А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 26
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+ X = fX (t) t 2 T g T = Oa b] Oc d]. @EZbcov(aZbcov(aZbX (t)dt =aZbX (s)ds X (t)) =X (s)dsZdcaEX (t)dtZbaX (t)dt) = r { .r(s t)ds s 2 TZ bZ dacr(s t)dsdt(11.73)(11.74)(11.75)G&% % % %% &, ( )) %%%< % % Oa b], a ! ;1 b ! 1.*, L2- X = fX (t) t 2 Oa b]g r, T h, T R, Y (t) =Z1;1h(t s)X (s)ds t 2 T:(11.76), Y (t) (11.76) ), MZ 1Z 1;1 ;1h(t s)r(s u)h(t u)dsdu:(11.77)J h, ) (11.76), $%.
C, , (11.76) h C s, Y (t) = h(t s). C , h(s t) t -, s. T = R , , Y (t) =Z1;1h(t ; s)X (s)ds t 2 R.. $. 9 $$% H (i) =Z1;1 h(s)e;isds 2 R:(11.78)(11.79)D h 2 L1(R), .. 3, H , , ). T ) . J h 2 L1(R), % ' 2 R X (s) = eis, s 2 R &% &% & (11.78), %< &% <H (i). 2 ) .225E. 11.34. + (11.78) X = fX (t) t 2 Rg, ) (11.9) - Z , G. C, 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 RZ1;1eitH (i)Z (d) t 2 R:(11.80)(11.81) " , , jH (i)j2% , % - % %< X ' ,%.E. 11.35. - $-, ..
, ) ) Oa b] (" , H (i) = 1 ab]())?* %, % - =< , dPn dt Y = X(11.82) X { , Pn (z) = a0zn + a1zn;1 + : : : + an { " , Y = fY (t) t 2 Rg n t 2 R.M (11.82) .. t 2 R. (11.9) X . 1 Y , X ( " Y , X ). 11.34 11.30, Z1;1eitPn (i)H (i)Z (d) =Z1;1eitZ (d) t 2 R (), Pn (i)H (i) = 1 .. G X .
1 , , X W_ (11.49). - , X (11.82) Qm(d=dt)X , Qm(z) = b0zm + b1zm;1 + : : : + bm { m " , Pn (i)H (i) = Qn (i)(11.83) H (i) = Qn(i)=Pn (i), 2 R, Pn (z) . " , Qm(z)=Pn (z) , (. O?, . 282{284]) , Y X .K Pn (d=dt)Y = Qm(d=dt)X , , 226(t 2 T Z) { ). = " .
O?]. ) , ) ) . M L2- X = fX (t) t 2 T Rg Ht(X ), t 2 T , ) X (s), s 6 t (s t 2 T ). 0, #) % Y = fY (t) t 2 T g, .. ), Ht(X ) = Ht(Y ) t 2 T:(11.84)] O?] X = fX (t) t > 0g, (11.84) . - O?] , fGn 1 6 n 6 N g B(R), G1 & G2 & : : : ), ) Zn , 1 6 n 6 N , ), EjZn()j2 = Fn(), 2 R, Fn { Gn , ) X = fX (t) t 2 Rg, H t (X ) =NXn=1Ht (Zn ) t 2 R:(11.85)2 >:X (t) =N Z tXn=1 ;1gn(t )dZn () t 2 R(11.86) gn , n = 1 : : : N N Z tXn=1 ;1jgn (t )j2dFn () < 1 t 2 R:(11.87)+ N 2 N f1g # % X .
0 (11.86) (. O?]).227 12. 4$ 4$5 A . !. # Z # !, !. F F, Z . ? F . !. % F . - L2. F . , ) , ) , ) , , , M, 3, M { 0, 3 { 0, C(. K.(. - "* " O?]). * ) 1 + O?]. , , ) " ". @, 10 ./, -% { % % 3% %, %.%, & <% %, %& %, %%, %, , ' , %, % .
' . ( (. " ).+-, (. It(f ) =Z(0t]f (s)dWs t > 0 f (s), s > 0, " " (. O?] O?]). K , " " d(fW ) = fdW + Wdf , RIt(f ) = f (t)W (t) ;Zt0f 0(s)Ws ds(12.1) 0t f 0(s)Wsds , (.. ! 2 E) M O0 t] f 0(s)Ws(!), s > 0. 'R, (0t] f (s)dWs 3 { 0 ! , 4.1 . 1944 -.* O?] ) " ", , ) ) " .+ , , & & .
K , (Ft)t>0 { (E F P ).+ fW (t !) t > 0 ! 2 Eg, (E F P ) " , , .. , 06s<t<1W (t) ; W (s)??Fs W (t) ; W (s) N (0 t ; s):(12.2)228@ 4.3 , . 2 , , -, d () , W (t) { . ' , W (t), t > 0, Fs Fs = f W (u) u 2 O0 s]g, s > 0.M (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. D 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) 7 ) K ( , D 2 Fs Fu, D n A 2 Fu . 2B , K <Z ((s t] A !) := (W (t !) ; W (s !))1 A(!):(12.5)=, 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. 1% 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)):3 , 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 ) < 1.229J f : S ! R, S = (0 1) E, , f 2 PredjB(R).
+ f S Re f Im f . *, f 2 L2(S Pred ) Jf =Zf ()Z (d) = (t !) 2 S(12.6) f g 2 L2(S Pred )EJf = 0 (Jf Jg) = hf gi(12.7) ( ) { L2(E F P ), h i { L2(S Pred ), ..hf gi = EZ10f (t !)g(t !)dt(12.8)( J (12.8) ). ( (12.6) If =Z10f (t !)dWt:(12.9)2 2.( , f () Z (d) (12.6) !. = " . T , $%Xm;1f (t !) =k=0fk (!)1 (tk tk+1](t) t 2 (0 1) ! 2 E(12.10) f : E ! 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) E : f (t !) 2 B g =m;1k=0f(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) 7 K. 2', f (12.10) L2(S Pred ) , fk 2 L2(E Ftk P ). 2 , Z10jjf (t !) 2dt =Xm;1k=0jfk (!)j2(tk+1 ; tk ):(12.12)A 12.4. 4 $% f (12.10) If =230Xm;1k=0fk (!)(W (tk+1 !) ; W (tk !)):(12.13)2 3 , fk , L2(E Ftk P ) NnXh(kn)(!) =j =0c(knj )1 A(kn) (!) c(knj ) 2 R A(knj ) 2 Ftk j = 0 : : : Nn n 2 Nj(12.14).. Ejf ; h(kn)j2 ! 0 n ! 1 (k = 0 : : : m ; 1).@ , f (n):=Xm;1k=0()h(kn)(!)1 (tktk+1](t) L;!f n ! 1:2()0, If (n) L;!If n ! 1. + I { (. 10.5), 2If (n)XX= I(m;1 Nnk=0 j =0c(knj )1 A(kn) (!)1 (tk tk+1](t)) =jXm; 1k=0h(kn)(!)(W (tk+1 !) ; W (tk !)):= , EjXm;1k=0fk (W (tk+1 !) ; W (tk !)) ;Xm;1k=0h(kn)(W (tk+1 !) ; W (tk !))j2 ==Xm;1k=0Ejfk ; h(kn)j2(tk+1 ; tk ) ! 0 n ! 1:4 , k 2 Ftk jB(R) Ejk j2 < 1, (k = 0 : : : m ; 1), 0lrkl := Ek l (W (tk+1) ; W (tk ))(W (tl+1) ; W (tl)) = Ej j2(t ; t ) kk 6==l:k k+1k(12.15)4 (12.15) , ..Ejk j2(W (tk+1) ; W (tk ))2 = Ejk j2E(W (tk+1) ; W (tk ))2 < 1(k 2 Ftk jB(R) W (tk+1) ; W (tk )??Ftk ), 2 L2(E), 2 L1(E) - { 1 { j.
- , E(jA) = E, ??A Ejj < 1, "rkk = Ejk j2E(W (tk+1) ; W (tk ))2jFtk ) = Ejk j2(tk+1 ; tk )(12.16)rkl = E(k l(W (tk+1) ; W (tk ))E(W (tl+1) ; W (tl)jFtl )) = 0 k < l rkl = 0 k > l, (k l = 0 : : : m ; 1). 2*, , (12.10) * (12.13). - , , " (12.7). @ , % '%, %, % 3% % (12.10) (12.13), % %%, & = .231', % % J ' & %%, %% ' , %, % , '% fOs t)A A 2 Fs0 6 s 6 t < 1g.
+ " f : O0 1)E ! R,, f (t !) =Xm;1k=0f (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 O0 1) E:C 2 Prog , C \ fO0 t] Eg 2 Bt Ft t > 0(12.18) - Bt = B(O0 t]).J f : O0 1)E ! R , f 2 ProgjB(R).* , t > 0 B 2 B(R)f(s !) 2 O0 t] E : f (s !) 2 B g 2 Bt Ft:(12.19)+" %, f %, f O0 t] E Bt FtjB(R) ; .". t > 0(12.20) 12.5. $% (12.17) .2 B 2 B(R) t > 0. D t 2 O0 t1], f(s !) 2 O0 t] E : f (s !) 2 B g = O0 t] f! : f (0 !) 2 B g 2 Bt Ft:D t > t1, N = maxfk : tk 6 tg, f(s !) 2 O0 t] E : f (s !) 2 B g = Nk=0;1fOtk tk+1) f! : f (tk !) 2 B g OtN t] f! : f (tN !) 2 B g 2 Bt Ft:4 , f! : f (tk !) 2 B g 2 Ftk Ft tk 6 t.
2+ L2 = L2(O0 1) E Prog ), = mes P , .. h 2 L2EZ10jh(s !)j2ds < 1:(12.21)D h { , , Re h Im h { .', f (12.17) L2 , f (tk ) 2 L2(E Ftk P ), k = 0 : : : m, (12.12).9 % f (.. (12.17)) fW (t) t > 0g() (12.2) (Ft)t>0) % 3% (12.13)If =232Xm;1k=0f (tk !)(W (tk+1) ; W (tk )):(12.22)2 . D (12.17)f (t !) =r ;1Xj =0g(sj !)1 sj sj+1 )(t) mes P { ..,P;1 g (s ! )(W (s ) ; W (s )) P { ..
0 = s0 < : : : < sr < 1, rj=0jj +1j (12.13). C,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 !)). +" , O0 1). @ " P { ..@ 12.6. (12.22) W (t) ( ).
= ), , "" , . 12.7. 4 $% f g 2 L2 (12.7).P2 + f (12.17), g(t !) = mj=0;1 g(tj !)1 tj tj+1)(t). -7 , , , ) O0 1). *XXm;1 m;1(If Ig) =k=0 j =0Evkj (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 9 u 2 L2O0 1) = L2(O0 1) B(O0 1)) mes) > 0 1Xu (t) =k=0u(k)1 k (t)(12.24) ^k = ^k () = Ok (k + 1)), k = 0 1 : : : u(0)= 0u(k)= ;1Zk;1u(t) dt k > 1:(12.25)+ -{1{j m > 1Zmju j(t) 2 dt =jj 6u(m) 2Zm;1ju(t)j2dt(12.26)233"Z10ju(t)j2dt = u () 2 L2O0 1).C, +u(t) =Xm;1j =01 ZXk=1 kju (t)j2dt 61 ZXZ1k=1k;1ju(t)j2dt = ju(t)j2dt0u () ! u() L2O0 1) ! 0.(12.27)(12.28)rj 1 tj tj+1 )(t) 0 = t0 < : : : < tm < 1Q rj 2 C j = 0 : : : m ; 1:(12.29)@ ^k;1 ^k Otj tj+1), u (t) = u(k) = rj = u(t) t 2 ^k .
@, u u e j = ^ij ^ij +1 tj , j = 0 : : : m. 9,() ^ tj , (12.28). @ , (12.26)ZSm e jju(t) ; u j(t) 2dt 6 2j=0ZSm e jjju(t) 2dt + 2j=0ZSm e jju jj=0(t) 2dt 6 4ZSm 0iju(t)j2dtj=01 ^ij ^ij +1 (^;1 = ?), u 2 L2 O0 1) R ju(^t)0ij2dt= !^i0j ;mes B ! 0.B+ u | L2O0 1). C " > 0 v(") (12.24), ku ; v(")k < ", k k | L2O0 1). @ku ; u k 6 ku ; v(")k + kv(") ; v(")k + kv(") ; uk 6 2" + kv(") ; v(")k < 3" .
4 , v(") ; u = (v(") ; u) , (12.27).4, %, h(t !) 2 L2 R ! 2 E = 1=n (12.24), (12.25) h1=n(t !). D h(t !) dt ), k . * (12.21) J , Z10jh(t !)j2dt < 1 ..R0, .. ! ) h(t !) dt k = 0 1 2 : : : . -kR, Jh(s !) ds Fk jB(C )- . 4 k;1 (12.20). + (12.27) (12.21), , h1=n(t !) { L2 ( , ). (12.27)Z10234Z1jh(t !) ; h1=n(t !)j2dt 6 4 jh(t !)j2dt ..00 (12.28) Z10jh(t !) ; h1=n(t !)j2dt ! 0 .. n ! 1.0, 3 Z1E jh(t !) ; h1=n(t !)j2dt ! 0 n ! 1:0= , (12.24) u (t N ) =NXk=0u(k)1 k (t) (12.28) > 0 u( N ) ! u() L2O0 1) N ! 1:0, h1=n(t !) n 2 N , Z1jh1=n(t !) ; h1=n(t !Q N )j2dt ! 0 N ! 1: 20A, %, I %% L2. 'EIf = l:i:m:Ifn(12.30)L f n ! 1, (12.30), , fn ;! L2(E).
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