А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 28
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* (12.46) (12.53) Xt = Xs + (Wt ; Wu)fj + (t ; u)gj t 2 Ou v]:(12.56)M Ou v] u = s(0n) <: : :< s(nn) = v, (n)(n)n = i=0max(s;si+1i ) ! 0 n ! 1::::n;1(12.57)Xs (!)) .. + aj (s !) = fj (!) @h(s@x Ou v] (Xs | .. , .(12.56), @h=@x ).
C, (12.49) (12.54) jaj (s !)j 6 M0jfj j 2 L2(E). - , aj (s !) Ou v] . 0, 12.10 ( u v)Zv @hn;1Xfj @x dWs = l:i:!m0: fj @h@x (si Xsi )(Wsi ; Wsi )nui=0+1(12.58)243 ) si s(in).*, , n;1 Xi=0h(si+1 Xsi+1 ) ; h(si Xsi ) ; fj @h@x (si Xsi )(Wsi+1 ; Wsi ) !Zv @h2h@h1@2! @t (s Xs ) + gj @x (s Xs) + 2 fj @x2 (s Xs ) ds ..
(12.59)u ) , . . fnk g, nk ! 0 k ! 1.@ , h(si+1 Xsi+1 ) ; h(si Xsi ) =(2)= (h(si+1 Xsi+1 ) ; h(si Xsi+1 )) + (h(si Xsi+1 ) ; h(si Xsi )) = ^(1)i + ^i : (12.60)(2)+ ^(1)i @ , ^i | ,(2) @hsi Xsi+1 )(si+1 ; si) + @h^(1)i + ^i = @t (e@x (si Xsi )(Xsi+1 ; Xsi ) +2+ 21 @@xh2 (si Xesi )(Xsi+1 ; Xsi )2 i = 0 : : : n ; 1 (12.61) esi = sei(!) 2 (si si+1), Xesi (!) | Xsi (!) Xsi+1 (!). =, , sei Xesi , @h s (!) X (!)) | . ^(1)isi+1i | , @t (e2h@K, 2 (si Xesi ) | ( Xsi (!) = Xsi+1 (!), @xXesi (!) = Xsi (!) (12.61)).+ (12.56), (12.61), , (12.59) n;1 X@h2h@@h1(esi Xsi+1 )(si+1 ; si) + @x (si Xsi )gj (si+1 ; si) + 2 @x2 (si Xesi ) @ti=02222 Ofj (Wsi+1 ; Wsi ) + 2fj gj (Wsi+1 ; Wsi )(si+1 ; si) + gj (si+1 ; si) ] :(12.62)C 12.13 912.14.
% Xs .. Ou v], $% G O0 1) R. M (GQ ) = supfjG(s Xs ) ; G(r Xq )j : s r q 2 Ou v]js ; rj < jr ; qj < g ! 0.. ! 0: (12.63)2 J Xs (!) .. ! Ou v]. 0, ! 2 E0 (P (E0 ) = 1) t 2 Ou v] jXt(!)j 6 L(!) < 1. C L > 0 G(s x) Ou v] O;L L]. +" " > 0 = (" L) > 0, jG(s x) ; G(r y)j 6 ", 244js ; rj 6 jx ; yj 6 . = , > 0 ! 2 E0 ) = ( !) > 0, supsq2uv]js;qj<(!)jXs(!) ; Xq (!)j 6 : 2(12.64)+ 12.13. + 12.14 @h=@t Xn;1n;1X@h@h(esi Xsi )(si+1 ; si) ;(si Xsi )(si+1 ; si) 6@t@ti=0i=0 @h +16 M @t Q n (v ; u) ! 0 ..
n ! 0: (12.65)9 @h=@t .. Xt, , n;1X@hZv @hi=0u@t (si Xsi )(si+1 ; si) !@t (s Xs ) ds .. n ! 0:(12.66) @h=@x , n;1X@h(si Xsi )gj (si+1 ; si) !@xi=0Zv @hu@x (s Xs )gj ds .. n ! 0:(12.67)1 Ou v], " Xn;1 2@h fj gje2 (si Xsi )(Wsi ; Wsi )(si+1 ; si ) 6@xi=0+16 C (!)jfj gj j(v ; u) sup jWy ; Wr j ! 0 .. n ! 0 (12.68)yr2uv]jy;rj6nC (!) = maxfj@ 2h(t x)=@x2j : t 2 Ou v] jxj 6 tmaxjX (!)jg:2uv] tK, Xn;1gj2 @ 2h2 (si Xesi )(si+1 ; si)2 6 gj2C (!)(v ; u)n ! 0i=0 @x..(12.69)(12.70)@ , ) fnkg Ou v]n;1 2X@he2 (si Xsi )(Wsi+1 ; Wsi !@xi=0)2Zv @ 2hu@x2 (s Xs) ds ..(12.71)+ 12.14 245 Xn;1 2n ;1 2X@h@h22e2 (si Xsi )(Wsi ; Wsi ) ;2 (si Xsi )(Wsi ; Wsi ) 6@x@xi=0 @ 2h i=0Xn;1+1+16 M @x2 Q ni=0(Wsi+1 ; Wsi )2 ! 0 .. nk ! 0 (12.72) , n;1Xi=0()(Wsi+1 ; Wsi )2 L;!v ; u n ! 0:2(12.73)C, ) , X2 Xn;1n;12E (Wsi ; Wsi ) ; (si+1 ; si) = D(Wsi ; Wsi )2 =i=0i=0n;1X+1+1=2i=0(si+1 ; si)2 6 2(v ; u)n: (12.74)0, ) fnk g, (12.73) ..C, Xn;1n ;1 @ 2h(si 2Xsi ) (Wsi ; Wsi )2 ; X @ 2h(si2Xsi ) (si+1 ; si) 6@x@xi=0i=0n;1X2+16 C (!)i=0j(Wsi ; Wsi ) ; (si+1 ; si )j: (12.75)+1C (!) (12.69), (12.74),Xn;1Ei=02j(Wsi ; Wsi ; (si+1 ; si)j 6 2(v ; u)n:+1)2+" ) fn0k g fnk g, (12.75) 0 ..
k ! 1 ( nk n0k ). (, @ 2h=@x2 n;1 2X@h2 (si Xsi )(si+1 ; si ) !@xi=0Zv @hu@x (s Xs) ds .. n ! 0:(12.76)@ , .. :::: fnk g k ! 1. = , nk ;! , nk ;! nk = nk.., = .. 3 12.13 . 2' %,% % 12.12. + 12.8 f (n) ! f n ! 1 L2(O0 1)).K 12.8, (12.47), g(n), ) .. ! g L1(O0 1)).246+ t > 0ZtZt00Xt(n) = X0 + f (n)(s !) dWs + g(n)(s !) ds:+ 12.13 .. z > 0 n 2 Nh(z Xz(n) ) ; h(0 X0 ) =+Zz @h0(12.77)Zz @h0(n) (n)@x (s Xs )f (s !) dWs +(n) ) + g (n) (s ! ) @h (s X (n) ) +(sXss@t@x1 (f (n)(s !))2 @ 2h (s X (n)) ds: (12.78)s2@x29', % % (12.78) % %,% fnk g % %% %%% % (12.52).
+ 12.11 (5.29), " z P ( sup jXs(n) ; Xs j > ") 606s6z Zs " Zs " (n)(n)6 P sup (f ; f )dWs > 2 + P sup (g ; g)ds > 2 606s6z06s6z00 2 2 Zz Zz"6 " E (f (n) ; f )2 ds + P00jg(n) ; gjds > 2 ! 0 n ! 1:(12.79) h , z > 0Ph(z Xz(n) ) ; h(0 X0 ) !h(z Xz ) ; h(0 X0 ):(12.80)0 (12.31), (12.49) Xt Xt(n)(n 2 N), 2 Zz @hZz @h(n)(n)E @x (s Xs )f (s !)dWs ; @x (s Xs)f (s !)dWs =002Zz @h@hE @x (s Xs(n) )f (n)(s !) ; @x (s Xs )f (s !) ds 6=02Zz @h(n)6 2 E @x (s Xs ) (f (n)(s !) ; f (s !))2ds +0 @h2Zz@h+ 2 E(f (s !))2 @x (s Xs(n)) ; @x (s Xs ) ds 606 2M02Zz0E(f (n);f )2ds + 2ZzE^n ds(12.81)0247 @h2@h^n; @x (s Xs) 6 4M02f (s !)2:(12.82)@h , (12.79) s > 0 - , @x@h (s X (n)) !P @h(12.83)s@x@x (s Xs ) ,(s !) = (f (s !))2(n) )(sXs@xP^n(s !) !0 n ! 1:(12.84)* (12.84) (12.82) , E^n(s !) ! 0, n ! 1, s > 0, E^n(s !) 6 4M 2E(f (s !))2.
@ , 3 Zt0E^(s !) ds ! 0 n ! 1:(12.85)@ , 3, ) (12.78), ) (12.52), ) Ou v].9, f (n) ! f L2(O0 1)), fnk g , Zz0(f (nk )(s !) ; f (s !))2ds ! 0 .. k ! 1:(12.86)+ (12.79), fnk g ( , fn0k g fnk g), XkP ( sup jXs(nk ) ; Xs j > 2;k ) < 1:06s6z(12.87)@ Xs(nk ) Xs .. O0 z]. +" 1 @h@h(n)"1(nk !) := sup @x (s Xs k ) ; @x (s Xs ) ! 0 k ! 1:(12.88)s20z]+ ", C12.14, @h=@x O0 z] O;L L]. 0,Zz @h@h(n) (s Xs k ) ; (s Xs) ds 6 z"1(nk !) ! 0@x@x0.. k ! 1:(12.89)+ fnk g , "2(nk !) :=Zz0248jg(nk )(s !) ; g(s !)jds ! 0 .., k ! 1(12.90)( fn00k g fn0k g, - nk ). @ (12.88) (12.90)Zz @h@hg(nk )(s !) (s Xs(nk )) ; g(s !) (s Xs) ds 6@x@x0z @hZ6 "1(nk !) jg(s !)jds + sup @x (s Xs(nk )) "2(nk !):s20z]0(12.91)9 (12.47), , (12.91) ..
k ! 1, (12.88) , @h @h(n)sup (s Xs k ) ! sup @x (s Xs) < 1 ..(12.92)s20z]s20z] @x0 ( (12.86) , )@h @ 2 h (12.92)) , k ! 1 @x@x2Zz 2h2h@@(f (nk )(s !))2 2 (s Xs(nk )) ; (f (s !))2 2 (s Xs) ds ! 0@x@x..(12.93)0@ C12.12 . 2@ 912.15. * ) *. (, X { Rm, { Rm ( - { ). 4 15.19, 15.21 O?]. ' , ) (12.49) ( (13.84) ). ) * (12.50) 13. 912.16. ZTWs dWs:(12.94)02 C12.12, Xt = Wt h(t x) = 12 x2. =, " . @ Yt = h(t Wt) = 21 Wt2 @h dW + 1 @ 2h (dW )2 = 0 + W dW + 1 (dW )2 = W dW + 1 dt:dYt = @hdt+tttt@t@x t 2 @x2 t2 t2(12.95)0,1 d 2 Wt2 = Wt dWt + 12 dt:@ , (.
(12.46)) . . W0 = 0, 1 W 2 = Z W dW + 1 t t > 0ss2 t2t(12.96)0249Zt0Ws dWs = 21 Wt2 ; 21 t: 2(12.97)E. 12.17. 0 ) * , f = f (t) C 1 (12.1). + f = f (t !), (12.1) .E. 12.18. + Xt = expft + Wtg, t > 0, fWt t > 0g { , 2 R. ( dXt . , E (W )t = eWt ;t=2 dE (W )t = E (W )t dWt .( 0 1 jfs 2 O0 t] : W (!) 2 (;" ")gjLt(!) := "lim(12.98)s!0+ 2" j j 3, (12.98) ( , )) L2(E F P ).A 912.19 (A). ( $jWtj =Zt0sgn(Ws)dWs + Lt t > 0(12.99)sgnx { x.C " * h(x) = jxj, ., ., O?, . 42], O?, ?].
+ ) * h 2= C 12 . O?], O?].= % && % 3% (. 13). * J1.:, f 2 J1, f : (0 1) E ! R, PZ t0f 2(s !)ds <1 = 1 t > 0:(12.100)* , L2 fn , n 2 N, It(fn) , Zt0P(f (s !) ; fn (s !))2ds !0 n ! 1:(12.101)= , fIt(fn )gn2N . 0, ) ,R It(f ),P, It(fn) !It(f ), n ! 1. + It(f ) (0t] f (s !)dWs , (f W )t.
T fBt t > 0g, (f B )t.E. 12.20. C, ) ffn n 2 Ng, (12.101). C, f 2 J1, ) .. It(f ), t > 0.=, f 2 J1 It(f ) & &%, %, % , .. ) , n " 1 .. (n ! 1) n "" Itn (f ) := It^n (f ), t > 0, .250E. 12.21. M f 2 J1 ZtZt1Zt = expf f (s !)dWs ; 2 f 2(s !)dsg t > 0(12.102)00 " ". C, dZt = Ztf (t !)dWt.K J1(O0 T ]) f : O0 T ] E ! R, ) (12.100) t 2 T .0) (. C12.5), , %% & .A 912.22 (0). X = X (!) FT jB(R)- T > 0, (Ft )t>0 { $% . #) .1. " EX 2 < 1, %f = (f (s !))s20T ] 2 L2(O0 T ]), X = EX +ZT0f (s !)dWs..(12.103)2. " EjX j < 1, (12.103) % f 2 J1 (O0 T ]).3.
" X (.. P (X > 0) = 1) EX < 1, % f 2 J1 O0 T ] , X = ZT EX , ZT (12.102) .E. 12.23. =7, X , ) C?? , .. X (!) = g(W (s !) 0 6 s 6 T ), g : C O0 T ] ! R g 2 B(C O0 T ])jB(R).* 12.22 ) ,%% %% &%.A 912.24 (0). M = (Mt Ft)t20T ] { , $% (Ft )t>0 { , 412.22. 1. = % f = (f (s ! ))s20T ] 2 L2 (O0 T ]) , Mt = M0 +Zt0f (s !)dWs t 2 O0 T ]:(12.104)2. " M , (12.104) % f 2 J1 (O0 T ]).3.
" M , % f 2 J1(O0 T ]) , Mt = M0 Zt , Zt { ,t 2 O0 T ].C C12.22, C12.24 - O?],* O?], C O?], , ., ., O?], O?, ?], O?].0 ) " . 0 251 fW (t) t 2 O0 1]g , V = infEjW ; 0maxW j2 :6s61 s4 , (, ) (, " , .. ) ) . ', Wp ) max06s61 Ws , EW = 0 E max06s61 Ws = 2= ( ).
+" Ve = a2infR EjW + a ; 0maxW j2:6s61 s3 ( "), Ve = V ; 2=:+St = 0maxW t 2 O0 1]:6s6t sA 912.25 (I). +. . $p = inf ft 2 O0 1] : St ; Wt = z 1 ; tg z 4Y(z) ; 2z(z) ; 3 = 0 Y { $% . z = 1:12 : : : , V = 2Y(z ) ; 1 = 0:73 : : : .- " max Ws = a +06s61 a = const ( " ) Z10f (s !)dWs pf (s !) = 2f1 ; Y((St ; Wt)= 1 ; t)g s 2 O0 1] ! 2 E:B&& < % 3% O?]. + " .A 912.26 (3%). t > 0 n 2 N $ZZ n=2pt :::dWs1 : : : dWsn = tn! Hn Wt06s1 6:::6sn 6t Hn { E n, ..dn (e;x2=2) n = 0 1 : : :Hn(x) = (;1)nex2=2 dxn252(12.105)= .
O?].D) && % , & & , ' & '.M M == (Mt)t2R+ (E F (Ft)t>0 P ) , M0 = 0. - Mc2. 1 , (Ft)t>0 ( F0 N , ) P - ).E. 12.27. C, M 2 Mc2, " O0 T ].2 , IT (X ) =ZT0Xt(!)dMt(!)(12.106) , 3 { 0.0 (Ft)t>0 ( , ) A = (At)t2R+ #), .. ! A0(!) = 0, At(!) {) t 2 O0 1) EAt < 1 t 2 R+. + A , EA1 < 1, A1 = limt!1 At.
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