А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 20
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* (9.19) (9.20) (9.17) .C t > 0 h < 0 (9.18), , h 2 O;t 0), ( "). 2A 9.4 ( ). $% P (t), t > 0, % ) $ % Q qij , i =6 jpij (h) = qij h + ij (h) (9.21)iij (h)=h ! 0 h ! 0 + : t > 0 ) P 0 (t) ()# p0ij (t)) P 0(t) = P (t)Q. .
i j 2 X t > 0Xp0ij (t) = pik (t)qkj :k(9.22)(9.23)(9.24)1672 + t > 0, h > 0 i j 2 X . @pij (t + h) ; pij (t) = p (t) pjj (h) ; 1 + 1 X p (t)p (h):ijhhh k6=j ik kj(9.25)C " > 0 j (9.21) h0(" j ) > 0, jkj (h)j=h < " k 0 < h < h0(" j ), " X (h) X pik (t) kj 6 " pik (t) 6 ":(9.26)hk6=jk6=jP* (9.21), (9.26) , h1 pik (t)pkj (h) 6 pij (th+ h) 6 h1 , , k6=j t > 0 i 2 X (9.24). 0 (9.25) h ! 0+, , (9.23) .0 h < 0 (9.25) ( ) ). 2@ 9.5. J %% % X % P (t)%%, % & % 0. C, i j 2 X , t > 01 X p (t) = 1 ; pii ijttj 6=i , 9.2 . +" (9.21), (9.22), , , .. i X .( Q (9.16) (9.23) (%& %&).% 9.6.
9.4, i j 2 X)lim p (t) = pjt!1 ij( 8.6).Xk(9.27) #pk qkj = 0j(9.28). . p | % QT (QT #% Q), #) #. 9.4, ,2 + , P j pk (t)qkj , pk (t) = P (Xt = k), k )Pp0j (t) =Xkpk (t)qkj :(9.29)PD pj = 0, . . pj = 0, (9.28), , . D pj 6= 0,jj pi(0) = pi , i 2 X (. 8.9). @pj (t) = pj (8.29). 0, p0j (t) = 0, (9.29) (9.28). 2168r r r r r r r r r rr h+o(h)kh+o(h)h+o(h)nh+o(h)? y? ky k+1?y 1?k ;101;t+o(h)n;1 n1;(+k)h+o(h)1;nh+o(h).
9.1M% %, , ( 7 ), % X = f0 1 : : : ng, %% pij (h) h ! 0+ <% , . 9.1. @ , , Q % & 0 n 0 1 n n ; 1, " ( | ). @ , %, pij (h) = o(h) h ! 0+. @ 0 ;BBBBBBBQ=BBBBBBB@;(+)2;(+2)...1CCCCCCCCCCCCCA0C...k...;(+k)...0......(n;1) ;(+(n;1))n;n+, 8.6. j0 = n. C i = 0 : : : npin (nt) > pii+1(t)pi+1i+2(t) : : : pn;1n (t)pnn (nt ; (n ; i)t) >> pii+1(t)pi+1i+2 (t) : : :pn;1n (t)(pnn (t))i:( t0 > 0, pk;1k (t) = t + o(t) > 0, k = 1 : : : n (o() k) pnn (t) > 1=2 0 < t 6 t0. h = t0=n, , (8.21). 9.5 P P -, ", ,pj = 1, . . X ( pij (t) = 1 (9.27)). *, jj p, 9.6. M QTp~ = 0:0 ;BBBBBBBBBBBB@0;(+)...2......;(+k) (k+1).........;(+(n;1))10CC0p 1CC0CBCp1 CBCC= ~0:BC.C.@A.CCpnCCn A;n- QT , : : : , k- | )169 ( 2 6 k 6 n ; 1). +0 ; 10BC; 2BCBC..BC..~p = ~0:..BCBC@; n A0 ;n@ ,;pk + (k + 1)pk+1 = 0 k = 0 : : : n ; 1:k0, pk+1 = pk =(k + 1), = =. = pk = k! p0, .
. Pn pk=0k= 1, p0 = Pnk=0;1k =k!k =k! k = 0 1 : : : n:pk = Pnj =j ! (9.30)j =0J (9.30) $ E.A, &;, %% % % =, %%, . 9.1.+ % & (04=), ) n . + " . 1 , , . . 1, 1 +2, : : : , fk g , . . 8.2. .M % &, . . , (%&). @ , , . D , ( , " " ), , ). D , t | " Xt .0 . +, " > 0. - , , ) , fk g, ) .B& )B &<% ajbjc, a k (-, k = 1 + : : : + k , k , k 2 N { ...), b ( { ...
, ) fk g), c { . + " " M . @, M jM jn. ' GjGjc , ("G" { general). ), , , k .. (., ., O?]).170D ft t > 0g | , fk g, Am(t t + h) | (t t + h] m , )m e;h m = 0 1 : : :P (Am(t t + h)) = P (t+h ; t = m) = (hm!+"P (A0(t t + h)) = e;h = 1 ; h + o(h) h ! 0+Q;h = h + o(h) h ! 0+QP (A1(t t + h)) = he(9.31)S;h;hPAm(t t + h) = 1 ; e ; he = o(h) h ! 0 + :m>2C (;xp (x) = e x > 00 x < 0, x y > 0P ( > x + y j > y) =R1 p (z) dzx+yR1 p (z) dzy;(x+y)= e e;y = e;x = P ( > x):(9.32)(9.33)@ , ) ) x, , " ( ,) , " ,) (9.33)). ', ph = P ( < h) = 1 ; e;h = h + o(h) h ! 0 + :- .
+" , k ) t l " h, Ckl plh (1 ; ph)k;l ( 1). 0, Bkl(t t + h) % k ) l (t t + h]&,P (Bk1(t t + h)) = k(1 ; e;h)e;h(k;1) =(9.34)= Sk(h + o(h))(1; (k ; 1)h + o(h)) = kh + o(h) h ! 0+QPBkl(t t + h) = 1 ; (1 ; kh) ; kh + o(h) = o(h) h ! 0 + :l> 20 Cij (t t + h), ) , ) i j (t t + h], ) ( (t t + h]). D 1 6 i 6 n ; 1, Cii+r = ArBi0 Ar+1Bi1 : : : Ar+iBii r = 1 : : : n ; iCii;r = A0Bir A1Bir+1 : : : Ai;r Bii r = 1 : : : i:( C0k Cnk , k = 0 1 : : : n, fAqg fBmk g. (, (9.31) (9.34), pij (h), ) .
9.1.1714 fXt t > 0g , Xt s Xs , t ; s " , ) .+, =, %% , < 2, Xt - " . ' , ) : ) ( Q -). =, " . " &% (9.30). C k = n pn , . . , n , pn ( 8.7 , pj (t) pj " ). 0 O?], ) :n = 2 = 03n = 4 = 06n = 6 = 09 9.2p2 ' 00335:p4 ' 00030:p ' 00003:62 , ) ) ( ). ( $$% , " (. (9.32)) E = 1=.
+" .' , ( )Pn kpk = (1 ; pn ).k=11. K. 0 , 2 &% %%, ) , &' %& -' 1=, " ". .9 % %, %Q = (qij )ij2X (%%) X = fX (t) t > 0g.+qi := ;qii < 1 i 2 X :(9.35)@ (7) X O0 1) , C2.15, ( R O0 1)), . + s > 0 i P (X (s) = i) 6= 0. @ t > 0 ( X ) P (X (u) = i s 6 u 6 s + tj X (s) = i) = P (X (1s) = i) P (X (u) = i s 6 u 6 s + t) =1721P (X (u) = i u = s + tk2;n k = 0 : : : 2n ) =!1P (X (s) = i) nlim;n ))2n := nlim(p(t2ii!1(9.36)* qi (1 ; pii(h))=h = qi + i(h) i(h) ! 0 h ! 0 + :0, t > 0, i 2 X n 2 N(pii(t2;n ))2n = (1 ; qit2;n + o(t2;n ))2n = expf2n log(1 ; qit2;n + o(t2;n ))g:+ log(1 + x) = x + (x)x2, j(x)j 6 1 jxj 6 1=2, , ;n ))2n = expf;qi tg t > 0:lim(p(t2iin!1(9.37)=, (9.36) (9.37) t = 0.3 (9.36) X i , t, , s i. 0 (9.36) (9.37), , s, .
+" X i, .*, ( 8.5)A 99.1. % X = fX (t) t > 0g # (9.35). % i qi ( , $#) (9.36)).0 i, 0 6 qi < 1, . = )#), qi = 0. D , ( (9.35) (9.36) qi = 0). 0 i , qi = 1. 2 7E. 9.2. + qi = 1. C, X " , .., " , .@ 4, ) , .* , O?] , .P+ , .. qi < 1 j6=i qij = qi i 2 X . @ qi 6= 0 qij =qi, j 6= i, ' %%%,, %% %% % i % j .@ , j 6= i, t > 0Fij (t) := P (X (s + t) = j j X (s) = i X (s + t) 6= i):3 , Fij (t) = pij (t)=(1 ; pii(t)) ! qij =qi t ! 0 + :(9.38)173PPD 0 < qi < 1 j6=i qij < qi, 1 ; j6=i qij =qi " ".C " Q , C.+ X { , X = fX (t) t > 0g Q qi 2 (0 1), i 2 X . 1 ( ) (E F P ) n X n R+ = O0 1).
+ 1 , t > 0, i 2 XP (1 > tj 1 = i) = e;qit(9.39) n > 1, i j i1 : : : in 2 X , x1 : : : xn 2 R+P (n+1 = j j 1 = x1 : : : n = xn 1 = i1 : : : n;1 = in;1 n = i) = qij =qiP (n+1 ; n > tj 1 = x1 : : : n = xn 1 = i1 : : : n = in n+1 = j ) = e;qj t: (9.40)E. 9.3. C, (E F P ) ) n n, ) (9.39) (9.40). +" 0 < 1(!) < 2(!) < : : : .. 4 , limn!1 n = 1.. n ! 1? C, , supi qi < 1.E. 9.4.
+ 0(!) = 0 .. * n , n , n 2 N, ) 9.3, Y = fY (t) t > 0g, Y (t !) = n(!) n;1 (!) 6 t < n (!) n 2 N( ! , n;1 (!) = n (!) n 2 N, Y (t !) = 0, t > 0).C, Y { , ) .-.., X , Q. =, , qi . 9.3 E. 9.5. C, supi qi < 1 , pii(t) ! 1 t ! 0+ i 2 X (, ", pij (t) ! ij t ! 0+ i j 2 X ).E. 9.6. + , (9.1).
+ , (9.1), (9.35) .*, 9.4 , , Q, , ) Q. C " ( 8) { " P (t),t > 0, . D P (t), t > 0, ) Q, 8.1 P (t) ( , . . ).E. 9.7. (. C9.10). + X N ", () P (t) . C, P (t) = exp(tQ) t > 0(9.41)174 N N - Q qij > 0 i 6= j Xjqij = 0 i:(9.42)=. D Q (9.42), (9.41) .@ Q = (qij )ij2X " ,qij > 0 i 6= j qi = ;qii > 0 Xjqij 6 0 i 2 X :(9.43)0, ) fP (t) t > 0g, Q { , ..P 0(0) = Q(9.44)( ). 9.2 (9.14) , (9.41).
D ) P (t), ) (9.42), Q -, . (9.16). @ , - % %% = % 0. 2 ) . J O?] 3. 3 M O?] , Q " ". - O?] ).=, %, = (9.42) & %% % PP(t), t > 0, - (8.19), (8.20) j pij (t) = 1, t > 0, i 2 X P j pij (t) 6 1, t > 0, i 2 X . M P Q-%, ). D , j pij (t) = 1 t > 0, i 2 X , P .
( ), j pij (t) 6 1, t > 0, i 2 X , { %. ) (, ) pij (t), t > 0, i j 2 X . ) , 1. K , t>0peij (t) = pij (t) i j 2 X pe1j (t) = 1j pei1 (t) = 1 ;Xjpij (t):(9.45)E. 9.8. C, (9.45) O0 1) Xe = X f1g, ) Pe(t) = (peij (t))ij2Xe, t > 0.* , (9.17) (9.24) : p0ij (t) t > 0 t > 0, . : pij (t) ( 3) t > 0.175E. 9.9. C, - ( ) ". C { " (.
O?]).A 99.10. (9.41). ) Q-%,#) , >.2 J %% % , .. X = f0 : : : ng, (9.16), .. " P (t) = etQ t > 0:(9.46)+ " (8.19) P (t) (9.23).C, (9.46) . = = minq C (t) = e;tP (t) t > 0:i2X ii@C 0(t) = e;tP 0(t) ; e;tP (t) = e;tP (t)Q ; C (t) = C (t)Q ; C (t) = C (t)B B = Q ; I = (bij )nij=0. +C (t) = etB1 kXtB k bij = qij ; ij > 0 i j = 0 : : : nk!k=0= (C (t))ij > 0 , , pij (t) > 0 i j = 0 : : : n. * (9.24) X(jXpij (t))0 =0,jp0ij (t) =XXjkpik (t)qkj =P pij (t) 6 P pij (0) = 1.jXkpik (t)Xjqkj 6 0:jM% %, % X , ..
X = Z+. = "- " -Pn0 (t) = QnPn (t) Pn (0) = In Pn(t) = ((Pn (t))ij ), Qn = (qij ), In = (ij ), i j = 0 : : : n, n 2 Z+. + Pn (t) = etQn t > 0:(9.47)C An = (aij )nij=0 ( ) An = (aij )1ij=0, aij = 0 i > n j > n.@ ', % ' t > 09 nlimP (t) = P (t)(9.48)!1 n(.. ) " Pn (t)) -%% %< %< <.C i j 2 X , n > maxfi j g t > 0 (Pn+1 (t))0ij176=n+1XnXk=0k=0(Pn+1 (t))ikqkj =(Pn+1 (t))ik qkj + (Pn+1 (t))in+1qn+1j C 0(t) = C (t)Qn + D(t) C (t) = ((Pn+1 (t))ij )nij=0, D(t) = ((Pn+1 (t))in+1qn+1j )nij=0.
4 , Pn (t) = etQn C 0(t) = C (t)Qn. +", , C (0) = In, C (t) = Pn (t) +Zt0Pn (t ; s)D(s)ds t > 0:(9.49)* (9.49), , Pn (u) D(s) " u s > 0, n 2 N, , (Pn+1 (t))ij > (Pn (t))ij i j 2 X n > maxfi j g:(9.50)- , Pn (t), t > 0,(Pn (t))ij 6Xj(Pn (t))ij 6 1 i j 2 X n 2 Z+:(9.51)+ &< %,%, % () , , (9.48) .( " 99.11.
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