! . @ 812.26 (2$). t > 0 n 2 N #
Z
252
Hn {
Z
n=2 pt ::: dWs1 : : : dWsn = tn! Hn W t 06s1 6:::6sn 6t D n, .. dn (e;x2=2) n = 0 1 : : : Hn(x) = (;1)nex2=2 dx n
(12.105)
< . N?]. C( %% $ , % % , & % &. L M = = (Mt)t2R+
(D F (Ft)t>0 P ) , M0 = 0. , Mc2. 0 , (Ft)t>0
( F0 N
, ( P - ). D. 12.27. B , M 2 Mc2, ! N0 T ]. 1 ,
IT (X ) =
ZT 0
Xt(!)dMt(!)
(12.106)
, 2 { / . / (Ft)t>0 ( , ) A = (At)t2R+ "( , . . ! A0(!) = 0, At(!) { ( t 2 N0 1) EAt < 1 t 2 R+. * A , EA1 < 1, A1 = limt!1 At. ( A = (At Ft)t2R+ , M = (Mt Ft)t2R+ E
Z
(0 t]
Ms dAs = E
Z
(0 t]
Ms; dAs 0 < t < 1:
< S { (Ft)t>0 Sa { , P ( 6 a) = 1, a > 0. 9 , X = (Xt Ft)t>0 D,
fX g 2S X 2 DL, 0 < a < 1 fX g 2Sa .
/ ; ; + $ ; $ + +$ $. @ 812.28 ( & 8 % { ( ). # $ (Ft)t>0 . ! X = (Xt Ft )t2R { X 2 DL, Xt = Mt + At 0 6 t < 1 (12.107) M = (Mt Ft)t2R { , A = (At Ft )t2R { +
+
+
"( $. $ A ( " , .. . . ). ! X 2 D, M { , A { $.
D. 12.29. * X = (Xt Ft)t>0 { . B , X 2 DL, ( . 1. X > 0 .. 2. X (12.107).
F $ $ $ M = (Mt Ft)t>0 2 Mc2. * hM i { ( ( ),
( B 253
{ 3 Mt2, t > 0 ( ?? ??). * X = fXt t > 0 ! 2 Dg , (
( ftng1n=0 t0 = 0 limn!1 tn = 1,
ffn (!)g1n=0 0 < c < 1, supn>0 jfn(!)j 6 c ! 2 D, fn 2 Ftn jB(R)
Xt (!) = f0(!)1 f0g(t) +
1 X k=0
fk (!)1 (tk tk+1](t) 0 6 t < 1:
(12.108)
, L0 X 2 L0
It(X ) =
n;1 X k=0
fk (Mtk+1 ; Mtk ) + fn (Mt ; Mtn ) tn 6 t < tn+1:
(12.109)
B , (12.109) ( ) L0 . ) , ! , hM it . . ! 2 . ,
, , (
M 2 Mc loc ,
ZT
P(
0
Xt2(!)dhM it < 1) = 1 T > 0:
3 N?], N?, ?], N?, ?], N?]. ( I { / .
254
13. ## ) / ' . - # E. > , B ' . !B {/ . = 1 " B " . 3 " B " .
* ( , . $ $+ & $ & . B !
mv_ = ;v + $% t > 0
(13.1)
m | , v | , > 0 (, ( , ), $% (
). * , (13.1),
W_ , W = (Wt Ft)t>0 { (. (12.2)). /
4.1, (
. & , ! 6 | - . ' ( _ t > 0 mv_ = ;v + W (13.2) & . * (13.2) _ t > 0 v_ = av + W
(13.3)
a = ;=m < 0 = 1=m > 0:
(13.4)
\ , ( ! a )
v_ = av + f t > 0
(13.5)
v_ = av, ( v(t) = ceat (c = v(0)), , . . ( v(t) = c(t)eat. ? ,
v(t) = v(0)eat +
Zt 0
ea(t;u)f (u) du t > 0
(13.6)
( (
t 2 N0 T ], , , f | N0 T ] ). 255
? , (13.5) -
dv = av dt + f dt t > 0 (13.7) (13.3) dv = av dt + dW (t) t > 0: (13.8)
, . *! $+ (13.3) (13.8)
+ ; + $ +
Zt 0
dv(s) =
Zt
Zt
av(s) ds + dW (s)
0
(13.9)
0
) ( , , L2(N0 T ]) T > 0). ! (13.9)
Zt
v(t) ; v(0) = a v(s) ds + W (t):
(13.10)
0
L , f (u) du (13.6) dW (u)
Zt
v(t) = v(0)eat + ea(t;u) dW (u)
(13.11)
0
) ( e;au 2 L2 (N0 t]) t > 0). *, v(t), (13.11), (13.9), (13.10). ? , &, $ $ $+ v(t) ; ; $ $< (13.10), $ % $ & t $$+; . <
. J , 12.11 D0 D, P (D0 ) = 1 ! 2 D0 Rt
(13.11) t. ? ! 2 D0 , v(s) ds
0 t ( t 2 N0 T ]). ' (13.10) .. , t .. / , . . ! . ,$ ( +) < $ $ (13.9)
& % $ $+ , $ $$+; % $ $ $ , ; $ (
). '
.
13.1. # $ g(s u),
N0 T ] N0 T ], 0 < T < 1, , g 2 L2(N0 T ]2) = L2(N0 T ] N0 T ] A ) | + N0 T ], A { B (N0 T ]) B (N0 T ]) , ]n =
ZTZT 0 0
256
(g(s u) ; gn(s u))2ds du ! 0
n ! 0
(13.12)
n>1
n;1 X
gn (s u) =
i=0
g(s u(in))1 (u(in) u(i+1 n) ] (u)
0 = u(0n) < u(1n) < : : : < u(nn) = T , n = i=0max (u(n) ; u(in)). ::: n;1 i+1
ZT ZT 0
g(s u) ds dW (u) =
0
ZT ZT 0
(13.13)
"
g(s u) dW (u) ds:
1
(13.14)
0
RT
2 * - u 2 N0 T ] f (u) = g(s u) ds B(N0 T ]) jB(R)-0
I . B
,
ZT ZT 0 0
jg(s u)jds du < 1
(13.15)
g 2 L2(N0 T ]2). , , -. . u
ZT 0
2
g(s u) ds 6 T
ZT 0
g2(s u) ds:
(13.16)
*! f 2L2(N0 T ]) , N0 T ]. / , (13.14) (
. RT I J (s !) = g(s u) dW (u) . . ! B(N0 T ]) j B(R)-. 0 1 I ,
ZT 0
EjJ (s !)j ds < 1:
*
, . .
j
(13.17)
ZT
j = g2(s u) du
E J (s !) 2
(13.18)
0
P | () B(N0 T ]) F . / (12.13)
ZT ZT 0
0
gn (s u) dW (u) ds =
ZT X n;1 0
i=0
=
n) ) gn (s u(in))(W (u(i+1 n;1 X i=0
n) ) (W (u(i+1
;
;
W (u(in))) ds = T ( n ) W (ui )) gn (s u(in))ds: 0
Z
(13.19) 257
RT
I g(s u) ds (
- u 2 N0 T ]. C 0
RT g(s u(n))ds (
, . i
0
* ) ,
ZT ZT 0
gn (s u) ds dW (u) =
0
ZT X n;1 ZT 0
i=0 0
=
g(s u(in))ds1 u(in) u(i+1 n) ) (u)
n;1 X i=0
n) ) (W (u(i+1
;
W (u(in)))
dW (u) =
ZT 0
g(s u(in))ds: (13.20)
) , g = gn (13.14) . / I ,
2 , ,{0 {h , (12.31) (13.12), ,
ZT ZT ZT ZT gn(s u) dW (u) ds 6 E g(s u) dW (u) ds ; 0 0 0 0 ZT ZT 6 E (g(s u) ; gn (s u))dW (u)ds 6 0 0 21=2 ZT ZT 6 =
E
0
ZT ZT 0
0
0
(g(s u) ; gn (s u)) dW (u)
(g(s u) ; gn
1=2
(s u))2du
ds =
ds 6 (T ]n)1=2 ! 0 n ! 1:
(13.21)
J
ZT ZT ZT ZT E g(s u) ds dW (u) ; gn(s u) ds dW (u) 6 0 0 0 0 T T
Z Z 21=2 6 E =
0
Z Z T
0
T
0
0
(g(s u) ; gn (s u)) ds dW (u)
2 1=2
(g(s u) ; gn (s u))ds du
=
6 (T ]n)1=2:
(13.22)
B
( ! 2 ( ) L2 ( )
. C n ;! , n L;!
n ! 1 n = n .. n 2 N, = .. 2 258
3 ; & . )
Zt
a v(s) ds = a v(0) 0
Zt
Zt Zs
easds +
0
0
ea(s;u)dW (u)
0
= v(0)(eat ; 1) + a
Zt Zt 0
ds =
ea(s;u)1
0
(0 s] (u)dW (u)
ds: (13.23)
13.1 t > 0
Zt Zt 0
ea(s;u)1
0
(0 s] (u)dW (u)
=
Z
t
e;au
Zt u
0
ds =
Zt Zt
eas ds
0
ea(s;u)1
0
(0 s] (u) ds
dW (u) =
Z Z 1 1 a ( t ; u ) dW (u) = a e dW (u) ; a dW (u): (13.24) t
t
0
0
3 , g(s u) = ea(s;u)1 (0 s](u), s u 2 N0 T ], (13.12) N0 T ] u(0n) u(1n) : : : u(nn), n ! 0 (n ! 1). B
, ea(s;u) N0 T ] N0 T ], ]n 6 "2n
X 06i
n) ; u(n) ) + e2aT (u(kn+1) ; uk(n))(u(i+1 i
n;1 X i=0
n) ; u(n))2 6 "2 T 2 + T e2aT (u(i+1 n i n
"n = supfjea(s;u) ; ea(x;y)j : js ; xj 6 n ju ; yj 6 n s u x y 2 N0 T ]g. / , (13.10) (13.23) (13.24)
v(0)(eat ; 1) +
Zt 0
ea(t;u)dW (u) ; W (t) + W (t) = v(t) ; v(0)
(13.25)
(13.11). ? ,
@ 13.2. 3 t > 0 ( (. . ) - + v jt=0 = v (0), # (13.11). ? 13.3. C (13.2)
( . . = 0, , a = 0 (13.4)), v(0) = 0 = 1 ( . . m = 1) (13.11) , v(t) = W (t). ? , $ $ < & . * (
Ev(0). ? , (12.31), (13.11) , Ev(t) = eatEv(0)
(13.26)
. . , (13.4), ! . 259
2 @ 13.4. v(0) N 0 , = ;a, 2
F0 j B(R)-
v(0)
. - + ! {# , . . $ v (t), t > 0, "( $ " # $ " 2 cov(v(s) v(t)) = 2 e;js;tj s t > 0:
(13.27)
2 ) (13.26) , Ev(t) = 0 t > 0. B s t > 0 2 Ev(s)v(t) = E(v(0))2ea(s+t) + 2a ea(s+t)(1 ; e;2a(s^t)):
(13.28)
3 , (12.33), (12.31) (12.8) E
Zs
Zs^t
Zt
ea(s;u)dW (u) ea(t;u)dW (u) =
0
0
0
ea(s;u) ea(t;u)du = e 2a (1 ; e;2a(s^t)) a(s+t)
(13.29)
, t > 0 (12.38) Ev(0)
Zt 0
ea(t;u)dW (u) = E
v(0)E
Zt 0
ea(t;u)dW (u) j F0
= 0:
(13.30)
2 C E(v(0))2 = ; 2a , (13.28) (13.27). 8 & $+ v(t), t > 0. B k 2 N 0 6 t1 < : : : < tk 6 T Rt
(v(0) X (t1) : : : X (tk )), X (t) = ea(t;u)dW (u), . 0 B
, tm (m = 1 : : : k) X (tm) L2(D) n ! 1 Ih(nm), h(nm) | (., (k ) , 12.10). (Ih(1) n : : : Ihn ) , . . Pk c Ih(nm) c 2 R, m = 1 : : : k, ( m m m=1 ). * , v(0) F0 jB(R)- , , (k) (v(0) Ih(1) n : : : Ihn ) | . 1
E exp iv(0)0 + i
k X m=1
mIh(nm)
i P Ih(m) = Eeiv(0)0 E(e m=1 m n k
i P m Ih(nm) iv (0) 0 = Ee Ee m=1 k
j F0) =
j 2 R j = 0 : : : k: (13.31) / L2(D),
. 2
; $$ < $$ $, $% $ $
,$ dt dW (t) $+ , . . -
( !. L dXt = b(t Xt) dt + (t Xt)dWt 0 6 t 6 T X0 = Z 260
(13.32)
Zt
Zt
0
0
Xt = X0 + b(s Xs ) ds + (s Xs ) dWs 0 6 t 6 T
(13.33)
b N0 T ] R. + < $ $ Xt(!), ; .. $ $, $ ;% t > 0 $ ; ; $ (13.33) $ $ $$+; 1. * ! , b(s Xs) (s Xs) | , (s Xs) 2 L2(N0 T ]), b(s Xs ) 2 . . !. $% , b(u x) (u x) N0 t] R ( .. BN0 t] B(R)jB(R)-) t 2 N0 T ], 2 : (
L > 0,
jb(t x) ; b(t y)j + j(t x) ; (t y)j 6 Ljx ; yj x y 2 R t 2 N0 T ]:
(13.34)
* c > 0
b(t x)2 + (t x)2 6 c(1 + x2) x 2 R t 2 N0 T ]:
(13.35)
& , 0, . . , (
(
,
. C, , b(t x) = b(x), (t x) = (x), (13.34) (13.35). B
13.5.
$ Ys (! )
N0 T ] ( # $ (Fs)s2 0 T ] F ). # $ a(s x) N0 T ] R t 2 N0 T ] N0 t] R, .. B (N0 t]) B(R)jB(R)- . $ a(s Ys (! )) ( # $ (Fs )s2 0 T ]).
2 B s 2 N0 t] ! 2 D (0 6 t 6 T ) (s !) 7! (s Ys (!)) B(N0 t]) FtjB(N0 t]) B(R)-. u 2 N0 t] B 2 B(R),
f(s !) 2 N0 t] D : (s Ys(!)) 2 N0 u] B g = = f(s !) 2 N0 t ^ u] D : Ys (!) 2 B g 2 B(N0 t ^ u]) Ft^u BN0 t] Ft: ?
1.2. ? , (s x) 2 N0 t]R (s x) 7! a(s x) B(N0 t])B(R)jB(R)-. < , . 2
@ 13.6. # $ b " - . Z F0 jB (R)- , EZ 2 < 1.
( - (13.33) X0 = Z , Xt 2 L2 (D) " t 2 N0 T ] # $ EXt2
N0 T ]. 261
I$$+ ( ) , . .
X1(t !) X2(t !) | (13.33) , ( .. , X1(t !) = X2 (t !) 0 6 t 6 T ..
(13.36)
2 0 (13.33) . Xt0 = Z , t 2 N0 T ] n > 1 Xt(n)
=Z+
Zt 0
b(s Xs(n;1))ds +
Zt 0
(s Xs(n;1))dW (s):
(13.37)
13.6. n 2 N # (13.37), (n) ( .. " # $ "). supt2 0 T ] EjXt j2 < 1, n 2 N (13.37) . . N0 T ].
13.7. (n ) $ fXt t 2 N0 T ]g,
2 I Ys (!) = Z (!), s 2 N0 T ], ! 2 D, , Z 2 F0jB(R). * 13.5 b(s Z (!)) (s Z (!)) (s 2 N0 T ]). (13.35) sup E2(s Z ) 6 c(1 + EZ 2) < 1:
s2 0 T ]
R
/ , f(s Z ) s 2 N0 T ]g 2 L2(N0 T ]) 0t (s Z )dWs t 2 N0 T ] , 12.11, .. . 8 fb(s Z ) s 2 N0 T ]g, (13.35), ZT E jb(s Z )jds 6 (Tc(1 + EZ 2))1=2 < 1: 0 R R T * I b(s Z )ds . . ! ! ! t b(s Z )ds 0
0
t 2 N0 T ]. < 12.12. ? 6 .. (13.37) , fX (n;1)(t) t 2 N0 T ]g sups2 0 T ] EjXs(n;1)j2 < 1. ? , E(Xt(n))2
63
63
EZ 2 + E
EZ 2 + Tc
ZT 0
Z t 0
2 Z t
b(s Xs(n;1))ds
E(1 + j
j
+E
Xs(n;1) 2)ds + c
ZT
0
E(1 + j
0 E Xs(n;1) 2
6 3 EZ 2 + c(T + 1)T 1 + sup j s2 0 T ]
(s Xs(n;1))dWs
j
j
2
Xs(n;1) 2)ds
6
6
< 1: 2
& $ +$ $ 13.6. < EjXt(n+1) ; Xt(n)j2. C n = 0, , (12.31) (13.35), t 2 N0 T ] t Zt 2 Z (1) (0) 2 EjXt ; Xt j = E b(s Z ) ds + (s Z ) dWs 6 0
262
0
Zt
6 2E
2
Zt
b(s Z ) ds + 2E
0
6 2ct(t + 1)(1 + EZ 2) 6 M
(s Z ) dWs
0
2
Zt
6 2(t + 1)E c(1 + jZ j2) ds 6
+ 1)(1 + EZ 2):
M1 = 2c(T B n > 1 t 2 N0 T ], (13.34), EjXt(n+1) ; Xt(n)j2 6
6E
1 t
Zt 0
b(s Xs(n))
Zt
6 2E
0
j
L Xs(n)
;
;
b(s Xs(n;1))ds +
Xs(n;1)
2
Zt 0
((s Xs(n))
;
0
(13.38)
(s Xs(n;1)))dWs
2
6
Zt
jds + 2E ((s Xs(n)) ; (s Xs(n;1)))2ds 6 0
Zt
6 2L2(1 + T ) EjXs(n) ; Xs(n;1) j2ds:
(13.39)
0
) (13.38) (13.39) , M = maxfM1 2L2(1 + T )g n+1 n+1 EjXt(n+1) ; Xt(n) j2 6 M t n = 0 1 : : : t 2 N0 T ]: (13.40) (n + 1)! ? , sup j
06t6T
Xt(n+1)
;
Xt(n)
ZT
j 6 jb(s Xs(n)) ; b(s Xs(n;1))jds +
Zt ( n ) ( n ; 1) + sup ((s Xs ) ; (s Xs ))dWs : 06t6T
0
0
* 12.11,
5.17 (13.40),
P ( sup j 06t6T
Xt(n+1)
;
Xt(n)
j > 2;n ) 6 P
ZT 0
j
b(s Xs(n))
;
j
2
b(s Xs(n;1)) ds
> 2;2n;2
+
Zt ( n ) ( n ; 1) ; n ; 1 6 + P sup ((s Xs ) ; (s Xs ))dWs > 2 06t6T 0 ZT
6 22n+2 T E(b(s Xs(n)) ; b(s Xs(n;1)))2ds +
+ 22n+2
ZT 0
0
j
E (s Xs(n))
;
j
(s Xs(n;1)) 2ds 6 22n+2L2(T
+ 1)
ZT M n sn 0
(4MT ) : ds 6 n! (n + 1)! (13.41)
* 0{, (13.41) , P ( sup jXt(n+1) ; Xt(n) j > 2;n ) = 0: 06t6T
n+1
(13.42) 263
*! . . ! (
N0 = N0(!): 8n > N0(!) sup jXt(n+1) ; Xt(n) j 6 2;n :
(13.43)
t2 0 T ]
/ ,
Xt(n)(!) = Xt(0)(!) +
n;1 X k=0
(Xt(k+1)(!) ; Xt(k)(!))
(13.44)
( .. N0 T ], 1 ! . B ! 2 D0 D (P (D0) = 1) {
, Xt(n) N0 T ] , Xt(!) = nlim X (n) (!). !1 t B ! 2 D n D0 t 2 N0 T ] Xt(!) = 0. < , Xt(!) !. Xt 2 FtjB(R) 4.7, Xt(n) , ( FtjB(R)- ( - ). $+ fXt t 2 T g $ $ ( 12.12) $ $ +$ (Ft)t2 0 T ]. ? , m > n > 0 t 2 N0 T ] (13.40)
j
(E Xt(m)
;
j
Xt(n) 2)1=2 6
X
m;1
k
Xt(k+1)
k=n
;
k
Xt(k) L2( )
6
1 X (MT )k+1 1=2 k=n
(k + 1)!
! 0 n ! 1: (13.45)
L2(D) (
L2(D) Xt(n) n ! 1. 1
Yt. < , Yt = Xt .. t 2 N0 T ]. *, Xt (13.33). ) (13.45) t 2 N0 T ] m 2 N
k
k
Xt(m) L2 ( ) 6
kZ kL ( ) + 2
1 X (MT )k+1 1=2 k=0
(k + 1)!
= kZ kL2( ) + c(M T ):
(13.46)
*!
kXtkL ( ) 6 kZ k + c(M T ) t 2 N0 T ]:
(13.47) * I 2 (13.46), (13.47) 2
E
ZT 0
(Xs ;
Xs(n))2ds =
ZT 0
E(Xs ; Xs(n) )2ds ! 0 n ! 1:
(13.48)
(13.34) (13.48) t 2 N0 T ] n ! 1
Zt
0
2 (s Xs(n)) dWs L ( )
;! (s Xs) dWs
Zt 0
264
Zt
L2 ( )
b(s Xs(n)) ds ;!
0
Zt 0
b(s Xs) ds:
(13.49)
(13.50)
? , (13.37) fnm = nm (t)g, (13.49) (13.50) .. , (13.33). & , Xt N0 T ]
2 L (D). B
, Xt ; Xs ! 0 .. t ! s (t s 2 N0 T ]) E(Xt ; Xs)2 6 const
(13.47). < 0 2 . 8 & $$+ <. * Xt { (13.33) X0 = Z , Xet { (13.33) Xf0 = Ze, EXt2 EXet2 N0 T ]. J (13.39) ,
Z t
EjXt;Xetj2 6 3EjZ ;Zej2+3E
0
(b(s Xs );b(s Xes)ds
6 3EjZ ; Zej
2 + 3(1 + t)L2
Zt 0
2
+3E
Z t 0
2 e ((s Xs);(s Xs)dWs 6
EjXs ; Xes j2ds:
(13.51)
/ ( 0 2 , Xt Xet N0 T ]
L2(D). ) , y(t) = EjXt ; Xetj2
Zt
y(t) 6 c0 + Q y(s) ds t 2 N0 T ]
(13.52)
0
c0 = EjZ ; Zej2, Q = 3(1 + T )L2. (
. 13.8 (! ). y { $ N0 T ] # $ , "( (13.52) c0 > 0 Q > 0.
y(t) 6 c0 exp(Qt) t 2 N0 T ]: (13.53) 2 8 (13.52), t > 0 ( ) 0 Zt QyR (t) 6 Q: (13.54) log c0 + Q y(s)ds = c0 + Q 0t y(s)ds 0 ) 0 t,
log c0 + Q / ,
Zt 0
y(s)ds ; log c0 6 Qt t 2 N0 T ]:
Zt
c0 + Q y(s)ds 6 c0eQt t 2 N0 T ]: 0 / , ( c0 = 0 y(s) = 0 s 2 N0 u], (13.54) ). 2 C Z = Ze .., c0 = 0 (13.52) t 2 N0 T ] , EjXt ; Xet j2 = 0. * jXt ; Xetj, , X Xe N0 T ], .. P (X (t !) = Xe (t !) t 2 N0 T ]) = 1: 2
L< $ $ + ;$ %< + & . 265
@ 13.9.
13.6. - $.
(13.32) 2
7.6
(Xt t 2 N0 T ]) , C 2 B(R), m 2 N 0 6 t1 < : : : < tm < u 6 t
P (Xt 2 C j X11 : : : Xtm Xu ) = P (Xt 2 C j Xu ): (13.55)
. , , - . * 2.3, , (13.55) C 2 B(R). * ,
4.5 ( (4.15)), , (: f 2 Lipb(R) ( . . f : R ! R)
E(f (Xt) j Xt1 : : : Xtm Xu ) = E(f (Xt ) j Xu ): (13.56) ? , 13.6 , N0 T ] Nu T ], 0 6 u < T < 1, . . (
Zt
Zt
u
u
Zt = Z + b(s Zs) ds + (s Zs) dWs
(13.57)
Zu = Z , Fu , fWs s > 0g { (Fs)s>0 . 1 Zt Zt(Z ), t 2 Nu T ]. B Xt , t 2 N0 T ], (13.33) t > u
Zt
Zt
Zu
0
Zt
0
u
Xt = X0 + b(s Xs) ds + (s Xs) dWs = 0
= X0 + b(s Xs ) ds + +
Zt u
Zu
b(s Xs) ds + (s Xs) dWs +
Zt
0
Zt
(s Xs) dWs = Xu + b(s Xs) ds + (s Xs) dWs : u
u
(13.58)
8 , Xu Fu jB(R)- (! , ( )
(13.57), , X0 Xt = Zt (Xu) t > u: (13.59)
Z (13.57) x 2 R. * , . . t 2 Nu T ],
(13.51) 13.8, , t ( ) Zt(y) L;! Zt (x) y ! x (x y 2 R): 2
266
(13.60)
B t 2 Nu T ] f 2 Lipb(R)
G(x) = Ef (Zt (x)) x 2 R: (13.61) (13.60) , G(x) | ( ) R. 13.10. 3 " t 2 Nu T ] " x 2 R Zt(x) - Fu . 2 L Zt(x) ( . . Zt(x !), t 2 Nu T ], x 2 R, ! 2 D) : Zt (x) .. Zt(n)(x), n = 0 1 : : : , n ! 1, Zt(0)(x) = x, n > 1 Zt(n)(x) = x +
Zt u
b(s Zs(n;1)(x)) ds +
Zt u
(s Zs(n;1)(x)) dWs t 2 Nu T ]:
(13.62)
2 , Zt(n)(x) F u t] jB(R)- n = 0 1 : : : , () - F u t] = fWs ; Wu s 2 Nu t]g. B
, Rt Rt x + b(s x) ds !, (s x) dWs u u ( , (
, ( ..). * , F u t] jB(R)-,
n = 0, 4.7. * 13.5 ( Nu T ] F u t], t 2 Nu T ]), F u t] jB(R)- Zt(n)(x) n > 1. ? , Zt(x) F u t] j B(R)-
4.7. < 4.3, Fu F u t] ( - , ). 2 & $ +$ $ 13.9. *
Y (n) = 2;n N2n Xu ] n 2 N N] | . < , Y (n)(!) ! Xu (!) n ! 1 ! 2 D:
(13.63) (13.64)
(13.63) Y (n) Fu jB(R)- n 2 N. *! Zt(Y (n)), t 2 Nu T ], n 2 N. B
Xt , t > 0, , (13.56) G(Xu ), G (13.61). A 2 fXt1 : : : Xtm Xu g, t1 : : : tm u t (13.55) (13.56). ? f 2 Lipb (R) Ef (Xt )1 A = nlim Ef (Zt(Y (n)))1 A : (13.65) !1 B
,
jf (x) ; f (y)jL 0jx ; yj x y 2 R:
? t 2 Nu T ], n 2 N A
267
jEf (Xt )1 A ; Ef (Zt(Y (n)))1 A j 6 L0EjXt ; Zt (Y (n))j 6 6 L0(EjZt(Xu ) ; Zt(Y (n))j2)1=2 6 a0(EjXu ; Y (n)j2)1=2 6 a02;n a0 = L0 expfQT=2g (13.51) (13.53) Q = 3(1 + T )L2.
* 2 13.10, E1 A f (Zt = =
(Y (n))) =
1 X k=;1
1 X
k=;1
1 X
k=;1
E1 A f (Zt (k2;n ))1 fY (n)=k2;ng =
E1 A\fY (n)=k2;n gEf (Zt (k2;n )) =
1 X k=;1
E1 A\fY (n)=k2;n gG(k2;n ) =
E1 A 1 fY (n)=k2;n gG(k2;n ) = E1 A G(Y (n)):
(13.66)
3 , f (Zt (k2;n )) Fu, A \ fY (n) = k2;n g = A \ fk2;n 6 Xu < (k + 1)2;n g 2 fXt1 : : : Xtm Xu g Fu : (13.67) * G, (13.65), (13.66) (13.64) E1 A f (Xt ) = E1 A G(Xu ): (13.68) *! E(f (Xt) j Xt1 : : : Xtm Xu ) = G(Xu ): (13.69) ' ,
(7.10) , E(f (Xt) j Xu ) = E(E(f (Xt ) j Xt1 : : : Xtm Xu ) j Xu ) = E(G(Xu ) j Xu ) = G(Xu ) (13.70) (13.56). ? 13.9 . 2 $ 13.11. $ ) - {I (. 13.4) ,
$.
2 B , < {8 2 (13.8) X0 N (0 2=2), .
13.4. 2
.
813.1. L $ $ $ dXt = rX r = const: t dt ? , dXt = rXt dt + Xt dWt r | . 268
(13.71) (13.72)
2 )
Zt dXs 0
Xs ds = rt + Wt t > 0:
(13.73)
h(t x) = ln x, x > 0. & , ) ( B12.12) ,
, h : N0 1) ( ) ! R, ;1 6 < 6 1,
( h(t x) N0 1) R), Xt(!) 2 ( ) t > 0 ! 2 D. ) , Xt > 0 t > 0,
1 2X 2 dt = dXt ; 1 2 dt t ; d(ln Xt) = X1 dXt + 12 ; X12 (dXt )2 = dX Xt 2Xt2 t Xt 2 t t
dXt = d(ln X ) + 1 2 dt: t Xt 2 / ,
(13.74)
Xt = r ; 1 2 t + W : ln X t 2 0
(13.75)
Xt = X0eHt Ht = (r ; (1=2)2)t + Wt:
(13.76)
2 , X0 fWt t > 0g (. (12.2)), EXt = EX0 expfrtg
(13.77)
. . ( , r < 0) EXt , (13.71). 2 fXt t > 0g (13.76), ( (13.72) ! r 2 R, > 0. * H = (Ht)t>0 r ; 2=2 ## 2. 2 H , 2 , { ". *. /! (N?]) , ! , "! ". ? 813.2. < , 13,
. ?, (13.32) , Xt = (Xt(1) : : : Xt(n)), b | , | , ,
b( ): N0 T ] Rn ! Rn ( ): N0 T ] Rn ! Rn m:
(13.78) 269
* ! Wt | m- , . . Wt(k), t > 0, k = 1 : : : m, P ,
1 0 (1) b (t Xt) dt b(t Xt) dt = B A @ ... C b(n)(t Xt) dt
0 10 1 (t Xt) : : : m (t Xt) dWt(1) ... C (t Xt) dWt = B @ ... AB @ ... C A: 11
(13.79)
1
n1 (t Xt) : : : nm (t Xt)
dWt(m)
) Rt , (s Xs) dWs - i- ,
Pm
0
k=1 intt0ik (s Xs) dWs(k) .
< ,
(Ft)t>0 , fWt t > 0g { . ? 13.6 , (13.34) (13.35) b Rn, Pn Pm 2 .
, , , jj2 = ik i=1 k=1
* $ 2$. * Xt = (Xt(1) : : : Xt(n) ) | n- $ 1 , .. ( N0 1) ,
dXt = f (t !) dWt + g(t !) dt
(13.80)
f g |
f = (fik (t !) i = 1 : : : nP k = 1 : : : m)P g = (g1(t !) : : : gn (t !)). Wt = = (W1(1) : : : Wt(m)),
dXt(i)
=
m X k=1
fik (t !) dWt(k) + gi (t !) dt i = 1 : : : n
(13.81)
t > 0, i = 1 : : : n k = 1 : : : m
Zt
P(
0
Zt
P(
0
jgi (s !)jds < 1) = 1
(13.82)
jfik (s !)j2ds < 1) = 1:
(13.83)
< , ) t 2 Nu v], 0 6 u < v < 1 t 2 Nu v), 0 6 u < v 6 1. 270
@ 813.3. # $ h : N0 1) Rn ! R , h 2 C 1 2, ..
( "
@h=@t @ 2h=@xi@xj , i j = 1 : : : n,
@h sup n @x (t x) < 1 i = 1 : : : n: t>0 x2R i
$ Yt = h(t Xt), $ 1 Xt ## $ , # n n n @h @h @ 2h (t X ) dX (i) dX (j) (i) 1 dYt = @t (t Xt) dt + @x (t Xt) dXt + 2 t t t i i=1 i=1 j =1 @xi@xj dXt(i) dXt(j ) , # "( (13.81), "
dWt(i)dWt(j) = ij dt dWt(i)dt = dt dWt(i) = 0:
X
1 , m n dYt = k=1 i=1
X X @h
XX
(13.84) (13.80), (13.85)
(k) @xi (t Xt)fik (t !) dWt +
@h n n m 2h X X X @h 1 @ (t Xt) fik (t !)fjk (t !) dt: + @t (t Xt) + @x (t Xt)gi (t !) + 2 @x @x i i j i=1 i j =1 k=1 (13.86) B
) ., ., N?, . 1, x 4e]. * ( ! h 2= C 1 2, ., ., N?], N?]. * , N?], (. * X { () fWt t > 0g. * h = h(x) :
h(x) = h(0) +
Zx 0
f (y)dy
() f = f (y) 2 L2loc (R), .. c > 0
Z
jyj6c
f 2(y)dy < 1:
< Nf (W ) W ] { " $ " f (W ) W ( X Nf (W ) W ]t = P ; lim (f (Wt(n)(m+1)^t) ; f (Wt(n)(m)^t)) (Wt(n)(m+1)^t ; Wt(n)(m)^t): m m
& P ; lim , T (n) = ft(n)(m) m 2 Ng, n 2 N { t(n)(m), .. t(n)(m) 6 t(n)(m+1) m 2 N n 2 N t > 0 sup(t(n)(m + 1) ^ t ; t(n)(m) ^ t) ! 0 n ! 1: m
, h(W ) , ( , , (
, ( Nf (W ) W ]t. < N?]
(
! . 271
@ 813.4 ( $$, 0 +, H).
# $
$
h f
#
t h(Wt) = h(0) + f (Ws )dWs + 12 Nf (W ) W ]t: 0 813.5. # $ h 2 C 2.
Z
Nf (W ) W ]t = #
- -
(13.87)
Zt 0
(13.87)
f 0(Ws )ds
# 1 .
$ 813.6. h(x) = jxj.
Nf (W ) W ]t = 2Lt (0), Lt (0) { , N0 t] (. (12.96)). , (13.87) # (12.97), 3 { : jWt j. D. 13.7. * h(x1 x2) = x1x2, x1 x2 2 R X (1) = fXt(1) t > 0g, X (2) = fXt(2) t > 0g { , ( ) . ? -
d(Xt(1)Xt(2)) = Xt(1)dXt(2) + Xt(2)dXt(1) + dXt(1)dXt(2): B , dXt(i) = b(i)(t !)dt + (i)(t !)dWt(i) i = 1 2 fWt(i) t > 0g, i = 1 2 { ,
d(Xt(1)Xt(2)) = Xt(1)dXt(2) + Xt(2)dXt(1)
dXt(i) = b(i)(t !)dt + (i)(t !)dWt i = 1 2
d(Xt(1)Xt(2)) = Xt(1)dXt(2) + Xt(2)dXt(1) + (1)(t !)(2)(t !)dt: D. 13.8. L dXt = Xtdt + dWt t 2 N0 T ] ( e;t d(e;tXt)). D. 13.9. * R t V = V (t x) {
R+ Rn, C 1 2. * t = 0 c(s !)ds, c = c(s !) { ,
Zt P ( jc(s !)jds < 1) = 1 t > 0: 0
' fe;t V (t Xt) t > 0g, Xt = fXt(1) : : : Xt(n)g n- ) , (13.81). ? 813.10. C 13.6 N0 T ),
! , ! Xt EXt2 , N0 T ). 272
D. 13.11. / ( ) , ( )
dXt = T;;Xtt dt + dWt t 2 N0 T ) X0 =
(13.88)
Z t dWs Xt = (1 ; t=T ) + t=T + (T ; t) T ; s :
(13.89)
0
D. 13.12. ( 13.11) B , Xt ! .. t ! T ;. ? , (13.88) N0 T ] $ X0 = XT = . / T = 1 = = 0. ? 813.13. /(
( 13.6 . ', (13.34) + - $ x: t > 0, n 2 N jxj 6 n, jyj 6 n jb(t x) ; b(t y)j + j(t x) ; (t y)j 6 L(n)jx ; yj L(n) > 0. ) ! . , , ( ) !
!, ! b " " ( : b = b(tP Xs s 6 t), = (tP Xs s 6 t). < ( ., ., N?], N?], N?]. ) ( , J.,. & N?],
(, $ + < $ $ +
dXt = b(t Xt)dt + dWt (13.90) $ $ <+ $ (t x) $ ,$ b(t x). D. 13.14. B , dXt = (Xt)dt + dWt (13.91) "" ! (x) = signx
. 813.15. L (13.91) dXt = (Xt)dBt X0 = 0 (13.92) , -, (x) = signx B = (Bt)t>0 { (
Bt Wt). B, (
, ! . , ,
! . 2 L
C N0 1] W
fWt t 2 N0 1]g, .. Wt(!) = !(t), !() 2 C N0 1]. ' ( (., ., N?]) 273
@ 813.16 ( ). B = (Bt Ft)t>0
{ (P -. .) , # (D F (Ft)t>0 P ). (Bt2 ; t Ft)t>0 , ..
E(Bt2 ; Bs2)jFs) = t ; s 0 6 s 6 t:
B = (Bt)t>0 12.11
Bt =
Zt 0
(13.93)
.
(Ws)dWs t 2 N0 1] (x) = signx
(13.94)
( , (Ws), s 2 N0 1]
L2N0 1], ( (FtW )t2 0 1], ..
W ). D. 13.17. * , (13.94) (13.93) s t 2 N0 1]. D. 13.18. I (13.94) , dBt = (Ws)dWs. B ,
Zt 0
(Ws)dBs =
Zt 0
2(Ws )dWs
(13.95)
.. (13.95)
. 8 , 2(x) 1, (13.95)
Zt 0
(Ws)dBs = Wt
(13.96)
.. Wt (13.92) . ', (;x) = ;(x) x 2 R,
Zt 0
(;Ws)dBs = ;
Zt 0
(Ws)dBs = ;Wt t 2 N0 1]:
? , ;Ws (13.92). < , ! . $ $&;, & 813.15. *, (13.92), , ! ,
Xt =
Zt 0
(Xs)dBs t 2 N0 1]
(13.97)
( - (FtB )t2 0 1], B ). ) 2 ( ), (Xt FtB )t2 0 1] . * ? (. (12.97))
jXt j = 274
Zt 0
(Xs)dXs + Lt(0)
Zt 1 Lt(0) = lim 1 fjXs j 6 "gds: "!0 2" 0
(13.98)
I (13.97) (13.92), ! ( )
Zt 0
(Xs)dXs =
Zt 0
2(Xs )dBs
=
Zt 0
dBs = Bt t 2 N0 1]
(13.99)
, (13.98) Bt = jXtj ; Lt(0). *, X (FtB )t2 0 1], FtX FtB FtjX j t > 0. <
( . D.jW13.19. B , W { , j W Ft Ft t > 0 . L B13.15 . 2 & , 3. 0 N?] , (13.92) = (x) > 0. ' B13.15 W ;W (13.92) C N0 1]. 1
6
. A . / (13.32) , { B(R), - N0 T ],
(D F (Ft)t2 0 T ] P ), W = (Wt Ft)t2 0 T ] X = (Xt Ft)t2 0 T ], , L(X0 ) = P .. t > 0
(13.32). L , N0 T ] Nu v], 0 6 u < v < 1 Nu v) 0 6 u < v < 1. ' X (Ft)t2 0 T ] 12.12. , $ $ + <, $ -
+$ $$ $ $ % &, % $ %:$ (
) ;$, $% $ <+, $% < +. < ,
.
% $$+ < (13.32) $, $ ;% < ( ) ;$ , .. .
813.20.
13.6 (. 3 13.13). - ( ) (13.32) .
ft Fet) (X~ t W~ t F~t) | (13.32). * 2 B , (Xet W Xt Yt | ! ,
(Wt Ft)t>0 ft Fet) (W~ t F~ t) (t > 0). * 13.6 , Xt = Xet ..
(W Yt = X~ t .. t > 0. ? , ,
.-.. Xt Yt, t > 0. *
, ( n ) ( n ) n 2 N Xt Yt , t > 0, (
Xt Yt . 2
C
(
! (13.32). < ! (. N?], N?]) ( . L dXt = b(Xt)dt + (Xt)dWt (13.100) 275
,
Z
jxj (dx) = EjX0j < 1 > 2:
C ! b = b(x) = (x) , (13.100) . ? 813.21. ! (x)
(
( ) , ! b(x) (. N?]). * ( , ! , ( . 3 & ; + $ $ ! % % ;$ ,
. * (D F (Ft)t>0 P ) {
, W = (Wt Ft)t>0 { m- , W = (W 1 : : : W m). * a = (at Ft)t>0 { m- , a = (a1 : : : am),
,
P
Z t 0
kask
2ds
< 1 = 1 t 2 N0 T ]
kask2 = (a1s )2 + : : : + (ams)2 T < 1. < Z = (Zt Ft)t2 0 T ],
Zt
Zt = expf (as dWs ) ; 21 0
(as dWs) :=
Zt
Pm ak dW k . k=1 s s
813.22 (. N?]). !
0
kask2dsg
(13.101)
(13.102)
< :
1 Z t
E exp 2
0
EZT = 1 - $ .
kask
2ds
< 1
(13.103)
Z = (Zt Ft)t2 0 T ]
-
Zt (P -..) EZT = 1, (D FT )
QT ,
QT (A) = E(1 AZT ) A 2 FT :
(13.104)
@ 813.23 (! ). 3 - $ W a Bt = Wt ;
Zt 0
asds t 2 N0 T ]:
B = (Bt Ft)t2 0 T ] m- # (D FT (Ft )t2 0 T ] QT ).
276
/ ( ! & <$+ $ %
< $ $ + dXt = a(t X )dt + dWt t 2 N0 T ] (13.105) a(t x), t 2 N0 T ], x 2 C N0 T ], , .. t 2 (0 T ], D 2 B(R) f(s x) 2 N0 t] C N0 T ] : a(s x) 2 Dg 2 B(N0 t]) B(C N0 T ]): < , , ( , a(t x()) = b(t x(t)), b : N0 T ]R ! R. ) , (D F (Ft)t2 0 T ] Q) X = (Xt Ft)t2 0 T ] B = (Bt Ft)t2 0 T ] , Q-.. t 2 N0 T ]
Xt =
Zt 0
a(s X )ds + Bt
( ! , , ). B X0 = 0 2 Rm. D = (C N0 T ])m, F = B((C N0 T ])m), Ft = B((C N0 t])m), t 2 N0 T ] Q = QT (. (13.104)). C ! , ( 9 ,
Zt Bt = Wt ; a(s W )ds t 2 N0 T ] 0
N0 T ]. < Xt = Wt, (13.105). < , . . dXt = b(Xt) dt + (Xt) dWt t > sP Xs = x (13.106) Wt | m- , b : Rn ! R : Rn ! Rn m
13.6,
: (
L > 0, jb(x) ; b(y)j + j(x) ; (y)j 6 Ljx ; yj x y 2 Rn (13.107) (jj2 =
Pn Pm 2 ), , c > 0
i=1 k=1 ik
jb(x)j + j(x)j 6 c(1 + jxj) x 2 Rn:
(13.108) <
(13.106) t > s Xts x . B ( n = 1. @ 813.24. 3 x 2 R - $ Xts x, t > s, $.
2 / u = t + v, ( ) x 2 R, t h > 0,
Xtt+xh
=x+
Zt+h
Zt+h
t
t
b(Xut x) du +
(Xut x) dWu
= x+
Zh 0
b(Xtt+xv ) dv +
Zh 0
(Xtt+xv )dW v (13.109) 277
W v = Wt+v ; Wt, v > 0, 4.3. / ,
Xh0 x
Zh
Zh
0
0
= x + b(Xv0 x) dv +
(Xv0 x) dWv :
(13.110)
? , Wv W v . *!
B13.20 , x 2 R (Xtt+xh )h>0 =D (Xh0 x)h>0
(13.111)
. . , ( (13.111), . 2 D. 13.25. ' fXts x t > sg, ( (13.106). ' ) + $ $ $ $ ( $ 0 { $ $ ),
Zt 0
f (s !) dWs (!)
(13.112)
fWs s > 0g { , f . S , f (13.112) (
X
N ;1 i=0
f (ti !)(Wti+1 ; Wti )
0 = t0 < : : : < tN = t ti = (ti + ti+1)=2, i = 0 : : : N ; 1. L t Wt(n)(!), n 2 N , Wt(n)(!) ! Wt(!) n ! 1 . . ! t . ? (. N?, . 27]) ! Xt(n)(!) dXt(n) = b(t X (n)) + (t X (n)) dWt(n) t t dt dt Xt(!) n ! 1 . . ! t . , (. N?], N?], N?]) Xt
Xt = X0 +
Zt 0
b(s Xs)ds +
Zt 0
(s Xs) dWs :
(13.113)
1 dXt = b(t X ) + (t X )W_ (13.114) t t dt " " W_ , (13.113), (13.33). , , / , ( ) (. (12.50) (13.85)). 1 278
(. N?], N?]). , / , , , ) ( ! ). * ! , (t x) x, $ (13.113) & -
$+ ; 2$
Zt Zt 1 0 Xt = X0 + b(s Xs)ds + 2 x(s Xs )(s Xs)ds + (s Xs)dWs : (13.115) 0 0 0 3 $$, (s x) $ $ x, $.. $ (s), $ % $$ (13.114) ;$. )
(
" ( ", (
(., ., N?]). * , ) (., 12.11), / !
. Zt
279
