А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 16
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+ , 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). 2B, & % '%% 7.6. X | % , # m 2 N s1 < : : : < sm 6 t 6 u ( T ) # C 2 BP (Xu 2 C j Xs : : : Xsm Xt) = P (Xu 2 C j Xt) ..(7.12)12 g = 1 C , (7.12) 7.5. C - , 7.1. 2)%, &%130A 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, 1121 . ( , 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=11 1 = Xs1 , 2 = Xs2 ; Xs1 , : : : , m = Xsm ; Xsm;1 , m+1 = Xt ; Xsm , = Xu ; Xt.0, .. (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 = Ui=1yi U | . C, h1(! z) = (!) h2(! z) = z, E R, , F B(R)jB(R)-. 0, g((!) + z) = g((h1 + h2)(! z)) ()F B(R)jB(R)- .
F B(R) P Q (P Q { F B(RR ), .. F B(R) ), , J (. O?, . 363]) g((h1 + h2)(! z))dP .@ , mX+1 E(g(Xu ) j Xs1 : : : Xsm Xt) = Ui=1i..@ , 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 Ui i = Ui .. 2i=1i=1i=1F ' Rm W (t) = (W1(t) : : : Wm (t)),t > 0, m Wi( m .". C O0 1)).131% 7.8. A Rm %.
% %.A, < <, % % <<, % .J 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 0 { Q:P (s x t B ) =ZXP (s x u dy)P (u y t B ):(7.14) 3) (7.14) s 6 u 6 t.:, fXt t 2 T g &% P (s x t B ), s 6 t (s t 2 T ), B 2 BP (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 . 9- , %% 7=8 % ,% % (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 { xP (s x t B ) = E(P (u Xu t B )jXs = x):(7.16)= (. O?, . 242]), (C ) = P (Xu 2 C jXs = x) ( s u x), g 2 BjB(R) E(g(Xu )jXs = x) =ZXg(y) (dy) PXs ; ..(7.17)*, (7.16) (7.17) %% =, & % % (7.14), PXs - x. 7.9. + W (t), t > 0, | m- .
@, (7.13), s < tP (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 ;kzk2Z ;kz;xk211= (2(t ; s))m=2 e 2(t;s) dz = (2(t ; s))m=2 e 2(t;s) dz:B ;x132B+ 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 < t1P (s x t B ) = (2(t ; s))m=2ZBe;ky;xk22(t;s) dy(7.18) 1){4), , ( (7.14) , ).4 fXt t 2 T g, ) P (s x t B ), , x 2 X , B 2 B, h > 0 s s + h 2 TP (s x s + h B ) = P (0 x h B ):(7.19)+ (7.19) P (x h B ), " B h, x.
0 (7.18),m- & ' { . C 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 ) =ZXP (x s dy)P (y t B ):(7.20)A& (7.20) % % % %, % % . + " " .B&% '< .-.. . 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 ) ..1330 ,ZXP (s x t dz)1 B (z) = P (s x t B ):PN0, g() = ck 1 Bk , Bk 2 B, k = 1 : : : N .k=1D 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) z2XE(g(Xt) j Xs)= klimE(g(X)jX)..,s!1 k tZXP (s x t dz)gk(z) !RZXP (s x t dz)g(z) (k ! 1):B j B(R)- P (s x t dz)gk (z) gk XR 2) , B jB(R)- P (s x t dz)g(z) X x s t 4.7 ( - ).
2 7.11. : 7.10 # n 2 N, s 6 t1 6 : : : 6 tn( T ) # B jB(R)- $% g1 : : : gnE(g1(Xt1 ) : : : gn (Xtn ) j Xs ) = U(Xs )U(x) =ZXZZXX(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 + .
C 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) =ZXP (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 . 2134A 7.12 (.-.. ).
7.10 # n 2 N, # s 6 t1 6 : : : 6 tn ( T ) B1 : : : Bn 2 BP (Xt1 2 B1 : : : Xtn 2 Bn) =ZXZZB1BnQs(dx) P (s x t1 dz1) : : : P (tn;1 zn;1 tn dzn )(7.24)Qs = PXs .2 C 7.11 gi = 1 Bi (7.23) (2.10). @EU(Xs ) =ZXU(x)Qs(dx): 2@ 7.13. J (7.24) (Xt : : : Xtn ) "" B1 : : : Bn, " Bn = B : : : B. + 7.2 7.11, ) (7.24). D v 6 s(v s 2 T ), B 2 B1ZQs(B ) = Qv (dx)P (v x s B ):(7.25)X2 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 ): 2X , T = O0 1), Qs s > 0 Q0 ( , 7 { ).@ , X = fXt t > 0g | %% (X B) ( t > 0), % %< <, % .-..
%,< <% , (. . Q0(B ) = P (X0 2 B ), B 2 B) P (s x t B ).D , , . , C7.1. .B % (7.1) -% ' && %, . + , ', 7, %' 7.7 { 7.9, % , <% < , X = fXt t 2 T Rg { , -% %% %% (E F P ), < ' t % %% (Xt Bt), t 2 T .
C , 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). (, - { T s 6 u 6 t (s u t 2 T ), x 2 Xs, B 2 Bt P (s x t B ) =ZXuP (s x u dy)P (u y t B )(7.26) (7.15) , B 2 Bt.G % < , % .+ (Xt Bt)t2T { . C n 2 N s0 6 t1 < : : : < tn( T ), Bk 2 Btk , k = 1 : : : n ' ( , ) C = B1 : : : BnQt1:::tn (C ) =ZXs0Qs0 (dx)ZXt1P (s0 x t1 dz1)1 B1 (z1) ZXtnP (tn;1 zn;1 tn dzn )1 Bn (zn)(7.27) Qs0 { Bs0 .
(7.27) , 7.10 . +, Qt1::: tn { B1 : : : Bn . D B1 : : : Bn = 1q=1B1(q) : : : Bn(q), 7 (Bk(q) 2 Btk , k = 1 : : : nQ q 2 N), 1 B1:::Bn (z1 : : : zn) =1Xq=11 B1(q) (z1) 1 Bn(q) (zn):(7.28)@ 1. 3 (. O?, . 348]), (7.27) (7.28). +" ' %%, Qt1:::tn ' Bt1 : : : Btn .< % % &% % 7.12 ( ).A 97.1.
(Xt Bt) { - Bt t 2 T R. . Qt1 :::tn , t1 : : : tn 2 Ts0 = T \Os0 1),n 2 N ( s0 2 T ), # , (E 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 XuZP (u x t dz) = P (u x t Xt) = 1:Xt136G PXs s 2 Ts . n = 1 t1 = s (7.27). @ZZZP (Xs 2 B ) = Qs (dx) P (s0 x s dz) = Qs (dx)P (s0 x s B ):00Xs0Xs0B0(7.29) , s = s0 P (Xs0 2 B ) = Qs0 (B ) 3) .9', % (7.15). 0 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)3 A 2 fXs g A = fXs 2 Dg D 2 Bs.
+(7.27), E1 fXt2Bg1 A = P (Xs 2 D Xt 2 B ) =ZZXs0Qs0 (dx) P (s0 x s dz)P (s z t B ): (7.31)D+ (7.30), J, ZEP (s Xs t B )1 fXs2Dg ==ZXsXsP (s z t B )1 D (z)=ZXs0ZZXs0P (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 BtnP (Xtn 2 Bn jXt1 : : : Xtn;1 ) = P (Xtn 2 BnjXtn;1 ):C " , A 2 fXt1 : : : Xtn;1 gE1fXtn 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 ): 2137% 97.2. % Ts = T \ Os0 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. + C7.1 ) X sx = fXtsx t 2 Ts := Os 1) \ T g, ) 0 , Xssx = x .. C7.1, - ( 1.7), , X sx (. (1.8)) (XTs BTs ), Qsx = L(X sx ). X sx E = XT Ytsx(!) := Xtsx(TTs!) t 2 Ts ! 2 XT (7.34) TTs T Ts;1 BT Psx = Qsx ;1 .(. . 8).
- F>s := TTTTss ssxE. 7.3. =7, Yt , t 2 Ts { (E F>s Psx).A 97.4. 4 s 2 T # x 2 Xs % Y sx = fYtsx(!),t 2 Ts, ! 2 Eg (E F>s Psx) $% P (s x t B ).> , Yssx = x .. Psx .2 M Yssx = x , Qs = x. +, u 6 t ( T ), B 2 BtPsx(Ytsx 2 B jYusx) = P (u Yusx t B ) Psx ; ..(7.35)C , D 2 BuPsx(Ytsx 2 B Yusx2 D) =ZfYusx 2DgP (u Yusx t B )dPsx:+ (7.34), (2.10), X sx (XTs BTs Qsx), ZfYusx 2DgP (u Yusx t B )dPsx=ZfXusx 2DgP (u Xusx t B )dQsx == Qsx(Xtsx 2 B Xusx 2 D) = Psx(Ytsx 2 B Yusx 2 D):4 Y sx (7.12) . + " .
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