А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 29
Текст из файла (страница 29)
) A = (At Ft)t2R+ , M = (Mt Ft)t2R+EZ(0t]Ms dAs = EZ(0t]Ms; dAs 0 < t < 1:= S { (Ft)t>0 Sa{ , P ( 6 a) = 1, a > 0.:, X = (Xt Ft)t>0 D, fX g 2S X 2 DL, 0 < a < 1 fX g 2Sa .0<< , % < %, ,%%.A 912.28 (' 9& { )). $% (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 { %.E. 12.29. + X = (Xt Ft)t>0 { . C, X 2 DL, ) .1. X > 0 ..2. X (12.107).G% % % M = (Mt Ft)t>0 2 Mc2.+ hM i { ) (), ) C253{ 4 Mt2, t > 0 ( ?? ??).+ X = fXt t > 0 ! 2 Eg , ) ) ftng1n=0 t0 = 0 limn!1 tn = 1, ffn (!)g1n=0 0 < c < 1, supn>0 jfn(!)j 6 c ! 2 E, fn 2 Ftn jB(R) Xt (!) = f0(!)1 f0g(t) +1Xk=0fk (!)1 (tktk+1](t) 0 6 t < 1:(12.108)- L0 X 2 L0 It(X ) =n;1Xk=0fk (Mtk+1 ; Mtk ) + fn (Mt ; Mtn ) tn 6 t < tn+1:(12.109)C , (12.109) ( ) L0 . * , " , hM it ..
! 3. -, , ) M 2 Mcloc , ZTP(0Xt2(!)dhM it < 1) = 1 T > 0:4 O?], O?, ?], O?, ?], O?]. ) J { 0.254 13. $$ *0 (!. . $ F. ? , C (!. "C{0 . > 2 # C # . 4# C # .+ ) , . % %, ' % ' . C " mv_ = ;v + %& t > 0(13.1) m | , v | , > 0 (, ) , ), %& ( ).+, (13.1), W_ , W = (Wt Ft)t>0 { (. (12.2)).
0 4.1, ) . ', " 7 | - . ( ) _ t > 0mv_ = ;v + W(13.2) '. + (13.2) _ t > 0v_ = 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 +Zt0ea(t;u)f (u) du t > 0(13.6)( ) t 2 O0 T ], , , f | O0 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) ,< , %, Zt0dv(s) =ZtZtav(s) ds + dW (s)0(13.9)0 * ( , , L2(O0 T ]) T > 0). " (13.9)Ztv(t) ; v(0) = a v(s) ds + W (t):(13.10)0M , f (u) du (13.6) dW (u) Ztv(t) = v(0)eat + ea(t;u) dW (u)(13.11)0 * ( e;au 2 L2 (O0 t]) t > 0). +, v(t), (13.11), (13.9), (13.10). @ ,', % %%, v(t) < < %%= (13.10), % &% ' t %%,<. = . K , 12.11 E0 E, P (E0 ) = 1 ! 2 E0 Rt (13.11) t. @ ! 2 E0 , v(s) ds 0 t ( t 2 O0 T ]). ( (13.10) ..
, t .. 0, .. ! . -% (,) = %% (13.9)' &% %, , % %%,< &% %% , < % ( ).( . 13.1. $% g(s u), O0 T ] O0 T ], 0 < T < 1, , g 2 L2(O0 T ]2) = L2(O0 T ] O0 T ] A ) | , O0 T ], A { B (O0 T ]) B (O0 T ]) , ^n =ZTZT0 0256(g(s u) ; gn(s u))2ds du ! 0n ! 0(13.12) n>1n;1Xgn (s u) =i=0g(s u(in))1 (u(in) u(i+1n) ] (u)0 = u(0n) < u(1n) < : : : < u(nn) = T , n = i=0max(u(n) ; u(in)).:::n;1 i+1ZT ZT0g(s u) ds dW (u) =0ZT ZT00(13.13) #g(s u) dW (u) ds:1(13.14)RT2 + - u 2 O0 T ] f (u) = g(s u) ds B(O0 T ]) jB(R)-0 J. C,ZT ZT0 0jg(s u)jds du < 1(13.15) g 2 L2(O0 T ]2).
- , -.. u ZT02g(s u) ds 6 TZT0g2(s u) ds:(13.16)+" f 2L2(O0 T ]) , O0 T ].0, (13.14) ).RTJ J (s !) = g(s u) dW (u) .. ! B(O0 T ]) j B(R)-.02 J, ZT0EjJ (s !)j ds < 1:+ , . .j(13.17)ZTj = g2(s u) duE J (s !) 2(13.18)0 P | () B(O0 T ]) F .0 (12.13)ZT ZT00gn (s u) dW (u) ds =ZT Xn;10i=0=n) )gn (s u(in))(W (u(i+1n;1Xi=0n) )(W (u(i+1;;W (u(in))) ds =T(n)W (ui )) gn (s u(in))ds:0Z(13.19)257RT J g(s u) ds ) - u 2 O0 T ].
D0RT g(s u(n))ds ), .i0+ *, ZT ZT0gn (s u) ds dW (u) =0ZT Xn;1 ZT0i=0 0=g(s u(in))ds1 u(in) u(i+1n) ) (u)n;1Xi=0n) )(W (u(i+1;W (u(in)))dW (u) =ZT0g(s u(in))ds: (13.20)*, g = gn (13.14) .0 J, 3, -{1{j, (12.31) (13.12), , ZT ZT ZT ZT Eg(s u) dW (u) ds ;gn(s u) dW (u) ds 60000ZT ZT6 E (g(s u) ; gn (s u))dW (u)ds 60021=2ZT ZT6=E0ZT ZT000(g(s u) ; gn (s u)) dW (u)(g(s u) ; gn1=2(s u))2duds =ds 6 (T ^n)1=2 ! 0 n ! 1:(13.21)K ZT ZTZT ZTEg(s u) ds dW (u) ;gn(s u) ds dW (u) 60000TT Z Z21=26 E=0Z ZT0T00(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)C ) "2 ()2 (). D n L;!, n L;! n ! 1 n = n .. n 2 N, = ..
22584 < '. *Zta v(s) ds = a v(0)0ZtZt Zseasds + 00 ea(s;u)dW (u)0= v(0)(eat ; 1) + aZt Zt0ds =ea(s;u)10(0s] (u)dW (u)ds: (13.23) 13.1 t > 0 Zt Zt0ea(s;u)10(0s] (u)dW (u)=Zte;au Ztu0ds =Zt Zteas ds0ea(s;u)10(0s] (u) dsdW (u) =ZZ11a(t;u)dW (u) = a edW (u) ; a dW (u): (13.24)tt004 , g(s u) = ea(s;u)1 (0s](u), s u 2 O0 T ], (13.12) O0 T ] u(0n) u(1n) : : : u(nn), n ! 0 (n ! 1).
C, ea(s;u) O0 T ] O0 T ], ^n 6 "2nX06i<k6nn) ; u(n) ) + e2aT(u(kn+1) ; uk(n))(u(i+1in;1Xi=0n) ; u(n))2 6 "2 T 2 + T e2aT (u(i+1nin "n = supfjea(s;u) ; ea(x;y)j : js ; xj 6 n ju ; yj 6 n s u x y 2 O0 T ]g.0, (13.10) (13.23) (13.24) v(0)(eat ; 1) + Zt0ea(t;u)dW (u) ; W (t) + W (t) = v(t) ; v(0)(13.25) (13.11).
@ , A 13.2. 4 t > 0 ) (.. ) . , v jt=0 = v (0), $ (13.11).@ 13.3. D (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 A 13.4. v(0) N 0 , = ;a, v(0) 2F0 j B(R)-. . , !"{$, . . % v (t), t > 0, #) %# $%#2cov(v(s) v(t)) = 2 e;js;tj s t > 0:(13.27)2 * (13.26) , Ev(t) = 0 t > 0. C s t > 0 2Ev(s)v(t) = E(v(0))2ea(s+t) + 2a ea(s+t)(1 ; e;2a(s^t)):(13.28)4 , (12.33), (12.31) (12.8)E Zs Zs^tZtea(s;u)dW (u) ea(t;u)dW (u) =000ea(s;u) ea(t;u)du = e 2a (1 ; e;2a(s^t))a(s+t)(13.29) , t > 0 (12.38)Ev(0)Zt0ea(t;u)dW (u) = Ev(0)E Zt0ea(t;u)dW (u) j F0= 0:(13.30)2D E(v(0))2 = ; 2a , (13.28) (13.27).9' %, v(t), t > 0.
C k 2 N 0 6 t1 < : : : < tk 6 TRt (v(0) X (t1) : : : X (tk )), X (t) = ea(t;u)dW (u), .0C, tm (m = 1 : : : k) X (tm) L2(E) n ! 1 Ih(nm), h(nm) | (., (k ), 12.10). (Ih(1)n : : : Ihn ) , . .Pk c Ih(nm) c 2 R, m = 1 : : : k, ( mmm=1 ). + , v(0) F0 jB(R)- , ,(k) (v(0) Ih(1)n : : : Ihn ) | . 2 E exp iv(0)0 + ikXm=1mIh(nm)i P m Ih(m)= Eeiv(0)0 E(e m=1 nki P m Ih(nm)iv(0)0= EeEe m=1kj F0) =j 2 R j = 0 : : : k: (13.31)0 L2(E), . 2< %% = %% %, %& %% -% dt dW (t) %, , .
. -) !. M dXt = b(t Xt) dt + (t Xt)dWt 0 6 t 6 TX0 = Z260(13.32) ZtZt00Xt = X0 + b(s Xs ) ds + (s Xs ) dWs 0 6 t 6 T(13.33) b O0 T ] R. , = % % Xt(!), < .. %%, % <& t > 0 % < < % (13.33) % % %%,< 1. +" , b(s Xs) (s Xs) | , (s Xs) 2 L2(O0 T ]), b(s Xs ) 3 ..
!.%&, b(u x) (u x) O0 t] R (..BO0 t] B(R)jB(R)-) t 2 O0 T ], 3: ) L > 0, jb(t x) ; b(t y)j + j(t x) ; (t y)j 6 Ljx ; yj x y 2 R t 2 O0 T ]:(13.34)+ c > 0b(t x)2 + (t x)2 6 c(1 + x2) x 2 R t 2 O0 T ]:(13.35)', 0, . . , ) ), . D, , b(t x) = b(x), (t x) = (x), (13.34) (13.35).C 13.5. % Ys (! ) O0 T ] ( $% (Fs)s20T ] F ). $% a(s x) O0 T ] R t 2 O0 T ] O0 t] R,.. B (O0 t]) B(R)jB(R)-. % a(s Ys (! )) ( $% (Fs )s20T ]).2 C s 2 O0 t] ! 2 E (0 6 t 6 T ) (s !) 7! (s Ys (!)) B(O0 t]) FtjB(O0 t]) B(R)-. u 2 O0 t] B 2 B(R), f(s !) 2 O0 t] E : (s Ys(!)) 2 O0 u] B g == f(s !) 2 O0 t ^ u] E : Ys (!) 2 B g 2 B(O0 t ^ u]) Ft^u BO0 t] Ft:@ 1.2.
@ , (s x) 2 O0 t]R (s x) 7! a(s x) B(O0 t])B(R)jB(R)-.= , . 2A 13.6. $% b # . . Z F0 jB (R)-, EZ 2 < 1. ) . (13.33) X0 = Z , Xt 2 L2 (E) # t 2 O0 T ] $% EXt2 O0 T ].261J%%, () , . . X1(t !) X2(t !) | (13.33) , ) .. , X1(t !) = X2 (t !) 0 6 t 6 T ..(13.36)2 1 (13.33) . Xt0 = Z , t 2 O0 T ] n > 1Xt(n)=Z+Zt0b(s Xs(n;1))ds +Zt0(s Xs(n;1))dW (s):(13.37) 13.7. 13.6. n 2 N(n)% fXt t 2 O0 T ]g, $ (13.37), (n)(.. # $%#). supt20T ] EjXt j2 < 1, n 2 N (13.37) .. O0 T ].2 J Ys (!) = Z (!), s 2 O0 T ], ! 2 E, , Z 2 F0jB(R). + 13.5 b(s Z (!)) (s Z (!)) (s 2 O0 T ]). (13.35) sup E2(s Z ) 6 c(1 + EZ 2) < 1:s20T ]R0, f(s Z ) s 2 O0 T ]g 2 L2(O0 T ]) 0t (s Z )dWs t 2 O0 T ] , 12.11, ..
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