А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 27
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fn 12.8, , (12.30) ) , ) 12.7. + " (. (12.7))(If Ig) = hf gi E(If ) = 0 f g 2 L2:(12.31)* ) :2If =Z10f (t !) dWt f 2 L2 :(12.32)C 0 6 t1 < t2 6 1 f 2 L2 Zt2t1f (t !)dWt =Z10f (t !)1 t1t2)(t) dt:(12.33)C f f 1 t1 t2) . ) L2, ) mes P -.. 4.7. t1 = t2 % % (12.33) <.+ (12.32), , 0 6 v < s < u 6 1Zuvf (t !) dWt =ZsvZuf (t !) dWt + f (t !) dWt ..s(12.34)235@ 12.9. C T > 0 (..) R f (t !) dW , (12.33) T0t * (12.17), )tn = T ( , f (t !) = ftn;1 (!) t 2 Otn;1 T ]). (12.18) " t 2 O0 T ].A 12.10.
$% f (t !), 0 6 t 6 T < 1, . ZTf (t !) dWt = ln:i!1:m:n;1Xf (t(kn) !)(W (t(kn+1) ) ; W (t(kn)))k=00 0 = t(0n) < : : : : : : < t(nn) = T , . . n = 06max(t(n) ; t(kn)) ! 0 (n ! 1):k6n;1 k+1n;12 J f (t(kn) !)1 t(kn)t(kn))(t) { L2 k=0(12.35)PIfn =n;1Xk=0f (t(kn) !)(W (t(kn+1) ) ; W (t(kn))):L2= , fn ;!f O0 T ].C,ZT0Ejf (t !) ; fn(t !)j2dt 6 T sup Ejf (t !) ; f (s !)j2 ! 0jt;sj6n n ! 0 , , , " ( ).
2+ " , , ZT0Wt dWt = l:ni:!m0:n;1X2W (t(kn))(W (t(kn+1) ) ; W (t(kn))) = W2T ; T2 :k=0+, & (12.35) f '%Ot(kn) t(kn+1) ) % ,< ,. (,n;12Xl:ni:!m0: W (t(kn+1) )(W (t(kn+1) ) ; W (t(kn))) = W2T + T2 :k=02J f 2 L Yt =Zt0f (s !) dWs t > 0R(12.36) (12.33) 01 f (s !)1 0t)(s)dWs (Y0 = 0). C (12.36) It(f ) (f B )t, B = fB (s) s > 0g { .236A 12.11. t > 0 # Yt, # $- (12.36), ( ), #) :1) (Yt Ft)t>0 | 82) .. ! Yt(!) O0 1)83) .$. Yt(!), 0 6 t < 1, .2 9' 1). + f 2 L2 f (t !) = 0 .. 0 6 t < s. @ ,E(If j Fs) = 0 ..(12.37)D f | (12.17) L2 s > 0, s = t1.@E(If j Ft1 ) =n;1Xk=1E(E(f (tk )(Wtk+1 ; Wtk ) j Ftk ) j Ft1 ) ==n;1Xk=1' , E(If j F0) =E(f (tk )E(Wtk+1 ; Wtk j Ftk ) j Ft1 ) =n;1Xk=0n;1Xk=1E(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 ;!n2L2= fn1 s1) 2 L . . f (t) = 0 .. 0 6 t < s, gn ;! f . 0,2 ()2 ()Ign L;!If n ! 1. ( E(Ign j Fs) L;!E(If j Fs). C, 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 j2:9, E(Ign j Fs) = 0 .., (12.37) ) .
(12.34) (12.37), s 6 t2 Z1E(Yt j Fs) = Ys + ERt0f (u !)1 st)(u) dWu j Fs = Ys ..(12.39)4 , f (s !) dWs f Ft jB(R)-0 (12.18), (12.22), (12.30) 4.7. *, 1) . + " , Yt , " (- ).9' 2). D f | , (12.17), 8f (t0)(Wt ; Wt0 ) = f (0)Wt t 2 O0 t1)>>m;1>XZt< f (tk )(Wtk+1 ; Wtk ) + f (tm)(Wt ; Wtm ) t 2 Otm tm+1)f (s !) dWs = > k=0m = 1 : : : n0>>>:nXk=0f (tk )(Wtk+1 ; Wtk )t > tn:(12.40)237LC f 2 L2 fn ;!f n ! 1. 9 Rt fn(s !) dWs0(. (12.40)), , " ! O0 1).* fn fnk , 2Z1E 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 ZT2Zt;2P sup fn (s !) dWs ; fm(s !) dWs > " 6 " E (fn ; fm)dWs :06t6T000(12.42)* (12.41), (12.42) (12.31) , Zt 1 X1P sup (fnk ; fnk )dWs > k2 6 k42;k < 1:06t6Tk=1k=11 X+10+" 1{- ..
! k > K(!) Ztsup (fnk ; fnk )dWs 6 k;2:06t6T+10D O0 T ] .. , " .. O0 T ] . @ , t > 0Zt0fn1 (s !)dWs +XZN ;1 tk=1 0(fnk+1 ; fnk )dWs =Zt0L2 ()fnN (s !) dWs ;!Zt0f (s !) dWs():: N ! 1. ( t 2 O0 T ] k (t) ;!(t) k (t) L;! (t), L2 ()(t) = (t) .. t 2 O0 T ] (.
. k (t) ;! (t), mj (t) ! (t) .., " (t) = (t) ..).@ , O0 T ] Yt.2 O0 n], n 2 N. @, Xt = Zt .. t 2 O0 T ] " .. O0 T ], P (! : Xt(!) = Zt(!) t 2 O0 T ]) = 1. C, O0 T ] MT . @ Xt Zt " > 0P (! : sup jXt (!) ; Zt(!)j > ") = P (! : sup jXt(!) ; Zt(!)j > ") = 0:2t20T ]t2MT* , 7 , , Yt ..
O0 1).0 2) .2389' 3). + 1) , Yt Ft j B(R)- t > 0 ( - ). - , O0 1) .. ! Yt , 12.12. % X = fX (t !) t 2 O0 T ] ! 2 Eg , .. O0 T ). " Xt 2 Ft jB (R) t 2 O0 T ], (Ft)t20T ] { $%, % ) $%.2 + m > 2X (t !) =Xm;1k=0X (tk !)1 k (t) t 2 O0 T ] ! 2 E(12.43) 0 = t0 < : : : < tm+1 = u, ^0 = O0 t1], ^k = (tk tk+1], k = 1 : : : m ; 1.
@, , . =) . K , ! 2 E0, P (E0) = 1, X O0 T ), Xn (t !) = X (q (t) !)nX (T !) qn (t) := (O2n t] + 1)2;n 6 T qn (t) > T t 2 O0 T ], ! 2 E, n 2 N, O] . @ Xn (12.43). +Vn (t !) = Xn (t !)1 0 (!) t 2 O0 T ] ! 2 E:3 , Vn t 2 O0 T ], ! 2 Elim V (t !) = X (t !)1 0 (!):n!1 n= 4.6. 2@ 12.11 . 2+c f 2 L2 Yt, ) 12.11sup EjYtj 6 sup(EjYtjt>0t>0R2 )1=2= sup(Et>0Zt0jjf (s !) 2ds)1=26 (EZ10jf (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(O0 T ] E Prog mes P ) T < 1, 12.11 t 2 O0 T ]. .2394 -& Pred Prog '% < % Xt , t 2 T R, % ' % t 2 T " % %& " (t !). + " ) ". ' , O0 1) ( (0 1)) T R ( " O?], x6.2). K ,Pred , -, (T \ (t 1)) B , t 2 T , B 2 Ft. 4 A T E Prog, A \ ((;1 t] E) 2 Bt Ft t 2 T Bt = B(T \ (;1 t]).
= , * ) (0 1), * O0 1). + " - , .. . O?], x6.2, - Pred Prog T E (T R), , , . 153 A 912.1. ,# $% # $% - (Ft)t2T R( Pred Ad, Prog Ad, # A T E - Ad T E, f! : (t !) 2 Ag Ft t 2 T )." T . t0 , # $%Xt (t 2 T ) , Xt () = const.E. 12.2. C, T { , , Pred Prog.E. 12.3. + , 0.E. 12.4. + fXt t 2 T Rg { - { ( (Ft)t2T R).
C, X (!)(!) f! : (!) < 1g - F B(R).B= %% , % 3% (0 1). +(Ft)t>0 { (E F P ), fW (t) t > 0g{ . M 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. @ ) (., ., O?, I, . 45]).A 912.5 (9&).
H L2((0 1) E A ), A { -Pred = mes P , mes { , (0 1).* 12.8 (12.10), , , 12.5.E. 12.6. C, f 2 L2((0 1) E A ), h 2 H , f (t !) = h(t !) .. .240E. 12.7. + { , (!) 6 T ! 2 E(T { ). = It(f ) (12.36).C, I (f ) = IT (f 1 (0 ]), I (f ) := I (!)(f ).E. 12.8. C, f : O0 1) ! R 0 DO0 1), , f { " cadlag- , .. ) t > 0 O0 1), It(f ),t > 0 { . ( .0 "" . 0 X = fXt t > 0g Rm , (), a > 0 b > 0, Law(Xat t > 0) = Law(bXt t > 0):(12.44)* , = (t ! at) % % ' ,%%, % = (x ! bx).
D - (12.44) a > 0 b = aH , X % 9 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:E. 12.9. =7, fBH (t) t > 0g { ].+ BH K.(. - 1940 .
O?], :. @ ( , "fractional") 1. 4 q. ( 1968 . O?], ) .A 912.10 (),&%, 4 G). 4 0 < H < 1 t > 0 Z0BH (t) = cH f;1O(t ; s)H ;1=2 ; (;s)H ;1=2]dWs+Zt0(t ; s)H ;1=2dWs g(12.45)2 (1) = 1 (., ., M?, #) cH , EBH. 281]), fWs s > 0g fBs = W;s g s 6 0 { %.J ( ) , , (. O?], O?]). 0 BH (t) 0 < H 6 1 , H = 1=2 (.. ) " (.. , , .
O?, . 4]). D (12.45) H Ht (.. jHt ; Hs j 6 cjt ; sj, > 0) (0 1), , $ . 2 O?].241E. 12.11. (. O?]). C, BH n ! 1 XHbn := ln(n;1 jBH (k=n) ; BH ((k ; 1)=n)j)= ln(1=n) ! Hnk=1..1 , .. O0 1) Xt $$%, 1 t > 0ZtZt00Xt = X0 + f (s !) dWs + g(s !) ds(12.46) f 2 L2(O0 1)), g : O0 1) E ! R ,P Z10jg(s !)j ds < 1 = 1(12.47)( fWt t > 0g (Ft)t>0, ) (12.2)). f g (12.46) .. .0 (12.46) dXt = f (t !) dWt + g(t !) dt:(12.48)C F F (t), Ft. ', , t 2 Ou v], 0 6 u < v < 1.*, ( ), .
* . ' '= % < ,%%, .A 912.12 ( 3%). % Xt, t > 0, $$% ( $% f g , #) . ). $% h : O0 1) R ! R , )# @h=@t, @ 2h=@x2, sup j@h(s x)=@xj 6 M0 < 1s>0 x2R= h(t Xt), t > 0, @h (t X )dX + 1 @ 2h (t X )(dX )2dYt = @h(tX)dt+t@t@x t t 2 @x2 t t (dXt )2 )# (12.48), :(12.49) % Ytdt dt = dt dWt = dWt dt = 0 dWt dWt = dt:2 , ..
242z>0(12.50)(12.51)Zzh(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)(sX)+gsss@t@x2 @x204 " , . 912.13. ?? $% f g , . .f (s !) =Xm;1j =0fj (!)1 tj tj+1)(s) g(s !) = 0 = t0 < t1 < : : : < tm = z , fj , Xm;1j =0gj (!)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 3 * (12.52) , $Ztj @h2h@h1@2h(tj+1 Xtj ) ; h(tj Xtj ) =@t (s Xs ) + gj @x (s Xs ) + 2 fj @x2 (s Xs ) ds +tjZtj @h+1+1+1+tjfj @x (s Xs) dWs j = 0 : : : m ; 1: (12.55) j 2 f0 : : : m ; 1g ) u = tj ,v = tj+1.
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