А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 31
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13.14. C, dXt = (Xt)dt + dWt(13.91) "" " (x) = signx . 913.15. M (13.91) dXt = (Xt)dBt X0 = 0(13.92), -, (x) = signx B = (Bt)t>0 { ( Bt Wt). C, ) , " . - , " .2 M C O0 1] W fWt t 2 O0 1]g, ..
Wt(!) = !(t), !() 2 C O0 1].( ) (., ., O?])273A 913.16 (). B = (Bt Ft)t>0{ (P -..) , $ (E F (Ft)t>0 P ). (Bt2 ; t Ft)t>0 , ..E(Bt2 ; Bs2)jFs) = t ; s 0 6 s 6 t:(13.93)B = (Bt)t>0 . 12.11 Bt =Zt0(Ws)dWs t 2 O0 1] (x) = signx(13.94) (, (Ws), s 2 O0 1] L2O0 1], ) (FtW )t201], .. W ).E. 13.17. +, (13.94) (13.93) s t 2 O0 1].E. 13.18. J (13.94) , dBt = (Ws)dWs. C, Zt0(Ws)dBs =Zt02(Ws )dWs(13.95).. (13.95) .9, 2(x) 1, (13.95) Zt0(Ws)dBs = Wt(13.96)..
Wt (13.92) . (, (;x) = ;(x) x 2 R, Zt0(;Ws)dBs = ;Zt0(Ws)dBs = ;Wt t 2 O0 1]:@ , ;Ws (13.92). =, " . % %'<, ' 913.15.+, (13.92), , ", Xt =Zt0(Xs)dBs t 2 O0 1](13.97) ( - (FtB )t201], B ). * 3 (), (Xt FtB )t201] . + @ (. (12.97))jXt j =274Zt0(Xs)dXs + Lt(0)Zt1Lt(0) = lim1 fjXs j 6 "gds:"!0 2" 0(13.98)J (13.97) (13.92), " ()Zt0(Xs)dXs =Zt02(Xs )dBs=Zt0dBs = Bt t 2 O0 1](13.99), (13.98) Bt = jXtj ; Lt(0). +, X (FtB )t201], FtX FtB FtjX j t > 0.
= ) .E.jW13.19.C, W { , jWFt Ft t > 0 .M C13.15 . 2', 4. 1 O?] , (13.92) = (x) > 0.( C13.15 W ;W (13.92) C O0 1]. 2 7 .B. 0 (13.32) , { B(R), . O0 T ], (E F (Ft)t20T ] P ), W = (Wt Ft)t20T ] X = (Xt Ft)t20T ], , L(X0 ) = P .. t > 0 (13.32). M, O0 T ] Ou v], 0 6 u < v < 1 Ou v) 0 6 u < v < 1. ( X (Ft)t20T ] 12.12. , % % , =, %- ,% %% %% & ', & % &;% ( ) <%, %&% =,, %& =,. =, .& %%, = (13.32) %, % <& = ( ) <% , ..
. 913.20. 13.6 (. 4 13.13). . ( ) (13.32) .ft Fet) (X~ t W~ t F~t) | (13.32). +2 C, (Xet WXt 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 . 2D ) " (13.32).= " (. O?], O?]) ) .
M dXt = b(Xt)dt + (Xt)dWt(13.100)275 , Zjxj (dx) = EjX0j < 1 > 2:D " b = b(x) = (x) , (13.100) .@ 913.21. " (x) ) ( ) , " b(x) (. O?]). + ) , " ,) .4'< , %% " & &<% , .+ (E 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),, PZ t0kask2ds< 1 = 1 t 2 O0 T ](13.101) kask2 = (a1s )2 + : : : + (ams)2 T < 1.= Z = (Zt Ft)t20T ], ZtZt1Zt = expf (as dWs ) ; 2 kask2dsg00 (as dWs) :=Pm ak dW k .k=1 s s(13.102) 913.22 (.
O?]). " =:1 Z tE exp 20 EZT = 1 . % .kask2ds< 1(13.103)Z = (Zt Ft)t20T ] - Zt (P -..) EZT = 1, (E FT ) QT , QT (A) = E(1 AZT ) A 2 FT :(13.104)A 913.23 ("). 4 . % W a Bt = Wt ;Zt0asds t 2 O0 T ]: B = (Bt Ft)t20T ] m- $ (E FT (Ft )t20T ] QT ).2760 ) " ' =%, % &= %% , dXt = a(t X )dt + dWt t 2 O0 T ](13.105) a(t x), t 2 O0 T ], x 2 C O0 T ], , .. t 2 (0 T ], D 2 B(R)f(s x) 2 O0 t] C O0 T ] : a(s x) 2 Dg 2 B(O0 t]) B(C O0 T ]):= , , ) , a(t x()) = b(t x(t)), b : O0 T ]R ! R.* , (E F (Ft)t20T ] Q) X = (Xt Ft)t20T ] B = (Bt Ft)t20T ] , Q-..
t 2 O0 T ]Xt =Zt0a(s X )ds + Bt( ", , ). C X0 = 0 2 Rm. E = (C O0 T ])m, F = B((C O0 T ])m),Ft = B((C O0 t])m), t 2 O0 T ] Q = QT (. (13.104)). D " , ) :,ZtBt = Wt ; a(s W )ds t 2 O0 T ]0 O0 T ]. = Xt = Wt, (13.105).= , . . dXt = b(Xt) dt + (Xt) dWt t > sQ Xs = x(13.106) Wt | m- , b : Rn ! R : Rn ! Rnm 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 > 0i=1 k=1 ikjb(x)j + j(x)j 6 c(1 + jxj) x 2 Rn:(13.108)= (13.106) t > s Xtsx .C ) n = 1.A 913.24.
4 x 2 R . % Xtsx, t > s, %.2 0 u = t + v, ( ) x 2 R, t h > 0,Xttx+h=x+Zt+hZt+httb(Xutx) du +(Xutx) dWu= x+Zh0b(Xttx+v ) dv +Zh0(Xttx+v )dW v (13.109)277 W v = Wt+v ; Wt, v > 0, 4.3.0 ,Xh0xZhZh00= x + b(Xv0x) dv +(Xv0x) dWv :(13.110)@ , Wv W v . +" C13.20 , x 2 R(Xttx+h )h>0 =D (Xh0x)h>0(13.111). .
, ) (13.111), . 2E. 13.25. ( fXtsx t > sg,) (13.106).( * ,% % %% ( % 1 { %%), Zt0f (s !) dWs (!)(13.112) fWs s > 0g { , f .T , f (13.112) ) XN ;1i=0f (ti !)(Wti+1 ; Wti ) 0 = t0 < : : : < tN = t ti = (ti + ti+1)=2, i = 0 : : : N ; 1.M t Wt(n)(!), n 2 N , Wt(n)(!) ! Wt(!) n ! 1 .. ! t . @ (. O?, .
27]) ! Xt(n)(!) dXt(n) = b(t X (n)) + (t X (n)) dWt(n)ttdtdt Xt(!) n ! 1 .. ! t .- (. O?], O?], O?]) Xt Xt = X0 +Zt0b(s Xs)ds +Zt0(s Xs) dWs :(13.113)2 dXt = b(t X ) + (t X )W_(13.114)ttdt " " W_ , (13.113), (13.33). - , 0 , ) * (. (12.50) (13.85)).2 278 (. O?], O?]).
, 0, , , * ( " ). +", (t x) x, % (13.113) ' -%, < 3%ZtZt10Xt = X0 + b(s Xs)ds + 2 x(s Xs )(s Xs)ds + (s Xs)dWs : (13.115)0004 %%, (s x) % % x, %.. % (s), %& %% (13.114) <%. * ) " )", ) (., ., O?]). + , * (., 12.11), 0 " .Zt279.
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