А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 30
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. 9 fb(s Z ) s 2 O0 T ]g, (13.35), ZTE jb(s Z )jds 6 (Tc(1 + EZ 2))1=2 < 1:0RRT+ J b(s Z )ds .. ! " ! t b(s Z )ds00 t 2 O0 T ]. = 12.12.@ 7 .. (13.37) , fX (n;1)(t) t 2 O0 T ]g sups20T ] EjXs(n;1)j2 < 1. @ , E(Xt(n))26363EZ 2 + EEZ 2 + TcZT0Z t02 Z tb(s Xs(n;1))dsE(1 + jj+EXs(n;1) 2)ds + cZT0E(1 + j0E Xs(n;1) 26 3 EZ 2 + c(T + 1)T 1 + sup js20T ](s Xs(n;1))dWsjj2Xs(n;1) 2)ds66< 1: 2' %,% % 13.6.
= EjXt(n+1) ; Xt(n)j2. Dn = 0, , (12.31) (13.35), t 2 O0 T ] t Zt2Z(1)(0)2EjXt ; Xt j = E b(s Z ) ds + (s Z ) dWs 602620 Zt6 2E2 Ztb(s Z ) ds + 2E06 2ct(t + 1)(1 + EZ 2) 6 M(s Z ) dWs02Zt6 2(t + 1)E c(1 + jZ j2) ds 6+ 1)(1 + EZ 2):M1 = 2c(TC n > 1 t 2 O0 T ], (13.34), EjXt(n+1) ; Xt(n)j2 66E1 t Zt0b(s Xs(n)) Zt6 2E0jL Xs(n);;b(s Xs(n;1))ds +Xs(n;1)2Zt0((s Xs(n));0(13.38)(s Xs(n;1)))dWs26Ztjds + 2E ((s Xs(n)) ; (s Xs(n;1)))2ds 60Zt6 2L2(1 + T ) EjXs(n) ; Xs(n;1) j2ds:(13.39)0* (13.38) (13.39) , M = maxfM1 2L2(1 + T )gn+1 n+1EjXt(n+1) ; Xt(n) j2 6 M(n +t1)! n = 0 1 : : : t 2 O0 T ]:(13.40)@ , sup j06t6TXt(n+1);Xt(n)ZTj 6 jb(s Xs(n)) ; b(s Xs(n;1))jds +Zt(n)(n;1)+ sup ((s Xs ) ; (s Xs ))dWs :06t6T00+ 12.11, 5.17 (13.40), P ( sup j06t6TXt(n+1);Xt(n)j > 2;n ) 6 P ZT0jb(s Xs(n));j2b(s Xs(n;1)) ds> 2;2n;2+ Zt;n;1(n)(n;1)6+ P sup ((s Xs ) ; (s Xs ))dWs > 206t6T0ZT6 22n+2 T E(b(s Xs(n)) ; b(s Xs(n;1)))2ds ++ 22n+2ZT00jE (s Xs(n));j(s Xs(n;1)) 2ds 6 22n+2L2(T+ 1)ZT M n sn0(4MT ) :ds6n!(n + 1)!(13.41)+ 1{- (13.41) , P ( sup jXt(n+1) ; Xt(n) j > 2;n ) = 0:06t6Tn+1(13.42)263+" ..
! ) N0 = N0(!): 8n > N0(!)sup jXt(n+1) ; Xt(n) j 6 2;n :(13.43)t20T ]0, Xt(n)(!) = Xt(0)(!) +n;1Xk=0(Xt(k+1)(!) ; Xt(k)(!))(13.44)) .. O0 T ], 1 " . C ! 2 E0 E (P (E0) = 1) { , Xt(n) O0 T ] , Xt(!) = nlimX (n) (!).!1 tC ! 2 E n E0 t 2 O0 T ] Xt(!) = 0. = , Xt(!) !. Xt 2 FtjB(R) 4.7, Xt(n) , ) FtjB(R)- ( - ). %, fXt t 2 T g %% ( 12.12) % % ,% (Ft)t20T ].@ , m > n > 0 t 2 O0 T ] (13.40)j(E Xt(m);jXt(n) 2)1=2 6Xm;1k=nkXt(k+1);kXt(k) L2()61 X(MT )k+1 1=2k=n(k + 1)!! 0 n ! 1:(13.45) L2(E) ) L2(E) Xt(n) n ! 1. 2 Yt.
=, Yt = Xt .. t 2 O0 T ]. +, Xt (13.33).* (13.45) t 2 O0 T ] m 2 N kkXt(m) L2 () 6kZ kL () +21X(MT )k+1 1=2k=0(k + 1)!= kZ kL2() + c(M T ):(13.46)+" kXtkL () 6 kZ k + c(M T ) t 2 O0 T ]:(13.47)+ J 3 (13.46), (13.47) 2EZT0(Xs ;Xs(n))2ds =ZT0E(Xs ; Xs(n) )2ds ! 0 n ! 1:(13.48) (13.34) (13.48) t 2 O0 T ] n ! 1Zt02(s Xs(n)) dWs L ();! (s Xs) dWsZt0264ZtL2 ()b(s Xs(n)) ds ;!0Zt0b(s Xs) ds:(13.49)(13.50)@ , (13.37) fnm = nm (t)g, (13.49) (13.50) .. , (13.33).' , Xt O0 T ] 2L (E). C, Xt ; Xs ! 0 .. t ! s (t s 2 O0 T ]) E(Xt ; Xs)2 6 const (13.47). = 1 3.9' %%, =.
+ Xt { (13.33) X0 = Z , Xet { (13.33) Xf0 = Ze, EXt2 EXet2 O0 T ]. K (13.39) , Z tEjXt;Xetj2 6 3EjZ ;Zej2+3E0(b(s Xs );b(s Xes)ds6 3EjZ ; Zej2 + 3(1 + t)L2Zt02Z t+3E02e((s Xs);(s Xs)dWs 6EjXs ; Xes j2ds:(13.51)0 ) 1 3 , Xt Xet O0 T ] L2(E). *, y(t) = EjXt ; Xetj2Zty(t) 6 c0 + Q y(s) ds t 2 O0 T ](13.52)0 c0 = EjZ ; Zej2, Q = 3(1 + T )L2. ) . 13.8 ("). y { % O0 T ] $%, #) (13.52) c0 > 0 Q > 0. y(t) 6 c0 exp(Qt) t 2 O0 T ]:(13.53)2 9 (13.52), t > 0 ( ) 0ZtQyR (t)log c0 + Q y(s)ds =6 Q:(13.54)c0 + Q 0t y(s)ds0* 0 t, log c0 + Q0,Zt0y(s)ds ; log c0 6 Qt t 2 O0 T ]:Ztc0 + Q y(s)ds 6 c0eQt t 2 O0 T ]:00 , ( c0 = 0 y(s) = 0 s 2 O0 u], (13.54) ).
2D Z = Ze .., c0 = 0 (13.52) t 2 O0 T ] , EjXt ; Xet j2 = 0. + jXt ; Xetj, , X Xe O0 T ], ..P (X (t !) = Xe (t !) t 2 O0 T ]) = 1: 2M= %% , <% &= , ' .265A 13.9. 13.6. .(13.32) %.2 7.6 (Xt t 2 O0 T ]) , C 2 B(R), m 2 N 0 6 t1 < : : : < tm < u 6 t P (Xt 2 C j X11 : : : Xtm Xu ) = P (Xt 2 C j Xu ):(13.55) .- , - .+ 2.3, , (13.55) C 2 B(R). + , 4.5 ( (4.15)), , ): f 2 Lipb(R) (.
. f : R ! R) E(f (Xt) j Xt1 : : : Xtm Xu ) = E(f (Xt ) j Xu ):(13.56)@ , 13.6 , O0 T ] Ou T ], 0 6 u < T < 1, . . ) ZtZtuuZt = Z + b(s Zs) ds + (s Zs) dWs(13.57) Zu = Z , Fu , fWs s > 0g { (Fs)s>0 . 2 Zt Zt(Z ), t 2 Ou T ].C Xt , t 2 O0 T ], (13.33) t > u ZtZtZu0Zt0uXt = X0 + b(s Xs) ds + (s Xs) dWs =0= X0 + b(s Xs ) ds ++ZtuZub(s Xs) ds + (s Xs) dWs +Zt0Zt(s Xs) dWs = Xu + b(s Xs) ds + (s Xs) dWs :uu(13.58)9, Xu Fu jB(R)- (" , ) ) (13.57), , X0 Xt = Zt (Xu) t > u:(13.59) Z (13.57) x 2 R.
+ , . . t 2 Ou T ], (13.51) 13.8, , t()Zt(y) L;!Zt (x) y ! x (x y 2 R):2266(13.60)C t 2 Ou T ] f 2 Lipb(R) G(x) = Ef (Zt (x)) x 2 R:(13.61) (13.60) , G(x) | ( ) R. 13.10. 4 # t 2 Ou T ] # x 2 R Zt(x) - Fu .2 M Zt(x) (. . Zt(x !), t 2 Ou T ], x 2 R, ! 2 E) : Zt (x) .. Zt(n)(x),n = 0 1 : : : , n ! 1, Zt(0)(x) = x, n > 1Zt(n)(x) = x +Ztub(s Zs(n;1)(x)) ds +Ztu(s Zs(n;1)(x)) dWs t 2 Ou T ]:(13.62)3 , Zt(n)(x) Fut] jB(R)- n = 0 1 : : : , () - Fut] = fWs ; Wu s 2 Ou t]g. C,RtRtx + b(s x) ds !, (s x) dWs uu (, ) , ) ..).
+ , Fut] jB(R)-, n = 0, 4.7. + 13.5 ( Ou T ] Fut], t 2 Ou T ]), Fut] jB(R)- Zt(n)(x) n > 1. @ , Zt(x) Fut] j B(R)- 4.7.= 4.3, Fu Fut] ( -, ). 2' %,% % 13.9. +Y (n) = 2;n O2n Xu ] n 2 N O] | . =,Y (n)(!) ! Xu (!) n ! 1 ! 2 E:(13.63)(13.64) (13.63) Y (n) Fu jB(R)- n 2 N.+" Zt(Y (n)), t 2 Ou T ], n 2 N.C Xt , t > 0, , (13.56) G(Xu ), G (13.61). A 2 fXt1 : : : Xtm Xu g, t1 : : : tm u t (13.55) (13.56). @ f 2 Lipb (R)Ef (Xt )1 A = nlimEf (Zt(Y (n)))1 A :(13.65)!1C, jf (x) ; f (y)jL 0jx ; yj x y 2 R:@ t 2 Ou T ], n 2 N A267jEf (Xt )1 A ; Ef (Zt(Y (n)))1 A j 6 L0EjXt ; Zt (Y (n))j 66 L0(EjZt(Xu ) ; Zt(Y (n))j2)1=2 6 a0(EjXu ; Y (n)j2)1=2 6 a02;n a0 = L0 expfQT=2g (13.51) (13.53) Q = 3(1 + T )L2.+ 3 13.10, E1 A f (Zt==(Y (n))) =1Xk=;11Xk=;11Xk=;1E1 A f (Zt (k2;n ))1 fY (n)=k2;ng =E1 A\fY (n)=k2;n gEf (Zt (k2;n )) =1Xk=;1E1 A\fY (n)=k2;n gG(k2;n ) =E1 A 1 fY (n)=k2;n gG(k2;n ) = E1 A G(Y (n)):(13.66)4 , f (Zt (k2;n )) Fu, A \ fY (n) = k2;n g = A \ fk2;n 6 Xu < (k + 1)2;n g 2 fXt1 : : : Xtm Xu g Fu :(13.67)+ G, (13.65), (13.66) (13.64) E1 A f (Xt ) = E1 A G(Xu ):(13.68)+"E(f (Xt) j Xt1 : : : Xtm Xu ) = G(Xu ):(13.69)(, (7.10) , E(f (Xt) j Xu ) = E(E(f (Xt ) j Xt1 : : : Xtm Xu ) j Xu ) = E(G(Xu ) j Xu ) = G(Xu )(13.70) (13.56).
@ 13.9 . 2% 13.11. % *.{J (. 13.4) , %.2 C , ={9 3 (13.8) X0 N (0 2=2), . 13.4. 2 . 913.1. M %% % dXt = rX r = const:tdt@ , dXt = rXt dt + Xt dWt r | .268(13.71)(13.72)2 *Zt dXs0Xs ds = rt + Wt t > 0:(13.73) h(t x) = ln x, x > 0. ', * ( C12.12) , , h : O0 1) ( ) ! R, ;1 6 < 6 1, ( h(t x) O0 1) R), Xt(!) 2 ( ) t > 0 ! 2 E.*, Xt > 0 t > 0, 1 2X 2 dt = dXt ; 1 2 dtt;d(ln Xt) = X1 dXt + 12 ; X12 (dXt )2 = dXXt 2Xt2 tXt 2ttdXt = d(ln X ) + 1 2 dt:tXt20,(13.74)Xt = r ; 1 2 t + W :ln Xt20(13.75)Xt = X0eHt Ht = (r ; (1=2)2)t + Wt:(13.76) 3 , X0 fWt t > 0g (. (12.2)), EXt = EX0 expfrtg(13.77). . (, r < 0) EXt , (13.71). 2 fXt t > 0g (13.76), ) (13.72) " r 2 R, > 0.
+ H = (Ht)t>0 r ; 2=2 $$ 2.3 H , 2 , { #. +. 0" (O?]) , " , "" ".@ 913.2. = , 13, .(1)@, (13.32) , Xt = (Xt : : : Xt(n)),b | , | , ,b( ): O0 T ] Rn ! Rn ( ): O0 T ] Rn ! Rnm:(13.78)269+ " Wt | m- , . . Wt(k), t > 0,k = 1 : : : m, Q , 10 (1)b (t Xt) dtb(t Xt) dt = BA@ ... Cb(n)(t Xt) dt0101 (t Xt) : : : m (t Xt) dWt(1)... C(t Xt) dWt = B@ ...AB@ ...
CA:11(13.79)1n1 (t Xt) : : : nm (t Xt)dWt(m)* Rt, (s Xs) dWs - i- , Pm0k=1intt0ik (s Xs) dWs(k) .= , (Ft)t>0 , fWt t > 0g{ . @ 13.6 , (13.34) (13.35) b Rn, Pn Pm 2 ., , , jj2 =iki=1 k=1+ % 3%. + Xt = (Xt(1) : : : Xt(n) ) |n- % 2, .. ) O0 1) ,dXt = f (t !) dWt + g(t !) dt(13.80) f g | f = (fik (t !) i = 1 : : : nQ k = 1 : : : m)Q g = (g1(t !) : : : gn (t !)). Wt == (W1(1) : : : Wt(m)), dXt(i)=mXk=1fik (t !) dWt(k) + gi (t !) dt i = 1 : : : n(13.81) t > 0, i = 1 : : : n k = 1 : : : mZtP(0ZtP(0jgi (s !)jds < 1) = 1(13.82)jfik (s !)j2ds < 1) = 1:(13.83)=, * t 2 Ou v], 0 6 u < v < 1 t 2 Ou v), 0 6 u < v 6 1.270A 913.3. $% h : O0 1) Rn ! R , h 2 C 12, ..)# @h=@t @ 2h=@xi@xj , i j = 1 : : : n, @h sup n @x (t x) < 1 i = 1 : : : n:t>0 x2Ri % Yt = h(t Xt), % 2 Xt $$%, $nn n@h@h@ 2h (t X ) dX (i) dX (j)(i) 1dYt = @t (t Xt) dt + @x (t Xt) dXt + 2tttii=1i=1 j =1 @xi@xj dXt(i) dXt(j ) , $#) (13.81), # dWt(i)dWt(j) = ij dt dWt(i)dt = dt dWt(i) = 0:X2 ,mndYt =k=1 i=1X X @hXX(13.84)(13.80),(13.85)(k)@xi (t Xt)fik (t !) dWt + @hnnm2hXXX@h1@(t Xt) fik (t !)fjk (t !) dt:+ @t (t Xt) + @x (t Xt)gi (t !) + 2@x@xiiji=1ij =1k=1(13.86)C * ., ., O?, .
1, x 4e]. + )" h 2= C 12, ., .,O?], O?].+, O?], ).+ X { () fWt t > 0g. + h = h(x) :h(x) = h(0) +Zx0f (y)dy () f = f (y) 2 L2loc (R), .. c > 0Zjyj6cf 2(y)dy < 1:= Of (W ) W ] { # %# f (W ) W ) XOf (W ) W ]t = P ; lim(f (Wt(n)(m+1)^t) ; f (Wt(n)(m)^t)) (Wt(n)(m+1)^t ; Wt(n)(m)^t):mm' P ; lim , T (n) = ft(n)(m) m 2 Ng, n 2 N { t(n)(m), .. t(n)(m) 6 t(n)(m+1) m 2 N n 2 N t > 0sup(t(n)(m + 1) ^ t ; t(n)(m) ^ t) ! 0 n ! 1:m , h(W ) , ) , , ) , )Of (W ) W ]t. = O?] ) " .271A 913.4 (%%, 1,, I).
. - $%%hf $th(Wt) = h(0) + f (Ws )dWs + 12 Of (W ) W ]t:0913.5. $% h 2 C 2. ZOf (W ) W ]t =Zt0(13.87)f 0(Ws )ds(13.87) $ 2.% 913.6. h(x) = jxj. Of (W ) W ]t = 2Lt (0), $ Lt (0) { , O0 t](. (12.96)). , (13.87) $ (12.97), 4 { ; jWt j.E. 13.7. + h(x1 x2) = x1x2, x1 x2 2 R X (1) = fXt(1) t > 0g,X (2) = fXt(2) t > 0g { , ) *. @ -d(Xt(1)Xt(2)) = Xt(1)dXt(2) + Xt(2)dXt(1) + dXt(1)dXt(2):C, dXt(i) = b(i)(t !)dt + (i)(t !)dWt(i) i = 1 2 fWt(i) t > 0g, i = 1 2 { , d(Xt(1)Xt(2)) = Xt(1)dXt(2) + Xt(2)dXt(1) dXt(i) = b(i)(t !)dt + (i)(t !)dWt i = 1 2d(Xt(1)Xt(2)) = Xt(1)dXt(2) + Xt(2)dXt(1) + (1)(t !)(2)(t !)dt:E. 13.8. M dXt = Xtdt + dWt t 2 O0 T ]( e;t d(e;tXt)).E.
13.9. +R t V = V (t x) { R+ Rn, C 12.+ t = 0 c(s !)ds, c = c(s !) { , ZtP ( jc(s !)jds < 1) = 1 t > 0:0( fe;t V (t Xt) t > 0g, Xt = fXt(1) : : : Xt(n)g n- *, (13.81).@ 913.10. D 13.6 O0 T ), " , " Xt EXt2 , O0 T ).272E. 13.11. 0 ) * , () dXt = T;;Xtt dt + dWt t 2 O0 T ) X0 = (13.88)Z t dWsXt = (1 ; t=T ) + t=T + (T ; t) T ; s :(13.89)0E. 13.12. ( 13.11) C, Xt ! ..
t ! T ;. @ , (13.88) O0 T ] % X0 = XT = . 0 T = 1 = = 0.@ 913.13. 0) ) 13.6 . (, (13.34) ,.% x: t > 0, n 2 N jxj 6 n, jyj 6 njb(t x) ; b(t y)j + j(t x) ; (t y)j 6 L(n)jx ; yj L(n) > 0. * " . - , ( ) " !, " b "" ( : b = b(tQ Xs s 6 t), = (tQ Xs s 6 t). = ) ., ., O?], O?], O?].* ) , K.-. ' O?], ), % ,= %% , dXt = b(t Xt)dt + dWt(13.90)%% =, % (t x) %-% b(t x).E.
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